==== TABLE OF CONTENTS ====
Global-quantum-geometric-Langlands-at-irrational-level//01-introduction.tex
Global-quantum-geometric-Langlands-at-irrational-level//02-foundations.tex
Global-quantum-geometric-Langlands-at-irrational-level//03-pseudoidentity.tex
Global-quantum-geometric-Langlands-at-irrational-level//04-whittaker.tex
Global-quantum-geometric-Langlands-at-irrational-level//05-deformation.tex
Global-quantum-geometric-Langlands-at-irrational-level//06-plancherel.tex
Global-quantum-geometric-Langlands-at-irrational-level//07-loop-categories.tex
Global-quantum-geometric-Langlands-at-irrational-level//08-w-algebras.tex
Global-quantum-geometric-Langlands-at-irrational-level//09-local-comparison.tex
Global-quantum-geometric-Langlands-at-irrational-level//10-localization.tex
Global-quantum-geometric-Langlands-at-irrational-level//11-unipotent-period.tex
Global-quantum-geometric-Langlands-at-irrational-level//12-ran-period.tex
Global-quantum-geometric-Langlands-at-irrational-level//13-equivalence.tex
Global-quantum-geometric-Langlands-at-irrational-level//main.tex
Global-quantum-geometric-Langlands-at-irrational-level//preamble.tex
Global-quantum-geometric-Langlands-at-irrational-level//references.tex
===========================

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}

Let $X$ be a smooth projective connected curve over $\mathbb C$, and let $G$
be a connected simple complex algebraic group. Its Langlands dual
$G^\vee$ is defined by the dual of the full root datum, so the global form
of $G$ is part of the data. Write $\operatorname{Bun}_G(X)$ for the stack
of principal $G$-bundles. The adjoint determinant line on this stack is
\[
  L_G|_E=\det R\Gamma(X,\mathfrak g_E).
\]
Thus the determinant of $H^1$ occurs with exponent $-1$. If $h^\vee$ is
the dual Coxeter number, set
\begin{equation}\label{intro:level}
  D_c(\operatorname{Bun}_G(X))
  :=D\!\left(L_G^{(c-h^\vee)/(2h^\vee)}\right).
\end{equation}
Here $D(L^s)$ is the cocomplete derived DG category of all twisted
$D$-modules for the indicated complex power of the line. The power specifies
a sheaf of twisted differential operators and does not require a root of
$L$. Precise descent and sign conventions are given in
Section~\ref{sec:foundations}.

Let $r$ be the lacing number of $G$: it is $1$, $2$, or $3$ according to
the ratio of the squared root lengths. Fix dual pinnings and a theta
characteristic on $X$. These choices fix the residue characters and the
normalizations of localization used below.

\begin{theorem}\label{thm:main}
For every $c\in\mathbb C\setminus\mathbb Q$ there is an equivalence of
presentable $\mathbb C$-linear DG categories
\begin{equation}\label{intro:equivalence}
  D_c(\operatorname{Bun}_G(X))
    \simeq
  D_{-1/(rc)}(\operatorname{Bun}_{G^\vee}(X)).
\end{equation}
The construction uses the given global forms and all connected components.
It is characterized by the localization--Whittaker comparison of
Section~\ref{sec:equivalence}, including its compatibility with families of
marked points and their collisions. It also intertwines central torsor actions
with the corresponding transgression local systems.
\end{theorem}

The normalization \eqref{intro:level} is the shifted-level convention of
\cite[Section~1.1.5, Conjecture~1]{BogdanovaBetti}. In particular,
Theorem~\ref{thm:main} gives its unramified de Rham equivalence in the
irrational range. The localization comparison and the central compatibility
are part of the construction: they identify the functor as the quantum
Langlands correspondence. Their definitions and formulas appear with the
functors they concern in Section~\ref{sec:equivalence}.

\subsection{Context and prior work}

Geometric Langlands relates sheaf-theoretic categories on a bundle stack to
categories built from local systems for the dual group. Its quantum form
compares twisted $D$-module categories at inverse shifted levels. The
gauge-theoretic interpretation through electric--magnetic duality, developed
by Kapustin and Witten~\cite{KapustinWitten}, explains the appearance of the
dual root datum and reciprocal parameter. An early formulation of the
quantum deformation, attributed there to Drinfeld, appears in Stoyanovsky's
inverse-parameter conjecture for twisted $D$-module
categories~\cite[Section~1.3, Conjecture~2]{Stoyanovsky}. The formulation in
terms of local categories, localization, and Whittaker coefficients is
developed in Gaitsgory's quantum Langlands program~\cite{GaitsgoryQGL}.

Several established results supply the local and classical parts of the
argument. Feigin--Frenkel duality identifies principal $W$-algebras at inverse
shifted levels~\cite{FeiginFrenkel}. Arakawa--Frenkel identify the reductions of Weyl modules and
their coweight twists, including concentration and simplicity at irrational
level~\cite{ArakawaFrenkel}. Raskin's categorical form of
Drinfeld--Sokolov reduction gives a precise renormalized $W$-module category
and generalized vacuum objects~\cite{RaskinW}. The fundamental local
equivalences of Campbell--Dhillon--Raskin give the required irrational-level
point categories for the actual global forms~\cite{CDR}. Gaitsgory's
local/global Whittaker comparison supplies the bridge between the affine
Grassmannian and global pole models, also over multipoint parameter
schemes~\cite{GaitsgoryWhit}.

On the global side, the proof of ordinary geometric Langlands provides the
spectral description of the tempered category and its enhanced
Whittaker--Poincar\'e comparison~\cite{GLCI,GLCII,GLCIII,GLCIV,GLCV}.
The arbitrary-level localization and unipotent-period constructions of
\cite{GLCII} supply further inputs. In
Sections~\ref{sec:localization}--\ref{sec:ran} we construct the period
comparison with the finite levels, arbitrary Grassmannian tests, and
moving-point operations needed below.
Drinfeld--Gaitsgory's finiteness and
constant-term results and Gaitsgory's miraculous duality furnish the
categorical and geometric framework for our twisted
duality~\cite{DrinfeldGaitsgoryFiniteness,DrinfeldGaitsgoryCT,GaitsgoryStrange}.
Lin's comparison of Poincar\'e series with miraculous duality supplies the
ordinary-level geometric maps that we deform~\cite{Lin}.

Recent work of Bogdanova constructs a quantum Langlands functor in the
Betti setting and specifies its central compatibility through
2-Fourier--Mukai duality~\cite{BogdanovaBetti}. Her work on quantum
Whittaker nonvanishing also makes explicit the finite local averaging
diagram used in our argument~\cite{BogdanovaWhit}. The two further
mechanisms developed here are a
weight argument controlling filtered deformation of a global kernel,
and an extension argument turning local pairings into coherent
comparisons across collisions.

\subsection{The two main comparisons}

Localization takes representations of loop Lie algebras to twisted
$D$-modules on the bundle stack and is a quotient functor. We first
compare the two sides on localized objects, using local pairings and a
global period identity. The quotient property then produces a functor
between the bundle categories; two global Whittaker identities provide
its left and right inverses. We explain the two main inputs to this
construction.

The first concerns global Whittaker coefficients. For a group $H$ and a
determinant exponent $s$, let $R_s$ denote the coefficient functor with
arbitrary moving sets of allowed poles. Its target remembers the
nondegenerate unipotent character, as well as the pole positions. Pairing
with the opposite character and applying the twisted pseudo-identity
produces a Poincar\'e functor $J_s$. We construct a comparison and prove
that it is an isomorphism:
\begin{equation}\label{intro:plancherel}
                 J_sR_s\longrightarrow\operatorname{Id}
                 \qquad(s\notin\mathbb Q).
\end{equation}
This comparison is defined on the entire twisted $D$-module category.
Its ordinary-level counterpart has an anti-tempered remainder. A fixed
finite Whittaker test annihilates that remainder and is conservative at
irrational exponent. It therefore suffices to prove that the tested
kernel cone vanishes after deformation.

Finite pole bounds and finite diagrams of marked-point maps give finite
approximations to this cone. We express them by ordinary geometric
complexes with a universal Kummer local system of monodromy $q$;
Fourier transform in auxiliary affine lines then introduces the
exponential characters. Quasi-unipotent monodromy makes cohomology ranks
and transition-map ranks constant away from roots of unity. The passage
to the colimit requires a further argument. Compatible logarithm
variations and a uniform upper weight bound imply that each cohomology
generator in the completion at $q=1$ is killed by a finite transition.
Constancy of transition ranks then gives the same vanishing at every
non-torsion $q$, hence at every irrational exponent. Ordinary geometric
Langlands supplies the cohomological bound used to make the weights
uniform. Section~\ref{sec:deformation} proves the deformation principles;
Section~\ref{sec:plancherel} applies them to the tested kernel cone.

The second comparison concerns local categories over families of points.
For $G$ and $G^\vee$, convolution followed by Drinfeld--Sokolov
reduction pairs a positive-loop-equivariant Kac--Moody module with a
Whittaker object on its affine Grassmannian. At a single point,
inverse-level $W$-algebra duality identifies reductions with the
representation and modification labels exchanged. To use these pairings
globally, one needs maps over the
whole parameter scheme, including collision diagonals, together with
their insertion and factorization compatibilities. We construct the two
pairings with a common $W$-module target and continuous relative right
adjoints. On each configuration stratum, the cohomology of the
semisimple group makes the adjunction units isomorphisms through degree
two on heart objects. Maps between heart objects can therefore be
recovered from their images in the common target. Restriction across a
collision divisor has amplitude $[0,1]$; the recovered maps identify its
boundary epimorphisms, extending the equivalence of hearts across the
divisor. Realization and the adjunctions give the equivalence of DG
categories. A separate extension argument, using translation of the
marked points and the horizontal vacuum, identifies the pairings with
both slots inserted. Sections~\ref{sec:loop}--\ref{sec:local} carry out
these steps.

\subsection{Organization and the global period comparison}

The category-theoretic setup begins in Section~\ref{sec:foundations}.
Irrational inertia weights eliminate sufficiently unstable
Harder--Narasimhan strata. This reduces the relevant bundle categories
to quasi-compact opens and allows ordinary categorical duals to be used.
Section~\ref{sec:pseudoidentity} proves twisted pseudo-identity using
compactifications, adjunctions, and the Levi boundary of the diagonal.
Section~\ref{sec:whittaker} fixes the Whittaker models and their dual
pairings.

After the two main comparisons, Sections~\ref{sec:localization}--\ref{sec:ran}
identify unipotent periods with integrated reduction complexes
(Theorem~\ref{loc:period-comparison}). The
proof keeps the determinant lines, finite-level shifts, and moving-point
operators throughout. Ran integration contracts the added point
variables and turns the local pairing comparison into a global period
identity.

Section~\ref{sec:equivalence} assembles the equivalence through the
localization quotient, using the period identity and the two instances
of \eqref{intro:plancherel}. It also derives the exchange of components
and central characters and the coherent action of central torsors.
==== END Global-quantum-geometric-Langlands-at-irrational-level//01-introduction.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//02-foundations.tex ====
\section{Twists, boundedness, and categorical duality}\label{sec:foundations}

The kernel calculations below take place in categories of all twisted
$D$-modules on bundle stacks.  Their two necessary foundations are a precise
choice of twist and a finiteness statement: at the irrational exponents in
question, restriction to a suitable quasi-compact open is an equivalence.
We establish these facts first, and then explain the resulting kernel
formalism, including its renormalized evaluation pairing.

\subsection{Root data, determinant lines, and level conventions}

All groups retain their given root data.  Fix dual pinnings and a theta
characteristic $\vartheta$ on $X$, together with an isomorphism
$\vartheta^{\otimes2}\simeq\Omega_X$.  For a connected reductive group $H$ write
\[
 Y_H=\operatorname{Bun}_H(X),\qquad
 L_H|_E=\det R\Gamma(X,\mathfrak h_E).
\]
Thus $\det H^1$ occurs with exponent $-1$.  Write $\rho_H$ for the half-sum
of positive \emph{coroots}, and put $P_0=\vartheta^{2\rho_H}$, regarded as an
$H$-bundle through its torus reduction.  This uses the integral cocharacter
$2\rho_H$ and makes sense for the actual group $H$.  In Grassmannian
constructions we rigidify the determinant line by replacing $L_H$ with
$L_H\otimes(L_H|_{P_0})^{-1}$.  A constant line does not change the sheaf of
twisted differential operators.  When tensoring by integral powers, either
rigidification gives the same formula after retaining the corresponding
constant one-dimensional factor.

For a line $L$ on a smooth stack $Y$ and $s\in\mathbb C$, write $D(L^s)$ for
the presentable DG category of twisted $D$-modules.  Here is the sign
convention.  On a smooth scheme, if a frame $e$ is replaced by $ae$, the
local differentiation operator changes by $-s\,d\log a$.  Equivalently,
on the frame torsor of $L$ we impose strong Kummer equivariance with the
connection
\begin{equation}\label{foundations:kummer}
 d-s\,d\log t,\qquad q=\exp(2\pi i s).
\end{equation}
The associated de Rham local system has monodromy $q$.  On stacks these
categories and their functors are defined by strong derived descent.
The opposite twist is $L^{-s}$.  Sums of twists use the tensor product of
frame torsors, or equivalently their ratio torsor; these conventions also
apply on singular auxiliary stacks by smooth descent and Kashiwara's lemma.

We use sheaf conventions for operations: $\omega_Y$ is the unit of
$\otimes^!$, $k_Y$ is the constant object, and a lower star denotes de Rham
direct image.  To express calculations with left $D$-modules on smooth
$Y$, write $M^{\langle Y\rangle}$ for the object corresponding to the left
module $M[\dim Y]$.  In particular,
\[
 \mathcal O_Y^{\langle Y\rangle}=\omega_Y.
\]
Inside these brackets, $!$-pullback is left crystalline pullback without
an additional shift.  The convention descends to smooth stacks, with their
stack dimension.  The exponential object of phase $\phi$ is the left
connection $d+d\phi$ in this dualizing normalization.  All tensor products,
functors, and mapping complexes are derived.  Categories are cocomplete
and $\mathbb C$-linear; functors between them are continuous unless an
adjoint or a partially defined functor is explicitly specified.

We now translate the shifted level of the problem into the four categories
used in the proof.  Use the basic invariant forms
$\operatorname{Kil}/(2h^\vee)$ and
$\operatorname{Kil}^{\vee}/(2\check h^\vee)$.  Their inverse forms give
long roots, respectively long coroots, squared length $2$.  Thus inverse
shifted forms have scalar parameters $a,d$ related by $d=1/(ra)$, where
$r$ is the lacing number.  Set
\begin{equation}\label{foundations:levels}
 a=-c,\qquad d=\frac1{ra}=-\frac1{rc},\qquad
 k_a=a-h^\vee,\qquad k_d=d-\check h^\vee.
\end{equation}
Our notation is
\begin{equation}\label{foundations:four-categories}
 \begin{aligned}
 A&=D\bigl(L_G^{-k_a/(2h^\vee)}\bigr),&
 A'&=D\bigl(L_G^{k_a/(2h^\vee)}\bigr),\\
 B'&=D\bigl(L_{G^\vee}^{-k_d/(2\check h^\vee)}\bigr),&
 B&=D\bigl(L_{G^\vee}^{k_d/(2\check h^\vee)}\bigr).
 \end{aligned}
\end{equation}
In the convention of \cite[Section~1.1.5]{BogdanovaBetti}, a shifted level
$c$ means the exponent $(c-h^\vee)/(2h^\vee)$.  Since
\[
 -\frac{k_a}{2h^\vee}
 =\frac{c-h^\vee}{2h^\vee}+1,
\]
tensoring by $L_G$ identifies the source category in that convention with
$A$.  Its target at shifted level $-1/(rc)$ is exactly $B$.
This is an integral translation by the determinant line; it requires
neither a square root of that line nor divisibility of the basic form in
the lattice of integral levels.  Because $c\notin\mathbb Q$, all four
exponents in \eqref{foundations:four-categories} are irrational.

We retain all connected components:
\begin{equation}\label{foundations:components}
 \pi_0(Y_H)=\pi_1(H)
 =X_*(T_H)/\mathbb Z\Phi_H^\vee.
\end{equation}
In particular, modification labels and dual weights always belong to the
lattices of the specified groups.  For semisimple $H$, its finite centre
acts trivially on $L_H$, since the adjoint representation has trivial
central action.  Central inertia therefore gives the usual decomposition
into character sectors.  These components and sectors are retained by the
constructions below.

\subsection{An irrational twist cuts off the unstable directions}

We prove the boundedness statement in a form that also applies to the
Levi subgroups occurring later.  Let $H$ be connected reductive and let
$V$ be a finite-dimensional self-dual representation.  Put
\[
 L(V)|_E=\det R\Gamma(X,V_E),\qquad
 Q(z,w)=\operatorname{Tr}_V(zw).
\]
Assume that $Q$ is positive definite on the real cocharacter space.
For a degree $u\in\pi_1(H)$ its rational image has a unique representative
$u_{\mathrm{cen}}\in X_*(Z(H)^0)\otimes\mathbb Q$.  We write $Q(z,u)$
for $Q(z,u_{\mathrm{cen}})$ when $z$ is central.

\begin{lemma}[Inertia vanishing]\label{foundations:inertia}
Let $Z$ be an algebraic stack carrying a line $L$.  Suppose a central
$\mathbb G_m$ in its inertia acts on $L$ with a fixed nonzero integral
weight $m$.  If $sm\notin\mathbb Z$, then $D_Z(L^s)=0$.
The assertion remains valid after restricting to a locally closed
substack and after taking a product with a spectator stack carrying any
external twist.
\end{lemma}

\begin{proof}
Pull an object to the frame torsor of $L$.  The weighted scaling of the
frame supplied by the inertial $\mathbb G_m$ is $2$-isomorphic to the
projection.  Strong Kummer equivariance consequently identifies the two
pullbacks of the object to the product with $\mathbb G_m$, one with a
constant factor and the other with the Kummer factor of exponent $sm$.
On a smooth chart, apply de Rham direct image along $\mathbb G_m$.
The nonintegral Kummer factor has zero de Rham cohomology.  For example,
in the Laurent-polynomial de Rham complex its differential has eigenvalues
$n-sm$ on $t^n$, all invertible.  The constant factor has the nonzero finite
complex $H^\bullet_{\mathrm{dR}}(\mathbb G_m)$, containing $\mathbb C$ as a
summand.  Tensoring with that complex detects zero, even for unbounded
complexes.  Hence the pullback object vanishes.  Smooth pullback and
frame-torsor pullback are conservative.  The calculation leaves all
spectator factors unchanged.
\end{proof}

\begin{proposition}[Bounded support in the categorical sense]\label{foundations:cutoff}
For $s\notin\mathbb Q$, the category $D_{Y_H}(L(V)^s)$ vanishes on every
component of nontorsion degree.  There is a quasi-compact open
$j:U\hookrightarrow Y_H$, contained in the union of the torsion-degree
components, such that
\begin{equation}\label{foundations:cutoff-equivalence}
 j^!:D_{Y_H}(L(V)^s)\xrightarrow{\ \sim\ }D_U(L(V)^s)
\end{equation}
with inverse $j_*$.  The same open works for $-s$, and the equivalence
persists after taking a product with any spectator stack and any external
twist there.
\end{proposition}

\begin{proof}
First suppose that $u$ has nonzero rational image.  Nondegeneracy on the
centre gives an integral central cocharacter $z$ with $Q(z,u)\ne0$.
Decompose $V$ into its central weight spaces.  Riemann--Roch gives the
weight of $z$ on $L(V)$ as
\begin{equation}\label{foundations:rr-weight}
 \sum_{\mu}\mu(z)\chi(X,V_{\mu,E})=Q(z,u).
\end{equation}
The term involving $1-g(X)$ is zero because $V$ is self-dual.
The right side is an integer, being the weight of an algebraic
$\mathbb G_m$-action on a line.  Lemma~\ref{foundations:inertia} proves the
first assertion, including its restriction to any locally closed bundle
condition in that component.

Now fix a torsion degree.  We use the characteristic-zero
Harder--Narasimhan theorem for principal bundles: canonical parabolic
reductions describe the strata, the HN types form a discrete set with
bounded denominators, and bounded HN regions admit quasi-compact open
neighbourhoods.  The associated bundle of a semistable Levi bundle under
a representation on which the connected centre acts by a character is
semistable.  These are the forms of the reduction and vanishing results
recalled in \cite[Sections~7.3--7.4 and~10.1]{DrinfeldGaitsgoryBun}.
Write $\nu$ for the dominant rational HN coweight.  Its central component
is zero because the degree is torsion.  Put
\[
 b_X=\max\{0,2g(X)-2\}.
\]
We claim that the twisted category on its stratum vanishes whenever
$\alpha_i(\nu)>b_X$ for some simple root $\alpha_i$.

Let $Q_i$ be the maximal standard parabolic obtained by omitting
$\alpha_i$, and let $M_i$ be its Levi subgroup.  The canonical parabolic
of $\nu$ is contained in $Q_i$.  Projecting the canonical reduction to
$M_i$ identifies the HN stratum with a stack of unipotent extensions over
the corresponding HN stratum of $Y_{M_i}$.  Filter the unipotent radical
of $Q_i$ by height in $\alpha_i$.  Each vector-group quotient, with its
filtration induced by the canonical reduction inside $M_i$, has
semistable associated graded pieces whose slopes are root evaluations
$\beta(\nu)$.  Every such root $\beta$ contains $\alpha_i$ with positive
coefficient; dominance therefore gives
\[
 \beta(\nu)\geq\alpha_i(\nu)>2g(X)-2.
\]
Serre duality and semistability imply $H^1=0$ for every piece, hence for
each successive extension bundle.  Riemann--Roch then makes the ranks of
$H^0$ constant on the fixed stratum.  Consequently these extension stacks
are, locally in families, successive classifying stacks of smooth
unipotent groups.  The section of split extensions is a smooth
surjective cover in the sense of stacks.

Choose a positive integral multiple $z$ of the $i$th fundamental
coweight, in the semisimple span, that belongs to $X_*(T_H)$.  It is
central in $M_i$.  On the split section, the determinant filtration
identifies $L(V)$ with the determinant line for the restricted
$M_i$-representation.  Formula~\eqref{foundations:rr-weight} now gives
its $z$-weight as $Q(z,\nu)$.  This is strictly positive: on each simple
factor the trace form is a positive multiple of the basic form, and the
inverse Cartan matrix has strictly positive entries.  The dominant
coweight $\nu$ is nonzero on the factor containing $\alpha_i$.
Lemma~\ref{foundations:inertia} thus kills the pullback to the split
section, and smooth descent kills the whole HN stratum.  The proof also
kills its restriction to any locally closed substack.

The remaining HN types satisfy $0\leq\alpha_i(\nu)\leq b_X$ for every
simple root.  Their central part is zero, so they lie in a bounded
region.  Bounded denominators make their number finite.  There are only
finitely many torsion degrees in $\pi_1(H)$.  Choose a quasi-compact HN
open $U$ containing all these types.  On the complement, restriction to
every HN stratum vanishes.  To pass from strata to arbitrary objects,
work on each bounded open in the HN exhaustion: its complement of $U$
has a finite HN stratification, and the localization triangles show
successively that its twisted category is zero.  The exhaustion detects
objects, so the same holds on the full complement.  The localization
triangle for $j$ then proves \eqref{foundations:cutoff-equivalence}, with
inverse $j_*$.  All arguments commute with an external spectator and
use only the nonintegrality of a nonzero integer times $s$; they apply
unchanged to $-s$.
\end{proof}

The assertion is about $!$-restriction of categories.  It makes no
set-theoretic claim about closures of supports of the underlying
complexes.  For the simple groups in the main theorem one takes
$V=\mathfrak h$.  For a Levi subgroup, the restriction of the original
adjoint representation supplies a self-dual representation with positive
trace form, including on the Levi centre.

\subsection{Operations with compatible twists}\label{foundations:operations}

A QCA stack is a quasi-compact algebraic stack with affine automorphism
groups; our stacks are locally of finite type over $\mathbb C$.
We use the $D$-module formalism of
\cite{DrinfeldGaitsgoryFiniteness}: compact generation, categorical
duality, and renormalized de Rham direct image with base change and the
projection formula.  A quasi-compact morphism is safe when the neutral
components of the automorphism groups in its geometric fibres are
unipotent.  Safe direct image agrees with ordinary de Rham $*$-image and
is continuous; schematic quasi-compact morphisms are examples
\cite[Sections~9--10]{DrinfeldGaitsgoryFiniteness}.

For correspondences with compatible twists the same formulas apply.
Compatibility identifies the relevant pullbacks of frame torsors, or
trivializes their ratio when twists are added.  Pulling to these torsors
reduces $!$-base change, the $!$-tensor projection formula, and continuity
of safe direct image to the ordinary formulas.  Their natural
identifications commute with frame scaling, so they descend with the
specified Kummer equivariance.  This also explains why one must retain
the compatibility of lines throughout a correspondence.

We use the six operations on locally holonomic objects wherever they
are defined.  In particular, for schematic separated finite-type maps,
shriek direct image on holonomic arguments is the partial left adjoint
to $!$-pullback, with the adjunction tested against arbitrary $D$-modules.
This assertion and Verdier duality, which negates the twist, can be
checked on smooth charts.  Riemann--Hilbert is used only for regular
holonomic objects, again on charts; Kummer connections are regular
singular.  For a smooth map $q$ of relative dimension $e$, our convention
is $q^*=q^![-2e]$.

\subsection{Compact generation and opposite-twist duality}

The cutoff has reduced our bundle categories to smooth QCA stacks.
We record explicitly why twisting preserves the finiteness and kernel
statements needed later.

\begin{proposition}\label{foundations:categories}
Let $Y$ be a smooth quasi-compact open in a bundle stack as above, and
let $\tau$ be a determinant-line twist.  Then $D_\tau(Y)$ is compactly
generated.  On every smooth finite-type scheme chart, the pullback of a
compact object has bounded coherent cohomology.  For two such
stacks and twists, external tensor product induces an equivalence
\begin{equation}\label{foundations:kunneth}
 D_{\tau_1}(Y_1)\otimes D_{\tau_2}(Y_2)
 \xrightarrow{\ \sim\ }
 D_{\tau_1\boxplus\tau_2}(Y_1\times Y_2).
\end{equation}
The ordinary categorical dual of $D_\tau(Y)$ is $D_{-\tau}(Y)$, with
evaluation and coevaluation
\begin{equation}\label{foundations:evaluation}
 (M,N)\longmapsto
 \Gamma_{\mathrm{dR}}^{\mathrm{ren}}(Y,M\otimes^!N),
 \qquad \Delta_*\omega_Y.
\end{equation}
The twist on the diagonal is canonically trivial.  These assertions
remain valid for the full bundle categories reduced by
Proposition~\ref{foundations:cutoff}.
\end{proposition}

\begin{proof}
\emph{The induction adjunction.}
We first specify the underlying quasi-coherent complex of a twisted
object.  Write $\tau=s\,d\log L$ and let $P=L^\times\to Y$ be the frame
torsor.  An object $N\in D_\tau(Y)$ is represented on $P$ by a strongly
Kummer-equivariant D-module.  Forgetting its connection in the normalized
left-module convention turns the Kummer connection
$d-s\,d\log t$ into the trivial line with its multiplicative frame $1$.
In this normalization, smooth $!$-pullback induces ordinary pullback of
the underlying quasi-coherent complexes, with no remaining dimension shift.
Its equivariance isomorphisms therefore give ordinary quasi-coherent
descent along $P\to Y$, including the descent coherences.  Denote the
result by $N^{\mathrm{oblv}}$.  This operation is conservative and
preserves colimits.  It retains the equivariance maps, rather than
replacing them by identity maps; in particular it retains their
infinitesimal isotropy actions.

If $Y$ has dimension $d$ and canonical line $\omega_Y^{\mathrm{can}}$,
this normalization is right-module oblivion tensored with
$(\omega_Y^{\mathrm{can}})^{-1}[-d]$ on line-trivializing charts.
We will construct $\operatorname{Ind}_\tau$ on $\operatorname{Perf}(Y)$
with the adjunction
\begin{equation}\label{foundations:induction-adjunction}
 \operatorname{RHom}_{D_\tau(Y)}
       (\operatorname{Ind}_\tau(F),N)
 \simeq R\Gamma(Y,F^\vee\otimes N^{\mathrm{oblv}}).
\end{equation}
Since perfect complexes compactly generate $\operatorname{QCoh}(Y)$,
this formula will give compact generators for $D_\tau(Y)$.  Its
external-product compatibility will give K\"unneth.  We verify the
evaluation pairing separately after these constructions.

\emph{Presentations and induction.}
The bounded Quot construction presents $Y$ as a quotient of a smooth
finite-type scheme by a product of general linear groups.  Indeed, a
faithful representation bounds the associated vector bundles; after a
uniformly large twist they have vanishing $H^1$ and are generated by
global sections.  Framing those sections gives a scheme atlas.  Reductions
of structure group are representable by schemes of sections because the
quotient by the reductive subgroup is affine.  We can therefore choose
a smooth presentation $p:Z\to Y$ which is a torsor for such a product.

Here is the twisted induction construction on this presentation.
It is the relative Lie-algebroid construction of
\cite[Section~6.3, especially Lemmas~6.3.15--6.3.16]{DrinfeldGaitsgoryFiniteness},
with the lift of the relative action to the twisting line retained.
For a smooth schematic chart $T\to Y$, let $A_T=T_{T/Y}$ be the relative
tangent Lie algebroid, with its anchor to $T_T$.  Relative differentiation
lifts $A_T$ to the Atiyah algebroid of the pulled-back line, and hence to
its twisted differential operators $\mathcal D_T^\tau$.  Explicitly, if
$a\in A_T$ has anchor $v_a$ and its lift acts on a local frame $e$ by
$b_ae$, the resulting operator is $v_a+s b_a$.  Under $e'=fe$ one has
$b'_a=b_a+v_a(\log f)$, while the differentiation generator changes by
$-s v_a(\log f)$; the operator is therefore independent of the frame.
For
$F\in\operatorname{Perf}(Y)$ define the chart expression
\begin{equation}\label{foundations:induction}
 \operatorname{Ind}_\tau(F)|_T
 =\left(\mathcal D_T^\tau
       \mathop{\otimes}^{L}_{U(A_T)}F_T\right)^{\langle T\rangle}.
\end{equation}
Here $F_T$ has the canonical relative connection coming from its
pullback from $Y$, and $U(A_T)$ denotes the enveloping algebra.  If the
anchor has a kernel, that kernel is retained: its action on the twisting
line supplies the corresponding scalar action in
$\mathcal D_T^\tau$.  Trivializing the line on $T$ does not discard this
isotropy action.

For a smooth map $T'\to T$ over $Y$, the relative tangent algebroid
$A_{T'}$ is the inverse-image algebroid.  Resolve first in
$T_{T'/T}$ using its relative Spencer complex.  The result is precisely
the transfer bimodule for smooth pullback applied to
\eqref{foundations:induction}.  Resolving successively for two smooth
maps gives the same comparison, by associativity of transfer tensor
products.  Thus the chart objects and their comparisons descend.
This argument remains valid when an anchor has a kernel: after smooth
base change by $Z\to Y$, it is the relative differentiation along
ordinary smooth morphisms over $Z$, with its descent data.

At zero twist, the induction used here is the induction of
\cite[Section~6.3.21]{DrinfeldGaitsgoryFiniteness}
applied, in its IndCoh normalization, to
$F\otimes\omega_Y^{\mathrm{can}}[d]$.
For example, on $BH$ one has $d=-\dim H$ and
$\omega_{BH}^{\mathrm{can}}=\det\mathfrak h$.
At zero twist these factors cancel the
$\det(\mathfrak h^\vee)[\dim H]$ in the atlas formula of
\cite[Section~6.4.8]{DrinfeldGaitsgoryFiniteness};
\eqref{foundations:induction} then gives precisely derived
$\mathfrak h$-coinvariants on the point atlas.  The relative-algebroid
construction uses the same normalization at nonzero twist.

\emph{The adjunction, including unbounded coefficients.}
The Spencer resolution and enveloping-algebra adjunction on charts give
the desired formula \eqref{foundations:induction-adjunction} locally.
We explain the descent assertion implicit here.  Restricting $N|_T$ to
$A_T$ gives the relative connection on the pullback of
$N^{\mathrm{oblv}}$, with its derived descent homotopies.  The twist is
canonically split along these relative derivatives.  The mapping
complex is consequently the totalization, over the smooth faces of the
\v Cech nerve of $p$, of the corresponding relative de Rham complexes
with coefficients in $F^\vee\otimes N^{\mathrm{oblv}}$.

This totalization works for unbounded coefficients as well.  Put
$E=p_*\Omega^\bullet_{Z/Y}$ and
$p_n:Z^{[n]}=Z^{\times_Y(n+1)}\to Y$.  The product formula for relative
forms and the affine projection formula give, for any coefficient
complex $M$,
\[
 Rp_{n,*}\bigl(\Omega^\bullet_{Z^{[n]}/Y}\otimes p_n^*M\bigr)
 \simeq E^{\otimes^L(n+1)}\otimes^L M.
\]
The cofaces insert the unit of $E$.
The invariant Reynolds functional on
regular functions of the group, extended by zero on positive-degree
forms, gives a chain map $E\to\mathcal O_Y$ splitting the unit.
Left and right invariance make it independent of torsor
trivializations.  Applying this retraction in the first tensor factor
contracts the augmented tensor-power diagram.  Tensoring this explicit
contraction with any coefficient complex proves the asserted
unbounded totalization formula: the homotopy lowers cosimplicial degree
by one, so it acts on the product totalization without an infinite sum.
It contracts the augmented diagram,
not the individual de Rham complex of the group.

The forgetful functor is conservative and preserves colimits.  Perfect
complexes compactly generate $\operatorname{QCoh}(Y)$ on these smooth
quotient stacks, and their external products generate on products
\cite[Sections~3--4]{DrinfeldGaitsgoryFiniteness}.  Formula
\eqref{foundations:induction-adjunction} therefore makes the induced
perfects compact generators of $D_\tau(Y)$.  Their finite Spencer
resolutions on charts consist of coherent $\mathcal D_T^\tau$-modules.
They therefore have bounded coherent cohomology on each smooth
finite-type chart.  Finite cones and retracts preserve this property,
so it holds for all compact objects.  Induction commutes with external
products.  Applying
\eqref{foundations:induction-adjunction} to external perfects, together
with the quasi-coherent K\"unneth formula, proves full faithfulness on
compact generators in \eqref{foundations:kunneth}; these generators also
generate the target.  Hence \eqref{foundations:kunneth} is an equivalence.

\emph{The evaluation pairing.}
For an untwisted QCA stack, the evaluation is renormalized de Rham
cohomology and the coevaluation is $\Delta_*\omega_Y$
\cite[Sections~8.4 and~9.1]{DrinfeldGaitsgoryFiniteness}.
For opposite twists use the same two formulas, with the ratio line
trivialized on the diagonal.  To check either triangle identity, pull
the output leg to a smooth chart on which the output twisting line is
trivialized.  Base change replaces the diagonal by the graph of the
chart, and the projection formula cancels the two twists along this
graph.  The remaining expression is the untwisted graph calculation.
The graph is schematic and safe, so its renormalized image is its
ordinary star image and the composite is the identity.  These
identifications descend, since they use only the canonical diagonal
trivialization.  This proves \eqref{foundations:evaluation}.

\emph{Return to the full bundle stack.}
Transport compact generation and duality across
\eqref{foundations:cutoff-equivalence}.  The spectator version of that
equivalence identifies products and diagonal kernels, so the same
formulas apply.  The renormalized pairing may equivalently be computed
on the common cutoff open; it is independent of enlarging that open.
For local coherence, one need not assert that an arbitrary open star
image preserves coherence.  Given a finite-type smooth chart of $Y_H$,
choose a larger quasi-compact HN open $V$ containing both its image and
$U$.  The same cutoff argument gives $D_\tau(V)\simeq D_\tau(U)$.
A compact object is thus compact on $V$, where the preceding Spencer
argument proves its coherence on the chosen chart.  This proves local
coherence on the full stack.
\end{proof}

In particular, $A'$ and $B'$ realize the ordinary categorical duals of
$A$ and $B$.  Formula~\eqref{foundations:evaluation} still uses the
star-diagonal kernel.  The shriek-diagonal kernel
$\Delta_!k_Y$, which defines pseudo-identity, is a separate object;
proving that it gives an equivalence is the next step.
==== END Global-quantum-geometric-Langlands-at-irrational-level//02-foundations.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//03-pseudoidentity.tex ====
\section{Twisted pseudo-identity}\label{sec:pseudoidentity}

The ordinary and extraordinary diagonal kernels play different roles in the
argument.  The former represents categorical duality, as established in
Section~\ref{sec:foundations}; the latter must itself define an equivalence.
We prove this by combining parabolic adjunctions with a compactification of
the diagonal.  The essential additional point at an irrational twist is that
central inertia removes every boundary stratum with nonzero defect.

Let $H$ be a connected reductive group, let $V$ be a self-dual representation
whose trace form $Q$ is positive definite on the real cocharacter space, and
put $L_H=\det R\Gamma(X,V_E)$.  Restrictions of $V$ will define the corresponding
lines for Levi subgroups.  Write
\[
 \mathcal C_H=D(L_H^s),\qquad \mathcal C'_H=D(L_H^{-s}),\qquad s\notin\mathbb Q.
\]
The duality of Section~\ref{sec:foundations} identifies
$(\mathcal C'_H)^\vee$ with $\mathcal C_H$.  The line ratio on the diagonal is
canonically trivial.  Consequently the kernel $\Delta_!k_{Y_H}$ defines a
continuous endofunctor $F_H$ of $\mathcal C_H$; its existence as a kernel uses
that the diagonal is schematic and that $k_{Y_H}$ is locally holonomic.

\begin{theorem}\label{pseudo:equivalence}
For every $H,V,s$ as above, the functor $F_H$ is an equivalence.  The assertion
holds on all components, including those on which the twisted category is
zero, and also with $s$ replaced by $-s$.
\end{theorem}

We first establish parabolic adjunctions and their compatibility with
$F_H$.  The compactified diagonal will then show that the cone of the
natural map $F_H\to\operatorname{Id}[-2\dim Y_H]$ has a finite
filtration whose quotients factor through proper Levi categories.  Induction handles objects obtained by
Eisenstein series, while $F_H$ acts by a shift on objects whose proper
constant terms vanish.  These two cases generate the category and will
prove the equivalence.

The geometric constructions are those of geometric Eisenstein series and second
adjointness \cite{BravermanGaitsgory,DrinfeldGaitsgoryCT}.  We include the twist
and coherence arguments, since an adjunction only on holonomic objects would
not suffice here.

\subsection{Parabolic correspondences and their compactifications}

For the moment suppose that $H$ is semisimple.  Fix a parabolic $P$, its Levi
quotient $M$, an opposite $P^-$, and a torsion degree $u\in\pi_1(M)$.  Set
$Y=Y_H$ and use the correspondence
\[
 Y_M^u\xleftarrow{\ q\ }Y_P^u\xrightarrow{\ p\ }Y.
\]
The weight filtration of $V$ associated to a cocharacter defining $P$ gives
an identification $p^*L_H\simeq q^*L_M$.  Normalize it to be the identity on
split $M$-reductions.  Use the same normalization for $P^-$.  Define
\[
 C_P=q_*p^!,\qquad E_P=p_*q^!,\qquad
 e=\dim(q)=(g(X)-1)\dim N_P.
\]
The last equality uses that $u$ is torsion; it gives the same value for $P^-$.
Primes on these functors will mean that they are taken at the opposite twist.

Here $p$ is schematic, separated, and relatively quasi-compact, whereas $q$
is smooth, relatively quasi-compact, and safe.  A positive grading of $N_P$
presents $q$, locally over the Levi stack, as successive fibrations of vector
stacks.  A vector-stack step has the form $[V^1/V^0]$, with $V^i$ vector bundles
coming from a two-term complex computing cohomology on $X$.  There are no
obstructions in degree two.  This description has three useful consequences.
First, the direct images defining $C_P,E_P$ are continuous.  Second, $q_*$
preserves local regular holonomicity.  Third, the partial holonomic operations
$q_!$ and $q^*$ have their actual adjunction property against arbitrary
D-modules.  For the last two assertions present each vector stack by its
vector bundle: descent along the vector group is fully faithful by its de
Rham acyclicity, and both assertions reduce to the schematic formulas.
More explicitly, for $a:V^1\to[V^1/V^0]$, the star image is
computed using $a^*$ followed by the schematic star image; the holonomic
shriek image uses $a^!$ followed by the schematic shriek image.  The
identities $a_*a^*\simeq\operatorname{Id}$ and
$a_!a^!\simeq\operatorname{Id}$ follow from the corresponding ordinary
and compactly supported cohomology of the vector-group fibers.  This
retains the shifts even when the vector stack has negative dimension.
These reductions are compatible with smooth descent and with adding a
spectator category.

Kernel duality gives
\[
 (C'_P)^\vee=E_P.
\]
Indeed both kernels are obtained from the diagonal unit by !-base change
followed by the indicated safe direct image.  Every calculation below may
be made on quasi-compact open exhaustions; the boundedness result of
Section~\ref{sec:foundations} identifies the twisted categories of the bundle
stacks with those on fixed quasi-compact opens.  Thus this calculation uses
ordinary categorical duals and does not introduce a co-category.

We will use the fine Drinfeld compactification
\[
 Y_P^u\xrightarrow{\ j\ }\widetilde Y_P^u,
 \qquad \bar q:\widetilde Y_P^u\longrightarrow Y_M^u,
 \qquad \bar p:\widetilde Y_P^u\longrightarrow Y.
\]
Its points are generalized reductions with their Levi datum retained, and
$\bar p$ is schematic and proper.  If the derived group is simply connected,
this is the compactification by generically nondegenerate sections of the
affine closure of $H/N_P$.  The boundary is stratified by positive defects.
On each stratum the saturated reduction is an actual $P$-bundle; its Levi
bundle is related to the originally specified one by a positive
modification.  The map from the stratum to this Levi Hecke datum is the
base change of the ordinary map from parabolic bundles to Levi bundles.
The stratification is locally finite on quasi-compact substacks.  These are
the geometric compactification and stratification results of
\cite{BravermanGaitsgory}; see also
\cite[Sections~3.2--3.3]{SchiederVinberg}.

For completeness, the same statements hold for the actual global form of
$H$.  Choose a central extension $H_1\to H$ with torus kernel $D$ and simply
connected derived group, and form the compactification upstairs.  Central
$D$-bundles act on all its data: each Pl\"ucker block transforms through its
central character.  Quotient by this action.  Lifts of an $H$-bundle exist
smoothly locally on the base.  On geometric fibers this follows from the
vanishing of the Brauer group of the smooth projective curve; infinitesimal
lifting follows from the vanishing of degree-two coherent cohomology.
Thus $Y_{H_1}\to Y_H$ and the corresponding parabolic and Levi maps are
torsors for $Y_D$ in this sense.

Fix a lift of the ambient degree and the original Levi degree downstairs.
There is a unique compatible degree for its lift.  To check this, use the
exact sequence of maximal tori: two possible Levi lifting degrees differ
by a cocharacter of $D$, and this cocharacter injects into $\pi_1(H_1)$.
The injectivity follows because $D$ meets the derived group in a finite
subgroup.  Compatibility of ambient degrees therefore forces the difference
to vanish.  Conversely the same sequence provides any compatible lift.
Properness, schematicity, and the stratum description consequently descend.
The descended Hecke base is the simultaneous central quotient of the
upstairs Hecke base; its unipotent-extension fibers are unchanged.  A
nonzero defect changes the Levi degree nontrivially over $\mathbb Q$: the
simple coroots omitted from $M$ remain linearly independent in the quotient
of the rational cocharacter space by the Levi coroot span.  This rational
statement, rather than a possibly enlarged integral defect monoid, is what
we shall use.

\subsection{Boundary smoothness and cleanness}

On $\widetilde Y_P^u$ take the twist
\[
 (\bar p^*L_H\otimes\bar q^*L_M^{-1})^s
\]
and the kernel $J=j_!\omega_{Y_P^u}$ defined using its splitting over the open.
We next prove two properties: $J$ is clean, and it has no nonzero
characteristic covector pulled back from the Levi base.  The second property
allows us to tensor $J$ with arbitrary coherent Levi coefficients.

Here is the elementary analytic observation that will be used twice.  Let
$U\subset Z$ be an open subset of a finite-type complex scheme, let
$\mathcal L$ be a local system on $U$, and let $j_!\mathcal L$ be its extension
by zero.  A derivation of $\mathcal O_Z$ preserving the ideal of $Z\setminus U$
has local analytic flows preserving the pair $(Z,U)$.  To see this even when
$Z$ is singular, embed it locally as a closed subscheme of a smooth scheme.
Lift the derivation to the ambient local coordinates.  Its preservation of
the ideals of $Z$ and of the boundary makes the ambient flow preserve their
analytic zero loci.  Along a sufficiently small time disk, flow transport
identifies the two pullbacks of $\mathcal L$, and hence those of
$j_!\mathcal L$.  These identifications commute with extraordinary pullback
along a compatible map of pairs.  On a smooth test scheme, a family of such
fields spanning the tangent space therefore makes the pulled-back complex
a local system: successive flows give local coordinates.  Likewise every
characteristic covector of the closed direct image annihilates each such
field.  This last assertion follows from the equality of characteristic
variety and microsupport for regular holonomic D-modules
\cite[Theorem~11.3.3]{KashiwaraSchapira} and the local product description
along the flow.

\begin{lemma}\label{pseudo:fiber-descent}
Let $i_\theta:S_\theta\hookrightarrow\widetilde Y_P^u$ be a defect stratum,
and let $f_\theta:S_\theta\to B_\theta$ be its Levi Hecke-data projection.
Its twist is pulled back from $B_\theta$, and there is a locally holonomic
complex $K_\theta$ on that base such that
\begin{equation}\label{eq:M2}
 i_\theta^!J\simeq f_\theta^!K_\theta.
\end{equation}
\end{lemma}

\begin{proof}
The determinant filtration for the saturated reduction identifies the line
ratio with the ratio of the two Levi determinant lines.  This proves the
assertion about twists.  We first show smoothness on each geometric
!-fiber of $f_\theta$.

Choose a point on the stratum, and a point $z\in X$ outside its defect divisor.
Shrink the parameter chart so that $z$ remains outside every defect.  On the
formal disk at $z$ the generalized reduction is actual.  Choose a frame of
this reduction, locally on a smooth chart; for any finite collection of
Laurent principal parts only a finite jet of this choice is needed.  Frames
of that jet lift smoothly, and compatible higher jets exist by formal
smoothness.  In the central-extension construction take the same charts
upstairs before descent.

A Laurent element $\eta\in\mathfrak n_P((t_z))$ modifies the framed bundle
infinitesimally by $1+\epsilon\eta$.  Gluing on the punctured disk produces a
first-order deformation of the generalized reduction, unchanged away from
$z$.  Since the data are actual near $z$, this gluing preserves their defining
relations.  A smooth scheme presentation $T'\to\widetilde Y_P^u$ admits a
local lift of this infinitesimal deformation to a derivation of $T'$; any
chosen value differing by a relative tangent vector can be prescribed.
The derivation preserves the boundary ideal.  More explicitly, the defect
incidence is unchanged away from $z$ and is empty near $z$.  Its scheme image
in the parameter space is therefore unchanged by the automorphism over
$\mathbb C[\epsilon]/(\epsilon^2)$ associated to the derivation.  Formation
of this image commutes with these flat pullbacks.  This proves ideal
preservation, not just preservation of boundary points.

Fix the Levi Hecke datum and pull $T'$ back to the stratum fiber, obtaining
$W$.  The map from $W$ to that fiber is smooth, being a base change of the
smooth presentation $T'\to\widetilde Y_P^u$.  The fiber itself is a smooth
tower of vector stacks over a point.  Thus $W$ is smooth, after a scheme
presentation when necessary.
The fields just constructed lift compatibly to $W$, because a modification
by $N_P$ keeps the Levi modification fixed.  Their values, together with
relative presentation directions, span its tangent space.  Indeed the
non-presentation directions are deformations of a parabolic bundle with
fixed Levi, hence are the image of
$H^1(X,(\mathfrak n_P)_{E_P})$.  Principal parts at $z$ surject onto this
space: for sufficiently large $n$, the exact sequence for
$(\mathfrak n_P)_{E_P}\subset(\mathfrak n_P)_{E_P}(nz)$ and Serre vanishing
give the surjection.  Its connecting map is precisely the gluing deformation
above.  The tangent-complex sequence supplies the remaining presentation
directions, including the quotient by infinitesimal automorphisms.

Trivialize the line-frame torsors locally analytically.  The kernel $J$ is
regular holonomic, and on the open it corresponds to a local system.
The flow observation now shows that its !-pullback to $W$ is a local system.
Changing a frame only tensors by a smooth Kummer local system, so does not
alter this conclusion.

It remains to turn fiberwise smoothness into descent.  Each step of
$f_\theta$ is a vector-stack fibration.  On a vector stack over a point, a
smooth regular holonomic complex is constant, since both the vector space
and the acting vector group are contractible.  For one fibration apply the
counit $f^*f_*\to\operatorname{Id}$ and test it on geometric !-fibers by
!-base change.  It is an isomorphism on each fiber and therefore globally.
Repeat for the tower and use smooth descent.  Since $f^*$ differs from $f^!$
by the smooth dimension shift, this gives \eqref{eq:M2}.
\end{proof}

\begin{lemma}\label{pseudo:clean-noncharacteristic}
The kernel $J$ is clean: $j_!\omega\simeq j_*\omega$ in the specified twist.
Moreover, on a smooth Levi chart $S\to Y_M^u$, choose a scheme presentation
of $\widetilde Y_P^u\times_{Y_M^u}S$ and a local closed embedding into a
smooth scheme over $S$.  The characteristic variety of the closed direct
image of $J$ contains no nonzero covector pulled back from $S$.
\end{lemma}

\begin{proof}
For $\theta\ne0$, simultaneous central inertia on the two Levi bundles of
$B_\theta$ acts on the determinant ratio with weight
$Q(z,u_{\rm sat}-u)$.  The degree difference is rationally nonzero, so
positivity of $Q$ provides an integral central cocharacter $z$ for which
this integer is nonzero.  At exponent $s$ its Kummer parameter is not
integral.  The inertia vanishing test of Section~\ref{sec:foundations}
therefore kills the entire twisted category on this base, including any
additional spectator.  Under a central extension one may first multiply
and lift $z$; the removed central directions act trivially on the ratio,
so the test descends.  Lemma~\ref{pseudo:fiber-descent} gives
$i_\theta^!J=0$.  Localization and the locally finite stratification prove
cleanness.  The same reasoning applies to the opposite twist and hence to
the Verdier-dual kernel.

For the characteristic assertion repeat the modification argument at $z$
with elements of the Levi loop Lie algebra.  Prescribe their lifts first in
$S$, then in the presentation of the compactification.  These derivations
again preserve the open and its boundary.  Principal parts in the adjoint
Levi bundle span its $H^1$, and the relative chart directions complete their
images to a spanning set of $T_sS$.  Extend the derivations in the chosen
smooth ambient embedding.  The flow observation shows that every
characteristic covector annihilates them.  A horizontal covector is pulled
back from $T_s^*S$; annihilating fields whose projections span $T_sS$ forces
it to be zero.  This proves the assertion.
\end{proof}

\begin{proposition}\label{pseudo:first-adjunction}
On degree $u$, the continuous functor $L_P=E_P[-2e]$ is a left adjoint of
$C_P$:
\begin{equation}\label{eq:M1}
 L_P=E_P[-2e]\ \dashv\ C_P.
\end{equation}
It computes the partially defined expression $p_!q^*$ on every object.
\end{proposition}

\begin{proof}
First let $E\in\mathcal C_M^u$ be compact, hence locally coherent by
Section~\ref{sec:foundations}, and form
\[
 J_E=J\otimes^!\bar q^!E.
\]
In the charts of Lemma~\ref{pseudo:clean-noncharacteristic}, the second
factor has horizontal characteristic covectors.  The absence of nonzero
horizontal covectors for the first factor is exactly the condition making
the diagonal noncharacteristic for their exterior product.  The
noncharacteristic inverse-image theorem, including compatibility with
duality \cite[Theorems~2.4.6 and~2.7.1(ii)]{HottaTakeuchiTanisaki}, proves that $J_E$ is
locally coherent and that Verdier duality commutes with this tensor product,
up to the smooth dimension shift.  The same calculation applies after
closed direct image from the possibly singular presentation, by the
projection formula.

Both $J_E$ and its Verdier dual have zero !-restriction to the boundary,
since this holds for $J$ and its dual.  Hence $J_E$ is simultaneously the
star and the extraordinary extension of its restriction to $Y_P^u$.
Here extraordinary extension has its full partial-adjoint meaning:
there are no maps from $J_E$ to an arbitrary object pushed from the
boundary.  We verify this for arbitrary targets using chartwise compactness
and descent.
On every affine scheme chart of the compactification, embed the chart as
a closed subscheme of a smooth affine scheme.  Its closed direct image
of $J_E$ is bounded coherent, and hence compact in the D-module category
of that smooth affine scheme.  By Kashiwara equivalence, an arbitrary
boundary object is a colimit of bounded coherent boundary-supported
D-modules.  Verdier duality and the vanishing of the boundary !-restriction
of the dual of $J_E$ kill maps to each such coherent object.  Compactness
then kills maps to their colimit.  Kashiwara equivalence identifies these
with the mapping complexes on the original, possibly singular chart.
Apply this argument also on scheme charts of every level of a smooth
hypercover.  Smooth descent expresses the stack mapping complex as the
limit of these zero complexes, so it vanishes.  This uses compactness only
in smooth affine ambient charts; no global compactness of $J_E$ is required.

Push by the proper map $\bar p$.  Using $q^*=q^![-2e]$, the simultaneous
extension gives
\[
 p_!q^*E\simeq p_*q^!E[-2e]=L_PE
\]
with its adjunction against every target object.  Compact objects generate
the Levi category.  The functor on the right is continuous, so the same
adjunction extends by colimits to every $E$.  This proves \eqref{eq:M1}.
\end{proof}

\subsection{Second adjointness with the determinant twist}

The first adjunction controls compact Eisenstein objects.  We also need the
opposite-parabolic adjunction, including its unit and counit, to control
objects orthogonal to Eisenstein series.

\begin{proposition}\label{pseudo:second-adjunction}
On the same torsion component there is an adjunction
\begin{equation}\label{eq:M3}
 C_{P^-}\ \dashv\ E_P.
\end{equation}
\end{proposition}

\begin{proof}
Choose an integral cocharacter $\lambda$ defining $P$.  Use the smooth graph
interpolation group $\widetilde H$ over $\mathbb A^1$, with general fiber
ordered as $(g,t^{-\lambda}gt^\lambda)$ and special fiber
$P^-\times_M P$.  The interpolation of bundle stacks is smooth over
$\mathbb A^1$, and its map to $\mathbb A^1\times Y\times Y$ is schematic of
finite type; these are
\cite[Section~2.4 and Proposition~2.5.2]{DrinfeldGaitsgoryCT}.  Let $T$ be the
open substack obtained by retaining degree $u$ at the special fiber and all
of the general fiber.

The determinant-line ratio of the two ends has its canonical splitting for
$t\ne0$.  It extends across $t=0$.  Indeed its order there is locally
constant, and the special fiber is connected: it is a tower of vector
stacks over the connected $Y_M^u$.  Test the order on split reductions,
using the constant diagonal Levi subgroup.  The order is the sum of the
$\lambda$-weights times Euler characteristics, namely $\pm Q(\lambda,u)=0$.
Smoothness of $T$ then extends the section as a nowhere-vanishing section.
Its value at zero equals the parabolic splitting: their quotient is a
unit, is constant along each vector-stack fiber, and equals one on the
split section.  The splitting is equivariant under dilation of the
parameter.  This holds away from zero, where inner conjugation identifies
the induced bundle action with the identity, and hence holds everywhere.

Push $\omega_T$ to $\mathbb A^1\times Y^2$ with this ratio splitting.  The
result is conic in the first factor.  Its !-fiber at one is the identity
kernel, and its !-fiber at zero is the kernel of $E_PC_{P^-}$.  The
cospecialization map between these fibers supplies
\[
 \eta:\operatorname{Id}\longrightarrow E_PC_{P^-}.
\]
To specify the cospecialization and its continuity, on the conic axis use
\[
 i_{1,*}i_1^!\longrightarrow\operatorname{Id}
 \longleftarrow i_{0,*}i_0^!
\]
and apply compactly supported image along $\mathbb A^1$.  Contraction makes
the second arrow invertible.  One can check this on holonomic conic
objects by dualizing: the star extension from the punctured axis has zero
compactly supported image.  Pullback of a compact object from
$[\mathbb A^1/\mathbb G_m]$ is holonomic, because its characteristic variety
is the zero section off zero and has dimension at most one over zero.
Continuous extension from these objects defines the construction on the
conic category used here.  It also tensors with any compactly generated
spectator category, by compactness of the holonomic axis objects.  In these
!-fiber conventions cospecialization for a constant family is the identity.

Define the counit
\[
 \varepsilon:C_{P^-}E_P\longrightarrow\operatorname{Id}
\]
by restricting the kernel on $Y_P^u\times_Y Y_{P^-}^u$ to the open where the
reductions are everywhere transverse.  This open is $Y_M^u$; restriction
followed by star image gives the arrow.  The line identification there is
the split-reduction identification.

We verify the two triangular identities rather than infer them from an
untwisted result.  Precompose $\eta$ with $E_P$.  The family of kernels is
then the fiber product $R=Y_P^u\times_YT$, with $Y_P^u$ on the input end.
Include the general fiber and the everywhere-transverse locus of its
special fiber in an open $R^o$.  It is $\mathbb A^1\times Y_P^u$: an extra
reduction is a flag in $H/P$; its open orbit under the first projection of
$\widetilde H$ has stabilizer identified by the second projection with the
constant $P$.  At zero this is the big orbit of $P^-$, and the graph of
contracting conjugation on $P$ gives the identification across zero.
The maps to the output $H$-bundle and input Levi bundle are constant on
this product.  So is their determinant splitting, since the two
trivializations agree at zero and a unit is constant along an affine line.
Restriction to $R^o$ identifies the unit followed by the counit with
cospecialization of this constant family, which is the identity.

For the other triangular identity use
$T\times_Y Y_{P^-}^u$, with the parabolic factor on the output end.  The
second projection of $\widetilde H$ acts on the flags; on the corresponding
open orbit its stabilizer is identified by the first projection with
$P^-$.  The open family is $\mathbb A^1\times Y_{P^-}^u$, with constant maps
and constant line splitting.  The same calculation gives the identity.
All kernel operations here are safe, or are the holonomic partial
operations described above; their extension with spectators and through
open exhaustions is therefore legitimate.  This proves \eqref{eq:M3}.
\end{proof}

The two adjunctions interact with the extraordinary diagonal as follows:
\begin{equation}\label{eq:M4}
 C_{P^-}F_H\simeq F_M(L'_P)^\vee,
 \qquad F_HE_{P^-}\simeq L_PF_M.
\end{equation}
For the first identity apply $C_{P^-}$ to the output factor of
$\Delta_!k_Y$.  In the following calculation $p,q$ are the maps for $P$.
The expression $q_!p^*$ is the partial left adjoint of $E_P=p_*q^!$
on locally holonomic arguments, with its adjunction against arbitrary
targets.  Proposition~\ref{pseudo:second-adjunction} provides the genuine
left adjoint $C_{P^-}$; uniqueness of left-adjoint values identifies its
value on this kernel with $q_!p^*$.  Base change now gives
$(p,q)_!k_{Y_P^u}$.  Applying $L'_P$ to the first factor of
$\Delta_!k_{Y_M^u}$ gives the same kernel, by \eqref{eq:M1}.
The second identity is its transpose at the opposite twist.  Notice that
$(L'_P)^\vee$ is the right adjoint of $E_P$.  Thus \eqref{eq:M4} identifies
actual continuous functors, not just their values on holonomic test objects.

\subsection{The boundary of the diagonal}

The preceding identities deal with the Eisenstein part of the category.
To complete the proof, we must show that the difference between $F_H$ and
a shift of the identity is built from that part.

\begin{proposition}\label{pseudo:boundary-factorization}
For semisimple $H$, there is a natural transformation
\[
 F_H\longrightarrow\operatorname{Id}[-2\dim Y]
\]
whose cone has a finite filtration with quotients factoring through constant
term, a functor between Levi categories, and Eisenstein series for proper
parabolics.
\end{proposition}

\begin{proof}
We first describe the required compactification, including its properness
for the given global form.  The Vinberg semigroup $\operatorname{Vin}_H$
is the affine Rees monoid of the enhanced group.  Its parameter space is
$T_{\mathrm{ad}}^+\simeq\mathbb A^{\operatorname{rk}H}$, with coordinates
indexed by simple roots.  It has matrices $h_\mu$ in every representation
of dominant highest weight $\mu$ in the \emph{actual} weight lattice of
$H$.  Finitely many generators suffice.  Cartan multiplication has no
parameter factor, while the component of highest weight $\nu$ in the
product for $\mu,\mu'$ is multiplied by $v^{\mu+\mu'-\nu}$.
The nondegenerate locus, where all the $h_\mu$ are nonzero, is smooth over
the parameter space; each fiber is homogeneous for $H\times H$.
At the standard point $c_P$, with coordinates one on Levi roots and zero
on the other simple roots, the matrices are projections to
$V_\mu^{N_P}$, and their stabilizer is $P^-\times_M P$.
These forms of the Vinberg statements, including the smooth stabilizer
family and standard section, are recalled in
\cite[Appendices~C--D]{DrinfeldGaitsgoryCT}.

Let $\bar Y$ classify two $H$-bundles and a section of their associated
Vinberg bundle which is generically nondegenerate on $X$, modulo the
Cartan torus $T_H$.  The parameter $v$ is constant on the complete curve.
Write $b:\bar Y\to Y^2$ for the projection.  Over $v\in T_{\mathrm{ad}}$
there is an open immersion $j_0$ and a finite \'{e}tale $Z_H$-torsor
$l:Y\to\operatorname{im}(j_0)$: fixing $v=1$ gives an isomorphism between
the two bundles, with the residual central scaling.  Thus $bj_0l=\Delta$.

The map $b$ is representable and proper.  Relative finite type follows by
writing the finitely many regular matrix maps $h_\mu$ and their equations.
The torus acts freely relative to the two fixed bundles: all matrices are
generically nonzero, and the dominant weights generate the character
lattice.  For the valuative criterion, fix extended bundles over a DVR.
A torus torsor over its fraction field is trivial, so choose matrix data
there.  The minimum valuation $a(\mu)$ of $h_\mu$ at the generic point of
the special curve is additive under Cartan multiplication.  Indeed the
Cartan product of two nonzero residue matrices is nonzero.  Explicitly,
let $A^+=\bigoplus_\mu V_\mu$ and
$A^-=\bigoplus_\mu V_\mu^*$ be the two Cartan algebras, realized as
coordinate rings of basic affine spaces for opposite Borel conventions.
Inside their tensor product, the diagonal graded subalgebra is
\[
 A=\bigoplus_\mu(V_\mu\otimes V_\mu^*)
   =\bigoplus_\mu\operatorname{End}(V_\mu).
\]
Its product on two matrices $a_\mu,a_{\mu'}$ is
$\pi_{\mu,\mu'}(a_\mu\otimes a_{\mu'})\iota_{\mu,\mu'}$, where
$\iota$ and $\pi$ are the normalized inclusion and projection of the
Cartan component.  This is exactly the matrix product in the Vinberg
Cartan relation.  The basic affine spaces are geometrically integral;
hence the coordinate ring of their product and the displayed subalgebra
are domains, also after
extending to the residue function field.  The nonvanishing assertion
follows.  Hence $a$ extends to
an integral homomorphism from the actual weight lattice to $\mathbb Z$,
that is, a cocharacter of $T_H$.  Rescale by its value on the uniformizer.
Every matrix then has valuation zero and extends over the curve family;
normality removes the remaining apparent poles of these bundle maps.
The parameter coordinates are integral as well.  For a simple root
$\alpha_i$, choose sufficiently large actual dominant weights $\mu,\mu'$
so that the tensor product contains the component
$\mu+\mu'-\alpha_i$.  In its product identity the left side is integral
and the matrix of that component has valuation zero, forcing $v_i$ to be
integral.  The extended matrices are generically nonzero on the special
curve.  The rescaling is unique modulo $T_H$ over the DVR, which proves
uniqueness in the valuative criterion and hence properness.

Stratify $\bar Y$ first by the parameter orbits, indexed by the finitely
many standard parabolics for a fixed Borel.
On the orbit of $c_P$ one may set $v=c_P$, retaining the residual $Z_M$
scaling.  Refine by Levi degrees and defect.  The base $B$ of a refined
stratum consists of two Levi bundles and a positive modification between
them, described by the closure $\bar M$ of $M$ in the Levi blocks of the
semigroup, modulo this residual scaling.  The stratum is the pullback of
\begin{equation}\label{eq:M5}
 q^-\times q:Y_{P^-}\times Y_P\longrightarrow Y_M\times Y_M
\end{equation}
over $B$.  For simply connected derived group this is the defect
stratification in \cite[Proposition~3.2.2]{SchiederVinberg}.  The following
description verifies what we need without changing the global form.

At the generic point of $X$, the homogeneous-orbit description produces
the two parabolic flags.  They extend over the curve by properness of the
flag varieties.  Every matrix factors through its Levi block because it
does so generically; the remaining regular maps give exactly a
$\bar M$-modification.  Conversely these data give the Vinberg section.
In families, take actual positive multiples $\mu_i$ of the fundamental
weights for omitted simple roots.  Their Levi blocks are lines.  Stratify
by lengths of the zero divisors of the generically rank-one matrix maps,
and flatten those zero loci.  On each piece the divisors are relative
Cartier divisors.  Divide out their equations: the resulting image lines
and dual image lines recover the flags.  Their ranks and Pl\"ucker
relations, and all block factorizations, follow from the generically
nondegenerate locus, which is schematically dense on the relative smooth
curve.  Passing to reduced strata suffices for D-modules.  These finitely
many weights embed the flag varieties; the remaining conditions concern
only the modification.  This constructs the claimed locally finite
stratification with fibers \eqref{eq:M5}.

A nonempty defect divisor gives a nonzero rational difference of Levi
degrees, by pairing with these Levi characters.  Conversely, if all these
divisors vanish, determinants of every Levi block are nonzero.  Indeed
powers of these determinants are Laurent monomials in the line characters,
which span the rational character space; after clearing denominators these
relations hold on the closure.  Thus the modification lies in $M$ and has
zero defect.  This proves both implications used below.

On $\bar Y$ take the kernel
\[
 J^b=(j_0)_!l_*k_Y
\]
with the ratio twist of its two ends and the diagonal splitting on the
cover.  Its !-restriction to a stratum is pulled back from $B$.  We give the
argument, since this assertion is the reason the degree vanishing test can
be applied to this kernel.  Choose $z\in X$ where the section is
nondegenerate.  Trivialize the torus quotient locally and, by an \'{e}tale
torus rescaling, set all nonzero parameter coordinates equal to one.
Frame the open-orbit datum at $z$.  The stabilizer group scheme is smooth
over the parameters.  Laurent directions in
$\mathfrak n_{P^-}\oplus\mathfrak n_P$ at $c_P$ therefore extend locally to
sections of its Lie algebra, using only finitely many jets for a chosen
finite set of directions.  Infinitesimal gluing at $z$ defines derivations
on smooth presentations of the ambient Vinberg bundle stack.  They
preserve $v$, hence the pair defining $j_0$, and lift compatibly to the
stratum fiber over $B$: there they change the two unipotent extensions and
keep the Levi modification fixed.  Principal parts on the two factors,
with the relative chart directions, span that fiber's tangent space.
The finite \'{e}tale image $l_*k_Y$ is a local system on the open.  Apply
the analytic-flow observation and then vector-stack descent exactly as in
Lemma~\ref{pseudo:fiber-descent}.  This proves the assertion for $J^b$.

On $B$ there are independent central-inertia cocharacters at its two ends:
the residual $Z_M$ quotient absorbs the difference of the two central
actions on the modification.  Their weights on the ratio line are
$Q(z,u_1)$ and $-Q(z,u_2)$.  The irrational Kummer test therefore kills the
twisted category unless both $u_1$ and $u_2$ are torsion.  Their difference
then vanishes rationally, so the defect must be zero.  Only the defect-zero
strata of torsion degree remain.  There are finitely many such degrees and
finitely many parabolics.

Set
\[
 D^b=\operatorname{cone}\bigl(J^b\longrightarrow(j_0)_*l_*k_Y\bigr).
\]
For a boundary stratum inclusion $i$, the identity
$i^!(j_0)_*=0$ gives $i^!D^b\simeq i^!J^b[1]$.  Thus the preceding
boundary calculation applies to the cone.  Use localization with
!-restrictions and star extensions from strata to filter $D^b$.
Within a fixed parameter orbit, the !-restrictions on every nonzero-defect
stratum vanish by the preceding argument; this is checked on
quasi-compact pieces using the locally finite stratification.  Thus the
cone has a finite filtration involving only the defect-zero strata for
proper parabolics.  Push it by the proper map $b$.  On a stratum over $B$
its contribution has the form
\[
 (p^-\times p)_*(q^-\times q)^!K
\]
for a kernel $K$ on $Y_M\times Y_M$, by !-base change from $B$.  The map
forgetting the modification is quasi-compact and safe on fixed degrees:
in the remaining defect-zero case it is the isomorphism datum modulo
central scaling, which acts freely relative to the two Levi bundles.
If $T_K$ denotes the functor represented by $K$, the displayed kernel
represents $E_P\circ T_K\circ C_{P^-}$; any shift from the cone is
absorbed into $K$.  This is the required factorization through both a
proper constant term and a proper Eisenstein image.  Finally, $bj_0l=\Delta$, $b$ is proper, and
$k_Y=\omega_Y[-2\dim Y]$.  The two pushed kernels are therefore
$\Delta_!k_Y$ and $\Delta_*\omega_Y[-2\dim Y]$, respectively.  This proves
the Proposition.
\end{proof}

\subsection{Induction and the reductive case}

We finish the proof of Theorem~\ref{pseudo:equivalence}.  The induction is
on semisimple rank, simultaneously for the two signs of $s$.  We first
explain the semisimple step, assuming the assertion for reductive groups
of smaller semisimple rank.

For every proper parabolic and torsion degree, choose compact generators
$w$ of $\mathcal C_M^u$ and put $x=E_Pw$.  Equation~\eqref{eq:M1} implies
that $x$ is compact.  Equation~\eqref{eq:M4} gives
$F_Hx\simeq L_{P^-}F_Mw$, which is compact by the inductive equivalence.
For every $y$, adjunction and \eqref{eq:M4} give natural isomorphisms
\[
 \operatorname{Hom}(F_Hx,F_Hy)
 \simeq \operatorname{Hom}(F_Mw,C_{P^-}F_Hy)
 \simeq \operatorname{Hom}(w,(L'_P)^\vee y)
 \simeq \operatorname{Hom}(x,y).
\]
Full faithfulness requires the actual map induced by $F_H$ to be
invertible.  The preceding adjunction calculation gives an isomorphism
natural in $y$, but has not yet shown that its inverse sends the identity
of $x$ to the identity of $F_Hx$.  By enriched
Yoneda, the inverse of the displayed natural isomorphism has the form
$f\mapsto F_H(f)b$ for an endomorphism $b$ of $F_Hx$.  Surjectivity at $y=x$
provides $a\in\operatorname{End}(x)$ with $F_H(a)b=1$.  A split epimorphism
endomorphism of $F_Hx$ is invertible: on affine charts its bounded coherent
cohomology modules are noetherian, and a surjective endomorphism of a
noetherian module is injective.  Hence $F_H(a)$ and then $b$ are invertible.
The actual $F_H$-comparison is consequently an isomorphism.

Let $\mathcal B$ be the right orthogonal of this set of compact $x$.
Equation~\eqref{eq:M1} identifies it with the objects whose proper
constant terms vanish.  Equation~\eqref{eq:M3} also makes these objects
left orthogonal to every proper Eisenstein image.  If $m\in\mathcal B$,
Proposition~\ref{pseudo:boundary-factorization} therefore gives both
$F_Hm\simeq m[-2\dim Y]$ and
\[
 \operatorname{Hom}(m,y)\xrightarrow{\ \sim\ }
 \operatorname{Hom}(F_Hm,F_Hy).
\]
The second assertion follows by applying $\operatorname{Hom}(m,-)$ to
the cone filtration and using naturality of its map.  Compact localization
says that the $x$ together with $\mathcal B$ generate under colimits.
Mapping out of colimits converts them to limits, so these comparisons
prove full faithfulness of $F_H$ on every source.  Its essential image is
closed under colimits.  It contains $\mathcal B$ because $F_H$ is a shift
there, and it contains every Eisenstein generator by \eqref{eq:M4} and
the inductive equivalence.  Hence it is the entire category.

It remains to justify the reductive groups used at the induction base and
as Levi subgroups.  For a reductive $H$, let $H_{\rm der}$ be its derived
group and $Z^0$ its connected center.  Their product maps to $H$ by a
central isogeny.  On torsion degrees, the induced map of bundle stacks
\[
 f:Y_{H_{\rm der}}\times Y_{Z^0}^{\,0}\longrightarrow Y_H^{\rm tors}
\]
retaining all contributing lifting components is surjective.  In fact the
abelianization degree of a torsion class is zero.  In the exact sequence
\[
 \pi_1(H_{\rm der})\longrightarrow\pi_1(H)
 \longrightarrow\pi_1(H/H_{\rm der})\longrightarrow0,
\]
the left image is finite and the right group is a free lattice, so the
left image is exactly the torsion subgroup.  Thus the degree lifts from
the derived group.  The degree-two obstruction for the finite central
isogeny is the class of this degree in the cokernel of the map on
fundamental groups; it therefore vanishes.  One may check this last statement topologically on the complex
curve and then use comparison for finite coefficients.  When a lift
exists, its choices form a torsor for bundles of this finite kernel.
There are finitely many such bundles and finite automorphism groups.
The map $f$ is therefore, \'{e}tale locally on the base, a finite union of
finite \'{e}tale gerbes.  Formal \'{e}taleness follows as well from the
isomorphism of Lie algebras.  In characteristic zero, $f$ is proper and
safe, $f^!=f^*$ is conservative, and pushforward and pullback are
ambiadjoint.

The twisting separates on this product.  Decompose $V$ into representations
of $H_{\rm der}$ tensored with central characters.  The determinant-of-
cohomology tensor formula separates each summand into its derived and
central factors; its mixed Deligne-pairing term vanishes because a
representation of a semisimple group has trivial determinant.  On the
neutral component of the torus stack, a rigidification at a point of $X$
identifies the stack with a Picard variety times $BZ^0$.  Writing $V=\bigoplus_i W_i\otimes\chi_i$, its central inertia weight
on degree zero is $(1-g(X))\sum_i\dim(W_i)\chi_i=0$, because $V$ is
self-dual.  Thus only the Picard factor is twisted.  Its
pseudo-identity is a shift, since the variety is smooth and separated.
For $BZ^0$ use the untwisted miraculous duality theorem
\cite[Theorem~0.1.6]{GaitsgoryStrange}, applied to the neutral component of
$\operatorname{Bun}_{Z^0}^{\,0}(\mathbb P^1)=BZ^0$.  The extraordinary diagonal kernel of the product is the exterior
product of those of its factors, so its pseudo-identity is
$F_{H_{\rm der}}\otimes F_{Z^0}$.  Thus rank zero is established, and
the semisimple assertion of any rank gives the product assertion of that
rank.

Finally that assertion descends along $f$.  The extraordinary diagonal
kernels give
\[
 f_*F_{H_{\rm der}\times Z^0}\simeq F_Hf_* ,\qquad
 F_{H_{\rm der}\times Z^0}f^!\simeq f^!F_H.
\]
For example both kernels for the first equality are
$\Gamma_{f,!}k$, where $\Gamma_f$ is the graph: use $f_*=f_!$ and then
star--shriek base change with $f^!=f^*$, which holds because $f$ is
\'{e}tale.  Transposition at the opposite twist gives the second equality.
Compact objects pushed from the product remain compact and generate
downstairs, since $f^!$ is continuous and conservative.  Their images
under $F_H$ are compact by the product assertion.  The displayed identities
and ambiadjunction give the same natural Hom isomorphism as above on these
generators.  The noetherian split-epimorphism argument identifies it with
the actual comparison map.  Full faithfulness follows by generation;
essential surjectivity follows because the pushed generators lie in the
image.  This completes the simultaneous induction and proves
Theorem~\ref{pseudo:equivalence}.
==== END Global-quantum-geometric-Langlands-at-irrational-level//03-pseudoidentity.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//04-whittaker.tex ====
\section{Whittaker categories in families}\label{sec:whittaker}

The Whittaker coefficient functor will be used both in duality and in
families in which marked points collide. We construct its projection on
all D-modules, establish its compatibility with those families, and fix the
pairing and t-structure normalizations. The finite averaging calculation
below is the common source of these properties.

\subsection{The pole and Grassmannian models}

Let $H$ be one of the semisimple groups under consideration. Fix a Borel
$B=TN$ and the Cartan bundle $T_0$ inducing the bundle $P_0$ of
Section~\ref{sec:foundations}. Write
\[
 N_0=N^{T_0},\qquad \mathfrak n_0=\operatorname{Lie}(N_0),\qquad
 \delta_N=-\chi(X,\mathfrak n_0)=\dim\operatorname{Bun}_{N_0}.
\]
Here $\chi(X,-)$ denotes Euler characteristic. Each simple-root line of
$\mathfrak n_0$ is $\Omega_X$. The pinning and the sum of residues therefore
define a nondegenerate additive character, also denoted $\chi$, on the
unipotent loop groups. We use both $\chi$ and $-\chi$. Our gluing convention
sends the inside frame to the outside frame.

A parameter scheme $S$ carries a nonempty tuple $(x_i)_{i\in I}$ of points
of $X$; auxiliary parameters are allowed. The principal examples are the
powers $X^I$ and their coincidence strata. All operations in the parameter
scheme use the D-module conventions of Section~\ref{foundations:operations},
without a perverse shift unless one is displayed. The ordinary Ran space
is presented by the $X^I$, with transition maps the diagonals indexed by
surjections of nonempty finite sets.

Let $\mathcal Y\to S$ be the following pole stack. An object consists of an
$H$-bundle $P$ together with Pl\"ucker maps
\[
 T_0^\lambda\longrightarrow V_P^\lambda
 \qquad(\lambda\text{ a dominant weight of }H),
\]
regular away from the marked points, generically nonzero, and satisfying
the Cartan-product relations. All representations are those of the given
group $H$. Thus the definition uses the affine closure of $H/N$ and allows
defects; it imposes no quotient by Cartan scalings. Denote the forgetful map
by $p_{\mathcal Y}:\mathcal Y\to\operatorname{Bun}_H$ and give $\mathcal Y$
the pullback of the determinant twist $L_H^s$.

We write $\mathcal Y_m$ for closed pole bounds. Choose effective bounds for
a finite set of highest-weight generators, compatible with Cartan
multiplication; multiples of a coweight strictly positive in the coroot
span give a cofinal choice. At colliding markings these bounds add. Each
$\mathcal Y_m$ is locally QCA, and its map to
$\operatorname{Bun}_H\times S$ is schematic, separated, and of finite type:
after bounding poles, the finitely many maps are sections of vector bundles
satisfying equations and open conditions.

On each $\mathcal Y_m$ take the full subcategory of
$D_{p_{\mathcal Y}^*L_H^s}(\mathcal Y_m)$ defined by the strong Whittaker
condition: its objects are equivariant with character $\chi$ for
$N$-modifications away from the marked points and on the nondefective
locus. The determinant twist is split along these modifications by the
Borel filtration and the relative determinant identification. Restriction
between closed pole bounds respects this character descent. Define
$\mathcal W_s^+(S)$ to be the limit of these full subcategories under
$!$-restriction, inside the ambient category
\[
 D_{p_{\mathcal Y}^*L_H^s}(\mathcal Y)
 =\lim_m D_{p_{\mathcal Y}^*L_H^s}(\mathcal Y_m).
\]
The opposite category $\mathcal W_{-s}^-(S)$ is defined with $L_H^{-s}$
and $-\chi$. If the derived group is not simply connected, the usual
Drinfeld compactification imposes an additional closed defect condition.
Whittaker objects on the Pl\"ucker model used here are supported on that
closed substack \cite[Remark~4.1.2]{GaitsgoryWhit}; hence the two models
define the same Whittaker category. The projection and continuity properties, and the equivalent
colimit presentation, will be proved below. One may impose this condition by choosing an auxiliary point $z$, passing
to the open where $z$ is neither marked nor defective, framing the
reduction at $z$, and using the unipotent loops there. We verify below that
the resulting condition is independent of these choices.

The Grassmannian $\operatorname{Gr}_{P_0,S}$ parametrizes modifications of
$P_0$ at the marked points. Such a modification supplies the Pl\"ucker
maps and hence a morphism
\[
 \mu:\operatorname{Gr}_{P_0,S}\longrightarrow\mathcal Y.
\]
Its twisting line is the relative determinant: the pullback of $L_H$
divided by its fixed value at $P_0$. This line factorizes over disjoint
formal discs and has the unipotent splitting just specified.

\begin{proposition}[Comparison of Whittaker models]
\label{whittaker:model-comparison}
For a connected semisimple complex group $H$, a smooth complete curve $X$,
and the fixed half-canonical datum defining $T_0$, the functor $\mu^!$
identifies the pole-model Whittaker category with the strongly
$\chi$-equivariant D-module category on
$\operatorname{Gr}_{P_0,S}$ with its relative determinant twist. This holds
for every exponent $s$, with arbitrary D-modules in the marked-point
parameters, and is compatible with the multipoint and Ran presentations.
The assertion also holds with both signs reversed.
\end{proposition}

\begin{proof}
For the untwisted sheaf theory this is Gaitsgory's local/global comparison,
\cite[Theorems~5.5.2 and~7.2.3]{GaitsgoryWhit}. We explain why its proof
applies to our determinant twist, so that the distinction between a
character twist and a twisting line is explicit.

Let $\mathcal L=p_{\mathcal Y}^*L_H$ and form the frame torsor
$\mathcal L^\times\to\mathcal Y$. Pull every correspondence in the
comparison proof back from its target to this torsor. On an open
action correspondence, the unipotent and arc splittings transport this
frame to the source. On a compactified correspondence we pull back only
the target torsor; the open source-frame identification is not extended
as a trivialization over its boundary. This single Cartesian pullback
preserves the proper maps in the compactified action diagrams.

There are two ingredients in the comparison proof
\cite[Sections~6.1--6.6]{GaitsgoryWhit}: full faithfulness after Ran
uniformization, and the equivalence obtained by restricting to the unit
configuration in the Whittaker category. The geometric input to the first
is universal homological contractibility. This means precisely that the
corresponding pullback is fully faithful after every prestack base change;
see \cite[Appendix~A.1.8--A.1.11]{GaitsgorySemiInfiniteII}. It therefore
applies after the frame-torsor base change as well as with the marked-point
parameters retained.

The second ingredient uses unipotent averaging, arcs, and conjugates of
positive-depth subgroups. The determinant central extension is canonically
split on arcs, and the Borel filtration splits it on unipotent
modifications. These splittings restrict to the finite quotients used in
averaging and respect multiplication. For the subgroups conjugated by a
multiple of the half-sum of positive coroots, take the cofinal sequence of
even depths. The conjugating cocharacter then belongs to the actual
cocharacter lattice of $H$, since twice that half-sum is the sum of
coroots. Conjugation of the arc splitting is consequently defined: choose
a lift of the conjugating loop to the central extension, locally if
necessary; any two choices differ centrally and give the same result.

There is also a family compatibility to check at collisions. In the
notation of \cite[Section~6.6.4]{GaitsgoryWhit}, the group
$\mathfrak L_I^+(H)^j$ is the inverse image of the preceding one-point
subgroup under
$\mathfrak L_I^+(H)'\to\mathfrak L_x(H)$. On this group the relative
determinant extension is pulled back from the distinguished disc at $x$:
the primed group allows poles only on the distinguished graph
$E=S\times\{x\}$. More explicitly, let $t$ cut out $E$ and let $A_I$
be the ring completed along the entire marked divisor, including $E$. On a bounded
piece choose $m$ so that
$t^m\Lambda\subset g\Lambda\subset t^{-m}\Lambda$. The finite
lattice quotients defining the relative determinant are supported on a
thickening of $E$. For every $r$,
\[
 A_I/(t^r)\simeq\mathcal O_{S\times X}/(t^r)
             \simeq\mathcal O_S[[t]]/(t^r),
\]
because the ideal of the entire marked divisor is nilpotent modulo
$t^r$. Thus restriction to the distinguished disc preserves the quotient
complexes and their determinant identifications, including on collision
and nonreduced parameter schemes. The splitting therefore pulls back from the one-point
subgroup. It agrees on overlaps with the canonical arc splitting;
on pro-unipotent kernels uniqueness follows from the absence of
algebraic characters to $\mathbb G_m$. These constructions commute with
coordinate changes and match the determinant ratio at $P_0$.

Finally, the clean-extension argument lifts as well. In its product
presentation $(g,y)$, the ratio between the target and source determinant
lines is the multiplicative line
$\mathcal R(g)=\det(g\Lambda:\Lambda)$. The splitting on the
conjugated congruence group descends this line through the quotient used
in \cite[Appendix~A.2]{GaitsgoryWhit}, and the unipotent splitting gives
its left $N$-action. The ratio line can remain nontrivial on the boundary;
only its equivariance is needed. On every finite-dimensional boundary
piece the proof supplies a unipotent
stabilizer on which the Whittaker character is nontrivial
\cite[Appendix~A.2--A.3]{GaitsgoryWhit}. This stabilizer fixes the source coordinate and acts on the ratio-line
fiber by an algebraic character to $\mathbb G_m$, hence trivially. It
therefore fixes the target frame as well, while retaining its nontrivial
Whittaker character. The same boundary-vanishing argument
applies to arbitrary objects on the frame torsor. Together with the
lifted averaging correspondences, this transports the units, counits,
and clean-extension isomorphisms of the unit-restriction equivalence.

Every lifted map commutes with multiplication of the determinant frame by
$\mathbb G_m$. Taking strong equivariance with the Kummer character of
exponent $s$ therefore descends the lifted equivalence to D-modules
twisted by $\mathcal L^s$. An equivariant equivalence induces an
equivalence on these descent categories, including their full
cosimplicial descent data. This yields the stated determinant-level
comparison, and preserves its base and Ran compatibilities.
\end{proof}

We will need two point inputs. At an irrational scalar level, the point
Whittaker Grassmannian category is a sum of copies of
$\operatorname{Vect}$ indexed by the dominant coweights of the actual
group. Here is the precise consequence of Campbell--Dhillon--Raskin that
we use. The irrational case of their integral-Weyl-group calculation
\cite[Theorem~3.5.6, Equation~(3.28)]{CDR} leaves one dominant label in
each contributing spherical highest-weight block. Its finite-length
one-simple highest-weight heart is therefore finite-dimensional vector spaces.
The derived-heart comparison of \cite[Section~2.9.4]{CDR}, together with
the compact generators used in the proofs of their Corollary~3.5.9 and
Theorem~3.7.3, identifies the presentable block with
$\operatorname{Ind}(D^b(\operatorname{Vect}^{\mathrm{fd}}))=
\operatorname{Vect}$. Thus higher self-extensions are excluded by the
derived comparison, not just by the absence of linkage. Their spherical
fundamental local equivalence \cite[Remark~3.7.5]{CDR} transfers this
statement to Whittaker D-modules. Their actual-form reduction
\cite[Section~4.2, Proposition~4.2.2, and Section~4.3]{CDR} retains the
actual weight and coweight lattices and treats products. An irrational
complex level of either sign is in their negative good-level regime.
For groups with simply connected derived group, the corresponding
irrational clean-standard statement was already proved in
\cite[Proposition~8.4]{GaitsgoryTwisted}.

At exponent zero we use the clean-standard theorem
\cite[Theorem~3]{FrenkelGaitsgoryVilonen}; its Introduction explicitly
includes characteristic-zero D-modules. Cleanliness and the mapping
complex calculation in Proposition~\ref{whittaker:standards} give the
corresponding DG semisimplicity. This is the categorical form of geometric
Casselman--Shalika needed here. When passing between invariant and
coinvariant conventions at a point, we use the Whittaker
invariant/coinvariant theorem \cite[Theorem~2.1.1]{RaskinW}. These are point
statements; the family conclusions used below will be deduced separately.

\subsection{Defect strata and a continuous Whittaker projection}

In a fixed pole bound, record actual poles and defects by
\[
 D=\sum_y\lambda_y y,\qquad T_D=T_0(-D),
\]
with poles positive. Orders of Pl\"ucker maps pair additively with dominant
weights. Indeed their leading terms have nonzero Cartan product, so their
valuations add. Using every highest weight of $H$ shows that each
$\lambda_y$ is a coweight of $H$. Away from the markings it belongs to the
negative rational coroot cone. At a marking it is bounded above by the
specified pole bound, in the order defined by that cone.

Stratify both the markings by their coincidences and the Pl\"ucker maps by
this divisor. Write $S_D$ for the resulting divisor-parameter scheme and
\[
 f_D:\mathcal Y_D\hookrightarrow\mathcal Y_m,\qquad
 q_D:\mathcal Y_D\longrightarrow S_D
\]
for the inclusion and the parameter map. Dividing the maps by their common
zeros gives a genuine Borel reduction with Cartan bundle $T_D$.
Conversely such a reduction recovers the point of the stratum. Thus
$q_D$ parametrizes $N^{T_D}$-bundles. It is a smooth fibration given by a
tower of vector stacks: filtering $N$ by root height reduces each stage
to the stack associated with the two-term complex of global sections of
a root line bundle. The determinant twist restricts to a line from $S_D$,
by the same filtration.

This description is valid in families. After clearing bounded poles,
flattening by the lengths of zero divisors makes them relative Cartier
divisors. One then refines by coincidences, and, when needed, passes to
\'etale charts enumerating the support. D-modules allow us to ignore
nilpotents in this stratification. Opens bounding the total zero lengths
exhaust $\mathcal Y_m$; over quasicompact parameter pieces they are
quasicompact and have only finitely many of these strata. Their
stabilizers are unipotent. In fact an automorphism preserving all
Pl\"ucker maps restricts generically to the unipotent stabilizer of a
reduction with fixed Cartan datum. Alternatively, on a bounded open a
sufficiently deep frame at a nondefective point kills the stabilizer.

A stratum is \emph{relevant} if every $\lambda_y$ is dominant. Dominance
and the negative-cone condition force all off-marking defects to vanish.
There are only finitely many relevant labels in each pole bound.
On a relevant stratum the simple-root lines are
\[
 \alpha(T_D)=\Omega_X\bigl(-\langle\alpha,D\rangle\bigr)
 \longrightarrow\Omega_X.
\]
The simple-root abelianization and the trace on $H^1(X,\Omega_X)$ define a
function $\psi_D:\mathcal Y_D\to\mathbb A^1$. Choose its sign so that
modification at a nondefective point changes $\psi_D$ by the residue
character of the modifying loop. In Grassmannian coordinates its phase
along $nt^\lambda$ is therefore $\chi(n)$.

\begin{proposition}[Projection and stratum tests]
\label{whittaker:strata-properties}
In every pole bound, inclusion of the strongly Whittaker category has a
continuous right adjoint $\operatorname{Av}_*$. Its image consists exactly
of the objects $F$ satisfying
\begin{enumerate}
\item $f_D^!F=0$ on every irrelevant stratum;
\item on a relevant stratum,
$f_D^!F=\operatorname{Exp}(\psi_D)\otimes^!q_D^!E_D$ for an object $E_D$
in the D-module category of $S_D$ with its induced twist.
\end{enumerate}
The analogous statement uses $-\psi_D$ for the opposite character.
Restriction by $!$ computes the projection on each stratum. These
projections commute with $!$-base change in the parameter schemes and
with tensoring a spectator category. They are compatible with
$!$-restriction between pole bounds. Closed direct images, both between
pole bounds and along closed parameter embeddings, preserve the
Whittaker subcategories.

If a bounded-zero open contains every relevant stratum of a pole bound,
restriction to that open and star extension identify its Whittaker
category with that of the entire bound.
\end{proposition}

\begin{proof}
We first construct the projection on a quasicompact bounded-zero open on
which an auxiliary point $z$ is unmarked and nondefective. Use root-height
coordinates for $N_0$ at $z$. Let $U_n$ be the unipotent loop subgroup
allowing pole order at most $n\operatorname{ht}(\alpha)$ in the
positive-root coordinate $\alpha$; $U_0$ is the arc subgroup. Root heights
add under multiplication, so these are subgroups. Each $U_n$ is
pro-unipotent and the homogeneous space $U_n/U_0$ is finite dimensional
and de Rham acyclic; $U_0$ need not be normal in $U_n$.

Let $\mathcal Z^{\mathrm{fr}}$ be the formally framed stack on this
open. It carries a $U_n$-action, and the unframed open is
$\mathcal Z^{\mathrm{fr}}/U_0$. Its averaging arrows form the \v Cech
groupoid of
\[
 \mathcal Z^{\mathrm{fr}}/U_0
   \longrightarrow\mathcal Z^{\mathrm{fr}}/U_n.
\]
Equivalently, these arrows are isomorphisms away from $z$ and
modifications by $U_n$ at $z$. The character descends to the groupoid
because $\chi|_{U_0}=0$; normality of $U_0$ is unnecessary. Write
$s_n,t_n$ for source and target. They are smooth, with
fiber $U_n/U_0$ of dimension $d_n$, and preserve the pole bounds and defect
strata. Indeed the Pl\"ucker maps agree away from $z$, and $N$ fixes
every highest-weight vector at $z$; the modified reduction is therefore
still nondefective there. Thus the actual defect divisor and the
bounded-zero open are preserved. If finite-type charts are desired, divide frames by an arc subgroup
sufficiently deep to be normal in $U_n$. The arrow splitting of the
determinant line defines the averaging functor
\begin{equation}\label{eq:W1}
 \operatorname{Av}_{n,*}(F)=
 t_{n,*}\bigl(\operatorname{Exp}(\chi)\otimes^!s_n^!F\bigr)[-2d_n].
\end{equation}
Smooth pullback along the unipotent fibers is fully faithful because
their de Rham cohomology is the ground field. Base change and additivity of
$\chi$ identify \eqref{eq:W1} with a strongly equivariant object for this
finite groupoid. Smooth star-pullback/star-image adjunction then identifies
it with the right adjoint to inclusion of that equivariant subcategory.
In particular its counit and the transition maps for larger $n$ are
canonical. The maps in \eqref{eq:W1} are finite-type and safe after taking
these quotients, so the functor is continuous on all D-modules.

We next compute the restrictions of \eqref{eq:W1}. On a defect stratum its
arrow stack maps to $\mathcal Y_D\times_{S_D}\mathcal Y_D$. For large $n$
this map is, smoothly locally, a tower of affine-space bundles. To see
this, lift isomorphisms between the two unipotent extensions in order of
root height. At the $\alpha$-stage, possible lifts form a torsor under
global sections of the corresponding root line with poles
$n\operatorname{ht}(\alpha)z$. The obstruction lies in its $H^1$ and
vanishes for large $n$. Apply relative Serre vanishing to the bounded
family of root lines on $X\times S_D$. Once $R^1p_*$ vanishes,
Riemann--Roch and cohomology-and-base-change make the spaces of sections
locally free of constant rank, compatible with arbitrary parameter base
change. Hence their torsors are smoothly locally affine-space bundles
even where the unmodified $H^0$ and $H^1$ jump. Higher-root lifts are
controlled by the same graded root lines. The height convention makes these successive extensions
compatible with multiplication.

On a relevant stratum the character on the arrow stack is
$t_n^*\psi_D-s_n^*\psi_D$, by the residue description of the trace.
Base change and de Rham integration along the preceding affine-space
towers therefore give
\begin{equation}\label{whittaker:stratum-projector}
 F\longmapsto\operatorname{Exp}(\psi_D)\otimes^!
 q_D^*q_{D,*}\bigl(\operatorname{Exp}(-\psi_D)\otimes^!F\bigr).
\end{equation}
For the shift, put $e=\dim(q_D)$ and let $h=d_n-e$ be the relative
dimension of the affine-space tower from the arrow stack to
$\mathcal Y_D\times_{S_D}\mathcal Y_D$. Integration of its $!$-pullback
contributes $[2h]$, so
\[
 [2h-2d_n]=[-2e],\qquad q_D^![-2e]=q_D^*.
\]
The two pullbacks have the same essential image, differing by this fixed
shift on the stratum. Pullback is fully faithful for the vector-stack tower
$q_D$, so \eqref{whittaker:stratum-projector} is its exponential-conjugate
projection.

On an irrelevant stratum there is a simple root $\alpha$ for which
$\alpha(T_D)$ admits a pole at some point other than $z$. A section with
sufficient pole allowance at $z$ can prescribe a nonzero residue at that
point and zero residues at the other allowed poles. The required
surjectivity is again the vanishing of $H^1$ after increasing $n$.
The residue theorem makes its residue at $z$ nonzero. Thus, after
integrating the higher-root coordinates, the averaging contains the de
Rham cohomology of a nonconstant linear exponential on an affine space;
this cohomology is zero. The calculation works over the parameter scheme,
\'etale locally if a support enumeration is needed.

There are finitely many strata in our open. Choose $n$ large enough for
all of them. On every stratum the adjunction transitions from larger
averaging functors to \eqref{eq:W1} are now isomorphisms: on relevant strata
all are the projection \eqref{whittaker:stratum-projector}, with its
canonical counit, and on irrelevant strata all are zero. The finite
stratification is conservative for $!$-restriction, so the transitions are
isomorphisms on the open. Equivalently, the counits testing equivariance
for larger $U_n$ become isomorphisms there. The stable functor is therefore
the right adjoint to the full Whittaker inclusion, with exactly the two
stated stratum tests.

All calculations were relative and used finite-type correspondences, so
they commute with $!$-base change and spectators before stabilization.
The same uniform finite choice proves that they still do so afterwards.
On overlaps of auxiliary-point opens the tests describe the same full
subcategory. The right adjoints and their counits consequently glue,
uniquely compatibly with the inclusions. Several or varying auxiliary
points give the same condition: repeat the calculation over their
parameter scheme, replacing $nz$ by the corresponding auxiliary divisor
and adding their characters. This also verifies the smooth-local version
of strong equivariance.

The bounded-zero opens exhaust the pole bound, and their projections glue
by these compatibilities. Closed images preserve the two tests by
$!$-base change, both for an increase of pole bound and for a closed
parameter embedding. Finally, outside an open containing all relevant
strata the $!$-restriction of a Whittaker object is zero. The localization
triangle identifies it with star extension from that open. Conversely
star extension preserves the tests and hence recovers a Whittaker object.
This proves the last assertion and completes the proof.
\end{proof}

\subsection{Duality and passage to the Ran space}

The projections just constructed allow duality to be established on the
full Whittaker categories. The opposite character is essential: it
cancels the exponential in the pairing.

\begin{proposition}[Whittaker duality]\label{whittaker:duality}
The categories $\mathcal W_s^+(S)$ and $\mathcal W_{-s}^-(S)$ are dual,
with the normalized pairing
\begin{equation}\label{eq:W2}
 \langle W,W'\rangle=
 \Gamma_{\mathrm{dR}}(W\otimes^!W')[-2\delta_N].
\end{equation}
In a pole bound this formula uses the bounded-zero star-extension
convention of Proposition~\ref{whittaker:strata-properties}. It then
extends continuously to the full pole model. Equivalently, one may place
one argument in a pole bound and $!$-restrict the other to that bound.
The analogous pairing relative to $S$ is defined by relative star image.
Closed images transpose to $!$-restrictions.

These statements hold for the Ran categories, formed as limits under
$!$-restriction along the diagonals, and are compatible with tensoring a
spectator category.
\end{proposition}

\begin{proof}
First work in a bounded-zero quasicompact open of a pole bound. Its
unipotent inertia makes ordinary de Rham pushforward safe. The twisted
QCA duality pairing is therefore ordinary de Rham global sections after
the twists cancel.

Here is a direct way to pass the smooth-stack duality of
Proposition~\ref{foundations:categories} to these pole stacks. On a
quasicompact open of $\operatorname{Bun}_H$, allow sufficiently many extra
poles that the relevant $H^1$ groups vanish. The spaces of Pl\"ucker maps
are then vector bundles, and their relations are closed conditions in a
smooth quotient stack. An open condition can be realized by restricting
the ambient stack to an open, so the bounded piece admits a closed
embedding into such a smooth ambient stack, with the twisting line
extended from it. For the parameters, work on affine opens embedded as closed subschemes
of smooth ambient schemes, using the graph of the marked-point map and
local trivializations of the twisting line. For a closed embedding $i$, Kashiwara's equivalence realizes the
supported category as the image of the continuous projector $i_*i^!$.
On a product the two external projectors commute and cut out precisely
the product support. Thus K\"unneth and the diagonal duality identities
pass to these closed supports. After cancellation of the twists,
renormalized pushforward in those identities agrees with ordinary
pushforward by safety. This supplies the stated pairing, also with
spectators.

Let $P_+$ and $P_-$ denote the projections for the two opposite characters.
For a Whittaker object $W$ and an arbitrary oppositely twisted object $F$,
the counit $P_-F\to F$ induces an isomorphism
\begin{equation}\label{whittaker:orthogonal-projection}
 \Gamma_{\mathrm{dR}}(W\otimes^!P_-F)
 \xrightarrow{\ \sim\ }
 \Gamma_{\mathrm{dR}}(W\otimes^!F).
\end{equation}
Indeed, on an auxiliary-point open insert the finite formula
\eqref{eq:W1} for $P_-$ with $n$ sufficiently large. The $!$-projection
formula moves $W$ to the arrow stack. Its positive-character equivariance
cancels the negative exponential in the averaging kernel. Integrating
along the source then cancels the dimension normalization of
\eqref{eq:W1} and gives the right side of
\eqref{whittaker:orthogonal-projection}. The identification is the one
induced by the counit. Open descent gives it on the whole quasicompact
open. The same argument interchanges the two characters.

Consequently the Whittaker subcategory on either side annihilates the
kernel of the projection on the other. Transposing the inclusion and its
continuous right adjoint under ambient duality gives adjoints in the
opposite order. The transposed projector is the identity on the opposite
Whittaker image and zero on its projection kernel, by
\eqref{whittaker:orthogonal-projection} and its counterpart. It is
therefore exactly the opposite projector. Splitting these projectors
identifies the two Whittaker categories as duals. The fixed shift
$[-2\delta_N]$ is an invertible normalization of this duality.

An open containing all relevant strata computes the Whittaker category of
the entire pole bound. The tensor product in \eqref{eq:W2} is likewise
star-extended from that open: its $!$-restriction to the complement is
zero. This proves the formula and duality on the bound.

For closed increases of pole bound, the continuous closed image is left
adjoint to the continuous $!$-restriction. The latter also preserves
limits. The limit of these right adjoints is the colimit of their left
adjoints in presentable categories. Formula~\eqref{eq:W2}, together with
the $!$-projection formula, transposes the closed image to the
corresponding restriction. Passing to this limit/colimit therefore
preserves the duality.

The Ran presentation has the same form. By
Proposition~\ref{whittaker:strata-properties}, its diagonal restrictions
and closed images are computed in the collision parameter schemes, after
a cofinal increase of pole bounds. Their adjunctions and transpose
identities are the ones just proved. Thus
\[
 \mathcal W_s^+(\operatorname{Ran})
   =\lim_{I,\,\Delta^!}\mathcal W_s^+(X^I)
   =\operatorname*{colim}_{I,\,\Delta_*}\mathcal W_s^+(X^I),
\]
with the corresponding statement for the opposite character. To see the
duality directly, compute maps out of the colimit by the diagram and use
the termwise dualities; the resulting dual is exactly the displayed
limit on the opposite side. Tensoring by a spectator category commutes
with this presentation because it commutes with the colimit, and the
continuous adjunctions tensor with it. The same argument gives the
relative pairing over $S$.
\end{proof}

\subsection{Standard objects and the t-structure}

We now specialize to an irrational exponent $s$ or to $s=0$. The point
semisimplicity recalled above makes the relevant strata into actual
summands. We must establish this over parameter schemes, since a
pointwise statement alone would not control arbitrary D-modules in a
family.

Let $S$ be a fixed coincidence piece, so that the distinct marked points
are indexed by a fixed finite set. A relevant label is a tuple of
dominant coweights at these points. For such a label $D$, the scheme
$S_D$ is identified with $S$; write $\mathcal L_D$ for the line on $S$
whose pullback is the determinant line on $\mathcal Y_D$. Set
\[
 a_D=\sum_y\langle 2\rho_{\mathrm{rt}},\lambda_y\rangle,
 \qquad 2\rho_{\mathrm{rt}}=\sum_{\alpha>0}\alpha.
\]
Riemann--Roch, applied to the root-height filtration, gives
\begin{equation}\label{whittaker:stratum-dimension}
 \dim(q_D)=-\chi(X,\mathfrak n^{T_D})=\delta_N+a_D.
\end{equation}

\begin{proposition}[Standard summands in families]
\label{whittaker:standards}
Suppose $s$ is irrational or zero. Over a fixed coincidence piece $S$,
the Whittaker category is the direct sum, in presentable DG categories,
of the categories $D(\mathcal L_D^s)$ indexed by relevant labels $D$.
The inclusion of the $D$-summand is star extension from its stratum;
with the normalization below it is
\begin{equation}\label{eq:W3}
 E\longmapsto
 f_{D,*}\bigl(\operatorname{Exp}(\psi_D)\otimes^!q_D^!E[-a_D]\bigr).
\end{equation}
The construction takes place in any pole bound containing the label and
is then carried to larger bounds by closed image. For holonomic
arguments in this formula, the canonical shriek-to-star map is an
isomorphism. Both assertions hold for the opposite twist and character.
\end{proposition}

\begin{proof}
Begin over a point. The star standard and its shriek counterpart are
Whittaker objects. For the former this follows from the stratum tests;
for the latter use the smooth modification arrows, equivariance, and
partial shriek on the holonomic exponential. Their endomorphism
complexes are the ground field: locally closed image is fully faithful,
and pullback along the unipotent stack $q_D$ is fully faithful.

For irrational $s$, the canonical map from shriek to star is nonzero,
since its restriction to the stratum is the identity. In the semisimple
point category supplied by the block theorem and
Proposition~\ref{whittaker:model-comparison}, an object with scalar
endomorphism complex is a shift of a single simple. The nonzero map
between these two objects is therefore an isomorphism. For $s=0$, the
same cleanliness is precisely the imported clean-standard theorem.
In either case, adjunction and cleanliness make the mapping complexes
between distinct standard objects zero: restriction of one clean
standard to the other stratum is zero. Moreover, partial shriek
adjunction against an arbitrary Whittaker object $W$ identifies maps
from the $D$-standard with the coefficient complex extracted from
$f_D^!W$ by exponential cancellation and the fully faithful $q_D^!$,
up to the displayed fixed shift. This extraction is continuous by
Proposition~\ref{whittaker:strata-properties}. Each standard is therefore
compact in the Whittaker category. The standards generate by the stratum
tests and finite stratification in each bound. These orthogonal compact
generators with scalar endomorphisms give the asserted presentable DG
summand description, including at exponent zero.

Over a fixed coincidence piece choose, \'etale locally, curve coordinates
near the distinct marked sections and compatible spin frames. If $z$ is
such a coordinate, $t=z-z(x_i)$ is an actual relative parameter; formal
\'etaleness identifies the completed graph neighborhood with
$\mathcal O_S[[t]]$. Thus no section of an arbitrary formal-coordinate
torsor is required. The spin frames give the corresponding Cartan
trivializations. The local Grassmannians, residue
characters, loop actions, and determinant twists are then products of
the point models with the parameter scheme, with the possible residual
line on the parameters retained. The point decomposition tensors with
$D(S)$. Taking invariants of this product commutes with that tensor
product because $D(S)$ is dualizable. This proves the family decomposition
in these coordinates, with all D-modules as coefficients.

To identify and descend its summands, take $!$-restriction along the
sections $t^\lambda$ in the Grassmannian model. Under the model comparison
this is evaluation of the corresponding exponential-conjugate $q_D^!$
object on the global label stratum, carrying exactly the twist
$\mathcal L_D^s$. The standard defined by $f_{D,*}$ has these restrictions
by base change. Hence the summands and their inclusions are intrinsic,
and descend under changes of coordinates and twisting-line
trivializations. This gives the asserted formula before its harmless
normalizing shift.

For a holonomic parameter object, shriek extension of the stratum object
is Whittaker by the same smooth-arrow calculation. Partial adjunction
computes its maps to Whittaker objects from their $!$-restrictions to the
label stratum. The summand description shows that star extension
represents the same functor. The canonical shriek-to-star map is the
resulting isomorphism. This proves the family cleanliness assertion.
\end{proof}

Use the usual stack perverse normalization on D-modules: $!$-pullback to
a smooth chart is shifted back by its relative dimension. On the pole
model translate the resulting t-structure so that its heart is the stack
heart shifted by $[\delta_N]$. This fixes the Whittaker t-structure used
from now on.

\begin{proposition}[Normalization and divisor amplitude]
\label{whittaker:t-structure}
This t-structure restricts to each Whittaker pole bound and extends to the
full pole model with t-exact closed transition images. With this
convention \eqref{eq:W3} is t-exact. On an affine chart where a smooth
divisor diagonal has inclusion $i$ and affine open complement $j$,
\[
 i_*,\ j^!,\ j_*\quad\text{are t-exact},
 \qquad i^!\quad\text{has t-amplitude }[0,1].
\]
The resulting t-structures on the parameter categories used here are
separated.
\end{proposition}

\begin{proof}
The strong character condition is preserved by truncation: its
isomorphisms are tested by the t-exact smooth pullbacks normalized by
relative dimension. Closed transition images are t-exact in the ambient
D-module categories and hence on their Whittaker subcategories.

An object in the full pole model is the filtered colimit of the closed
images of its $!$-restrictions to pole bounds. Truncate these objects in
the bounds. The transition maps between their truncations are compatible
because the closed images are t-exact. This constructs the two
truncation terms in the colimit category. Orthogonality follows by
adjunction: restriction to any fixed bound is left t-exact and commutes
with filtered colimits. The resulting t-structure is compatible with
filtered colimits.

On a fixed coincidence piece, Proposition~\ref{whittaker:standards}
decomposes a pole bound into summands, and its t-structure decomposes
with them because its aisles are closed under direct summands. For one
label, work in an open where its stratum is closed. Open pullback and
closed image are t-exact. That open pullback is conservative on the
summand, so the t-structure is detected there. The smooth normalization
and \eqref{whittaker:stratum-dimension} now give the precise shift:
$q_D^![-(\delta_N+a_D)]$ takes a parameter heart object to the stack
heart, and translating the latter by $[\delta_N]$ gives exactly
$q_D^![-a_D]$. Exponential tensoring preserves this normalization. This
proves the t-exactness of \eqref{eq:W3}; closed transition images give it
in the entire pole model.

For a smooth divisor and its principal affine complement, the asserted
exactness is the ordinary D-module exactness for a closed inclusion and
an affine open inclusion, applied on the ambient charts. The localization
triangle then places $i^!$ in amplitude $[0,1]$. Base change with closed
images and filtered colimits passes these assertions to the pole
presentation. Finally, $!$-restrictions to the finitely many parameter
coincidence strata have bounded t-amplitude and jointly detect objects.
On each such stratum the standard decomposition reduces separation to
that of D-modules on a scheme. This proves separation in the parameter
categories at issue without an assertion about arbitrary ind-scheme
t-structures.
\end{proof}

\subsection{The coefficient functor}

We can now define the continuous Whittaker coefficient functor with all
its required parameter operations:
\begin{equation}\label{whittaker:coefficient-functor}
 R_H:D(L_H^s)\longrightarrow\mathcal W_s^+(\operatorname{Ran}).
\end{equation}
At marked parameters it is $p_{\mathcal Y}^!$ followed by
$\operatorname{Av}_*$ of Proposition~\ref{whittaker:strata-properties}.
The base-change and closed-image compatibilities of that proposition
assemble the marked-point functors into \eqref{whittaker:coefficient-functor}.
We use the analogous functor for $-s,-\chi$, and the same definition at
$s=0$. Proposition~\ref{whittaker:duality} supplies the duality for these
actual presentable Whittaker categories. Thus the adjoints and transposes
used below are defined on all D-modules in the base, while shriek
operations on holonomic standards serve only to establish their stated
cleanliness.
==== END Global-quantum-geometric-Langlands-at-irrational-level//04-whittaker.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//05-deformation.tex ====
\section{Deformation of the Whittaker counit}\label{sec:deformation}

Fix one of the groups $H$ under consideration.  Throughout this section and
Section~\ref{sec:plancherel}, put
\[
 Y=Y_H,\qquad b=\dim Y,\qquad D_s=D(L_H^s).
\]
Write $R_s$ for its enhanced Whittaker coefficient and $R'_s$ for the
opposite coefficient on $D_{-s}$.  We use the duality of
\eqref{eq:W2} to interpret $(R'_s)^\vee$, and define
\[
 J_s=F_H\circ(R'_s)^\vee[2b+2\delta_N].
\]
Thus $J_s$ applies a Whittaker functional to the first slot of the
$!$-diagonal kernel, whose two twists are $(-s,s)$.  Our objective is
\begin{equation}\label{eq:P1}
                         J_sR_s\simeq\operatorname{Id}_{D_s}
                         \qquad(s\notin\mathbb Q).
\end{equation}
The present section constructs the comparison and proves the finite
deformation statements needed to study it.  Section~\ref{sec:plancherel}
proves that its cone vanishes.

Since $F_H$ is an equivalence, it suffices to identify the kernels of
$J_sR_sF_H$ and $F_H$.  The latter is $\Delta_!k_Y$, with twists
$(-s,s)$.  Its first variable is a spectator, while $J_sR_s$ acts on the
second.  On a smooth affine chart $w:W\to Y$ in the first variable,
trivialize the determinant line and pull back the kernel.  Interchange
the factors and denote the resulting object on $Y\times W$ by $F_W$,
so the output variable now comes first.  Smooth base change describes
$F_W$ as the graph $!$-image, with
the smooth dimension conversion.  We construct the comparison on these
charts, naturally under smooth pullback, and then descend it to a map
of kernels.

\subsection{The ordinary-level inputs}\label{deform:classical-inputs}

We use the following established statements at exponent zero.  In each
statement the group is a connected reductive group over $\mathbb C$, the
curve is smooth and complete, the actual root datum is retained, and
the theta characteristic converts the half-density convention to the
ordinary $D$-module convention used here.

\begin{enumerate}
\item The geometric Langlands equivalence identifies $D(Y)$ with
$\operatorname{IndCoh}_{\operatorname{Nilp}}
(\operatorname{LS}_{\check H})$, where local systems are de Rham.
Under this identification the tempered inclusion is
\[
 \Xi:\operatorname{QCoh}(\operatorname{LS}_{\check H})
       \hookrightarrow
       \operatorname{IndCoh}_{\operatorname{Nilp}}
                         (\operatorname{LS}_{\check H}).
\]
The cone of the counit from Ran $!$-Poincar\'e applied to the enhanced
Whittaker coefficient lies in the kernel of the right adjoint to this
inclusion.  We call this kernel the \emph{anti-tempered category}.
The $!$-Poincar\'e images of the vacuum and its fixed-point Satake
translates are, up to one fixed shift and constant line, respectively
$\Xi(\mathcal O)$ and $\Xi$ of the tensor products of the evaluation
representation bundles.  These are the enhanced Whittaker and spectral
localization statements in the geometric Langlands theorem
\cite{GLCI,GLCII,GLCIII,GLCIV,GLCV}: Ran Casselman--Shalika identifies
$!$-Poincar\'e with spectral localization followed by $\Xi$, and the
right adjoint of spectral localization is fully faithful
\cite[Section~18.1.1, Proposition~18.1.6 and Equation~(18.4)]{GLCII}.
Together with \cite[Section~5.1 and Equation~(5.1)]{GLCI}, the
equivalence identifies the normalized counit with that of the
tempered inclusion.  In the
Drinfeld model the coefficient is $!$-pullback followed by Whittaker
projection; its left adjoint on holonomic finite-pole arguments is
$!$-image.

\item Ordinary geometric Satake and categorical Casselman--Shalika
identify the action of a Satake intersection complex on the vacuum
with the corresponding standard Whittaker object.  In the fusion
version these intersection complexes extend over the whole tuple
power of the curve, are universally locally acyclic over that power,
and specialize on diagonals to tensor products.  On a finite Hecke
correspondence they are relatively universally locally acyclic for
both projections; this follows locally from disk frames, with finite
jets sufficient in a fixed bound.  Pulling the correspondence to a
pole model gives the same action by changing the extension at the
marked points.  Both character signs are allowed: Cartan conjugation
over $\mathbb C$ interchanges them, also in families.

\item Lin's comparison between Poincar\'e series and miraculous
duality identifies the two geometric arrows in
\eqref{eq:P5} below, for ordinary holonomic Whittaker arguments.
We use the construction in
\cite[Propositions~2.2.4--2.2.5, Section~2.2.6,
Section~8.1.3 and Proposition~8.1.4]{Lin}, including
the version retaining the marked parameters.

\item The Whittaker characterization of tempering gives a finite
Whittaker test at a point that annihilates the anti-tempered category
\cite{FaergemanRaskin}.  Section~\ref{sec:plancherel} states the
precise finite-level functor and verifies its use in our situation.
\end{enumerate}

The passage from these ordinary-level inputs to irrational exponent uses
algebraically constructible calculations with a variable rank-one local
system.  Our deformation lemmas require mixed Hodge origin and
quasi-unipotent monodromy, so we first encode each Whittaker character by
an algebraic translation on an auxiliary affine line.  These translations
belong to the ordinary geometric diagrams to which the lemmas apply.
Fourier transform then turns them into the required exponential factors.

\subsection{An auxiliary axis and Fourier transform}

For a space $Z$ in a finite diagram, introduce $Z\times\mathbb A^1_u$.
We call a complex there a \emph{pre-complex}.  Fourier transform is
performed only after specialization of the coefficient parameter to a
field.  Our relative algebraic $D$-module Fourier transform is
\[
 \mathsf F(K)=p_{t,*}\bigl(p_u^!K\mathbin{\otimes^!}
                                      \operatorname{Exp}(ut)\bigr).
\]
Here the projections are from $Z\times\mathbb A^1_u\times\mathbb A^1_t$.
This normalization gives $\mathsf F(\delta_{u=0})=\omega_t$;
$\mathsf F[-1]$ is exact for the perverse convention.  The latter is
the Weyl-algebra Fourier theorem, applied relatively on affine charts
and with closed supports when the parameter scheme is singular.

\begin{lemma}[Fourier comparisons]\label{deform:fourier}
On holonomic arguments, the $!$ and $*$ integrals in the axis defining
$\mathsf F$ agree canonically.  Consequently Fourier commutes with
both pullbacks and both direct images for maps independent of the
axis, whenever these holonomic operations are defined.  These
identifications respect adjunctions, the $!$-to-$*$ map, and diagonal
orientation maps pulled back from the diagram without the axis.
\end{lemma}

\begin{proof}
Compactify the $u$-line and put $z=u^{-1}$ at infinity, retaining $t$
as a variable.  Extension across $z=0$ of the Fourier integrand is
clean.  Here is the local algebraic verification, which also explains
why retaining $t$ is necessary.  In a left-module presentation,
sections obtained from the module before tensoring with the
exponential, multiplied by Laurent functions of $z$ and polynomials
of $t$, are killed by powers of $z\partial_t-1$.  They generate the
meromorphic extension.  On a module supported at $z=0$, however,
$z\partial_t$ is locally nilpotent, so $z\partial_t-1$ is invertible.
The meromorphic star extension therefore has no quotient supported at
the boundary.  Duality excludes a boundary submodule of the
shriek extension.  Thus the cokernel and kernel, respectively,
of the clean comparison vanish.  Since both
extensions across the affine Cartier complement are perverse exact,
their comparison is an isomorphism.  The assertion for complexes
follows by perverse truncation.

Use the $*$ integral to commute $!$-pullback and $*$-image with
Fourier, and the canonically equal $!$ integral to commute $*$-pullback
and $!$-image.  Tensoring by the exponential connection is compatible
with either pullback.  Base change and the projection formula give
the resulting identifications.  Their transitivity, or equivalently
their adjunction descriptions, proves compatibility with units and
counits.  The same compactification identifies both constructions of
the $!$-to-$*$ map.  Finally a diagonal orientation class is external
to the axis; its pullback and tensor morphism use exactly these
adjunctions.  This proves compatibility also with orientation maps.
\end{proof}

We specialize Fourier transforms by $!$-pullback at $t=1$.  All
diagrams where this is done carry an algebraic torus action rescaling
$u$ through a nonzero power; the action may also move the other
variables.  Equivariance up to a Kummer factor suffices on frame
torsors.  Fourier gives the inverse action on $t$.  After a finite
cover of $\mathbb G_{m,t}$ and the coordinate change supplied by the
action, the diagram is an external product in $t$.  Base change at
$t=1$ therefore identifies its operations with the operations on the
specialized diagram.  For $*$ operations we use the smooth dimension
conversion in this product.  In particular,
$i_1^!\omega_t=k$, and covariant translation
$\operatorname{Trans}_{\psi}$ of the $u$-axis gives tensoring with
$\operatorname{Exp}(t\psi)$ after Fourier transform.  This is a
geometric base-change argument, rather than an interchange of a
nonproper image and an arbitrary specialization.

Let $p:Z\times\mathbb A^1_u\to Z$.  The projection that discards the
constant part in the axis is
\begin{equation}\label{eq:pre-axis-projection}
 C_!^u(K)=\operatorname{Fib}\bigl(K\longrightarrow p^!p_!K\bigr).
\end{equation}
The functor $p^!$ is fully faithful.  Its image Fourier-transforms to
objects supported at $t=0$, and the localization triangle gives
\[
 \mathsf F C_!^u(K)\simeq j_{t,!}j_t^!\mathsf F(K),
 \qquad j_t:\mathbb G_{m,t}\hookrightarrow\mathbb A^1_t.
\]
Thus $C_!^u(K)=0$ is the precise test for vanishing of Fourier on the
punctured axis.  A comparison whose cone is killed by $C_!^u$ will be
called an isomorphism \emph{modulo constants in $u$}.  Operations
independent of $u$, of either kind, preserve the constant subcategory,
as does translation by a function of the other variables.  Hence
these comparisons can be carried through subsequent operations.
We do not assert that $C_!^u$ commutes with a mixture of $!$ and $*$
operations.  When there are two axes the constructions and their
dilation actions are applied separately.

\subsection{Finite constructible calculations and Mellin ranks}

Put $A_0=\mathbb Q[q,q^{-1}]$, and let $M_q$ be the multiplicative
rank-one $A_0$-local system on $\mathbb G_m$ with monodromy $q$ and
its unit rigidification.  Twist calculations take place on frame
torsors.  A factor $M_q(r)$ means ordinary pullback and ordinary
tensor product; equivalently one can use its dualizing normalization
with $!$-tensor product.  Specialization at
$q=\exp(2\pi i s)$ and Riemann--Hilbert, after complexification,
recover the regular-holonomic calculation with exponent $s$.

We make explicit the finiteness scope used below.  A finite geometric
calculation consists of a finite diagram of bounded algebraically
constructible complexes, built by the six operations, cones, and
finite totalizations from constant complexes, intersection complexes,
and the displayed Kummer factors.  The maps in the diagram are the
geometric maps of these operations.  Over an output chart the maps
are finite type; maps used for $!$-image are schematic or separated
representable.  Smooth unipotent-extension stack projections are
also permitted.  Their two-term vector-stack presentations reduce
the calculation to schematic ones, with the specified shifts and
Tate twists.

For $*$-image we also allow finite-type Artin stacks with unipotent
identity components of stabilizers and finite component groups.
Stratify such a stack into gerbes over algebraic spaces, after
flattening inertia and rigidification.  Smoothly or \'etale locally
these gerbes have the calculation of a classifying stack.  A
connected unipotent group is constructibly acyclic, and pullback
along its torsors is fully faithful; one may further stratify to
use split unipotent coordinates.  Finite component groups contribute
exact invariants in characteristic zero.  Gluing the strata by
localization triangles proves bounded constructibility and the
coefficient formulas.  The same argument, using safe base change,
applies to holonomic $D$-module $*$-images.  It also extends
Lemma~\ref{deform:fourier} to the stack operations just listed.
Locally finite-type non-quasi-compact stacks are used only through
these relative finite constructions over charts.

Every stalk of a finite calculation is a perfect $A_0$-complex, and
coefficient specialization is derived tensor product.  For schemes
these facts follow by finite algebraic triangulations and the same
finite link calculations for pullback and extension.  The reductions
above establish them for the allowed stacks.

We shall also use the mixed enhancement of these diagrams.  Saito's
theory supplies the six operations, pure intersection complexes,
weight inequalities, and quasi-unipotence of nearby cycles
\cite{SaitoMHM}; the enhanced formalism, including stacks, supplies
the coherent diagrams of operations and their comparisons
\cite[Sections~1 and~3]{TubachMHM}.  All weight conclusions will be
tested on finite schematic charts.  In particular, cones and
adjunction maps below are formed in the mixed category, not by
choosing arbitrary maps after forgetting the mixed structure.
Our weight convention is
\[
 K\text{ has weight }\le w
 \quad\Longleftrightarrow\quad
 {}^pH^j(K)\text{ has weights }\le w+j\text{ for every }j.
\]
The dualizing complex of a smooth scheme has weight zero.
Ordinary pullback and $!$-image preserve upper weight bounds, as does
the smooth conversion $[2e](e)$ in relative dimension $e$.

\begin{lemma}[Non-torsion rank test]\label{deform:rank}
Consider a finite geometric diagram as above.  Assume that its
$q$-dependence is introduced by Kummer factors $M_q(r)$ for algebraic
invertible functions $r$, and that the corresponding diagrams with
these factors recorded as graphs have mixed Hodge origin.  At every
output stalk, the dimensions of its specialized cohomologies and
the ranks of the induced maps are constant as $q$ ranges over
complex numbers that are not roots of unity.  In particular, a
finite output complex which vanishes at $q=1$ vanishes at every such
$q$.
\end{lemma}

\begin{proof}
To compare non-torsion specializations, introduce a recording coordinate
$z\in\mathbb G_m$, independent of the Fourier axis.  Replace an
initial factor $M_q(r)$ by the graph
$z=r$ and apply at the end
\begin{equation}\label{eq:P2}
                K\longmapsto p_!\bigl(K\otimes M_q(z)\bigr),
\end{equation}
where $K$ is independent of $q$ and $p$ forgets $z$.
For a complex of mixed Hodge origin, the nearby-cycle monodromies
at $z=0,\infty$ are quasi-unipotent.  Tensoring with $M_q(z)$ for
non-torsion $q$ leaves no eigenvalue $1$ there.  Thus both invariants
and coinvariants vanish, and the extension over these endpoints is
clean.  Consequently the $!$ integral in \eqref{eq:P2} is canonically
the $*$ integral.  This criterion holds relatively, with charts and
spectators retained.  The argument of Lemma~\ref{deform:fourier}
therefore moves \eqref{eq:P2} past either pullback, either direct
image, and their geometric comparison maps.

For external products multiply the recording coordinates and use
$!$-image along multiplication, since $M_q$ is multiplicative.  This
also explains cancellation of ratio twists.  Before
\eqref{eq:P2}, a frame change rescales the recording coordinate by
the corresponding positive or negative power.  Two occurrences of
an inner frame have opposite scalings.  Their product recording
coordinate is unchanged, so the complex descends through that
frame torsor before the inner frame is removed.  In local
trivializations this is simply invariance of the product under
opposite changes of variables by a unit.  The graph construction
and its maps respect these equivariances.  We have therefore obtained
\eqref{eq:P2} as a presentation of the whole specialized diagram for
non-torsion $q$, including its maps.  At $q=1$ and with logarithm
coefficients, we retain the original geometric diagram: the recording
presentation is used only on the non-torsion locus.

Pull to an output stalk.  If $P$ is perverse on $\mathbb G_m$ and
of mixed Hodge origin, Artin vanishing and equality of $!$ and $*$
integration imply that
$R\Gamma(\mathbb G_m,P\otimes M_q)$ is concentrated in degree zero.
Its dimension is its Euler characteristic, independent of $q$:
stratify $P$ and use that tensoring with a rank-one local system
does not alter the Euler characteristic.  The same reasoning
applies to mixed subquotients.  Taking perverse cohomology now
proves constancy in each degree for a complex and exactness on the
perverse subquotients.  It also proves constancy for maps, either
by their images in perverse cohomology or by the long exact
sequence of their cones.  Finally, perfectness of the original
$A_0$-complex and vanishing of its derived fibre at $q=1$ give
vanishing at the generic parameter.
The constant-rank conclusion on the non-torsion locus gives the
last assertion.
\end{proof}

The preceding lemma concerns finite calculations.  A zero colimit at
$q=1$ says that each cohomology class dies under some later transition;
it does not say that any finite-stage complex vanishes.  We therefore
cannot apply the finite rank test directly to an increasing collection
of pole bounds.  The next lemma uses a uniform upper weight bound to
turn vanishing at a finite logarithm length into a transition that
kills the entire completed cohomology module.

\subsection{Logarithm coefficients and passage to colimits}

Write $B_n=\mathbb Q[h]/h^n$.  Substitution $q=\exp(h)$ identifies
$A_0/(q-1)^n$ with $B_n$.  The length-$n$ logarithm variation
$\mathcal L_n$ on $\mathbb G_m$ has this monodromy and the usual
Tate convention for $h$.  Its quotients and ideals fit into exact
sequences of admissible variations
\begin{equation}\label{eq:logarithm-sequence}
 0\longrightarrow\mathcal L_{n-a}(a)
  \xrightarrow{\ h^a\ }\mathcal L_n
  \longrightarrow\mathcal L_a\longrightarrow0
                         \qquad(n\ge a),
\end{equation}
with the first term zero when $n=a$.  One can construct these
variations as symmetric powers of the Kummer extension; their
successive quotients are $\mathbb Q(j)$, $0\le j<n$.
In particular multiplication by $h$ lowers weights by two.

\begin{lemma}[Logarithm colimit test]\label{deform:colimit}
Let $\{K_i\}$ be a filtered system of finite geometric calculations
with outputs on a fixed finite schematic chart.  Suppose that each
calculation and all transition maps are linear in a single input
$M_q(r)$.  Assume their substitutions by $\mathcal L_n(r)$ lift to
mixed diagrams compatibly with \eqref{eq:logarithm-sequence}.
For every stalk $x$ and every degree $d$, suppose that
\[
 \underset{i}{\operatorname{colim}}\,
 H^d\bigl((K_i)_{x,q=1}\bigr)=0
\]
and that the weights of these groups have a common upper bound
$C_{x,d}$, independent of $i$.  Then the same cohomology colimit
vanishes at every non-root-of-unity value of $q$.
\end{lemma}

\begin{proof}
Fix $x,d$ and write $C=C_{x,d}$.  Denote the stalk calculation with
length-$n$ logarithm coefficients by $C_{i,n}$.
Linearity makes \eqref{eq:logarithm-sequence} an exact triangle of
these calculations.  Induction on $n$, the long exact sequence,
and exactness of filtered colimits show that
\[
 \operatorname*{colim}_i H^j(C_{i,n})=0
                  \quad\text{for every }j,n.
\]
The same filtration shows that the weights of $H^d(C_{i,n})$ are
at most $C$ for all $i,n$: each successive term comes from
$H^d(C_{i,1})(a)$ and has weights at most $C-2a$.

For a fixed $i$, perfectness and the Mittag--Leffler property give
\[
 M_i:=\varprojlim_n H^d(C_{i,n})_{\mathbb C}
   =H^d\bigl((K_i)_x\mathbin{\otimes}^{L}_{A_0}
                                             \mathbb C[[h]]\bigr).
\]
This is a finite $\mathbb C[[h]]$-module.  The coefficient exact
sequence identifies $M_i/hM_i$ with the image of $M_i$ in
$H^d(C_{i,1})_{\mathbb C}$; it need not be all of that group.
For precision, these images carry canonical mixed Hodge
structures.  Put
$P_i=(K_i)_x\otimes^L_{A_0}\mathbb C[[h]]$, and choose $e$
such that $h^e$ annihilates the $h$-power torsion of
$H^{d+1}(P_i)$.  The universal
coefficient sequence is
\[
 0\longrightarrow M_i/h^nM_i\longrightarrow
 H^d(C_{i,n})_{\mathbb C}\longrightarrow
 H^{d+1}(P_i)[h^n]\longrightarrow0.
\]
Under transition from length $m$ to length $n$, the last map
on these torsion terms is multiplication by $h^{m-n}$.
For $m\ge n+e$ the image is therefore exactly $M_i/h^nM_i$:
containment follows from that vanishing, and surjectivity
onto it follows from $M_i/h^mM_i\to M_i/h^nM_i$.
Thus the latter modules are stabilized-image subobjects of
the finite-length mixed Hodge structures, with surjective
mixed transition maps.

Deligne's functorial complex splitting of mixed Hodge structures
\cite{DeligneHodge} permits a finite set of generators of this
image to be chosen homogeneous for the weight splitting and
lifted to weight-homogeneous elements of the inverse system.
Indeed, choose a basis in Deligne's bigrading at length one
and lift recursively in the same bigraded components of
the stabilized images.  Strictness gives surjectivity on
each such component.  The resulting elements lift
coherently and remain homogeneous in weight.  Nakayama's
lemma says that their pro lifts generate $M_i$.

Let $v$ be such a pro generator, of weight $w$.  Choose $a$ with
$C-2a<w$.  Since the length-$a$ colimit is zero, a transition
$i\to i'$ kills the length-$a$ component of $v$.  For every
$n\ge a$, exactness of \eqref{eq:logarithm-sequence} puts its
image at length $n$ in the image of
\[
                 H^d(C_{i',n-a})(a)
                       \longrightarrow H^d(C_{i',n}).
\]
The source has weights at most $C-2a$, whereas the image of $v$
is homogeneous of weight $w$.  Strictness forces that image to
vanish.  Its components at lengths below $a$ vanish as well.
Thus the whole pro generator is killed by this transition.
There are finitely many generators, so one further transition
kills $M_i$.

The substitution $q=\exp(h)$ embeds $\mathbb C(q)$ into
$\mathbb C((h))$: a nonzero Laurent polynomial cannot
vanish as a formal exponential polynomial.  After extension
to $\mathbb C((h))$, the preceding conclusion says that the
cohomology map from stage $i$ to a sufficiently late stage is
zero at the generic coefficient parameter.  Lemma~\ref{deform:rank}
applied to this finite transition makes its rank zero at every
non-torsion $q$.  Each stage in each degree is consequently
killed later in the system, which is precisely the required
vanishing of the filtered colimit.
\end{proof}

\begin{lemma}[Stalk detection for the colimits used here]
\label{deform:stalks}
On a finite smooth chart, the filtered colimits of regular-holonomic
complexes occurring above are detected by constructible stalks.
The analogous holonomic Ind-category has a separated perverse
$t$-structure and embeds fully faithfully into all $D$-modules.
\end{lemma}

\begin{proof}
Bounded holonomic complexes on the chart are compact as
$D$-modules.  Hence their Ind-category, and likewise the Ind-category
of regular-holonomic complexes, embeds fully faithfully.  The
perverse $t$-structure extends to a separated $t$-structure there.
For regular-holonomic complexes, the bounded comparison between
the perverse and ordinary constructible $t$-structures passes to
Ind, so the latter is separated as well.

It remains to justify stalk detection in its heart.  The ordinary
algebraically constructible heart is Noetherian.  Indeed, consider
an ascending sequence of subsheaves of one constructible sheaf.
Choose a smooth connected open stratum where the ambient sheaf
is lisse.  The generic ranks stabilize.  A term of that rank is
lisse as a subsystem on a smaller open.  Every later term agrees
with it there: the quotient is a subsheaf of a local system with
zero generic rank, hence is zero.  Induction on the dimension of
the closed complement proves stabilization.  A finite
stratification deals with several top-dimensional components.
Every object of the Ind-heart is a filtered union of its
Noetherian subobjects.  If all its stalks vanish, each such
subobject has zero stalks and hence is zero.  Stalk detection
and separatedness prove the claim for complexes.  We have used
constructible stalks only inside this Ind-category, not as a
test for arbitrary all-$D$-module objects.
\end{proof}

\subsection{Finite Whittaker stages}\label{deform:finite-stages}

We now apply the deformation lemmas to the kernel $F_W$.  Let
$S_I=X^I$, where $I$ is a nonempty finite set, and choose a common
pole bound $m$ at its marks.  Write
\[
 \pi:\mathcal Y_m\longrightarrow Y,\qquad
 \pi_S:\mathcal Y_m\longrightarrow S_I
\]
for the pole model of Section~\ref{sec:whittaker}; when appropriate
$\pi$ retains the marked positions.  Spectator variables are
carried throughout.  For a relevant stratum $\sigma$, write
$f_\sigma,q_\sigma,\psi_\sigma$ for its embedding, unipotent-stack
projection, and character.  The base of $q_\sigma$ is the
corresponding configuration open $S_\sigma$ in a partition
stratum of $S_I$, and set $e_\sigma=\dim(q_\sigma)$.

Lift $F_W$ to its constructible graph formula on frame torsors.
Its only coefficient input is $M_q(r)$, for the relative frame
ratio determined by the chosen spectator trivialization.  The
formula is the smooth pullback of $\Delta_!k_Y$, so its smooth
dimension shifts have the corresponding mixed Tate twists.
Put
\[
             \widehat F=\pi^!(F_W\boxtimes\delta_{u=0}).
\]
For the sign giving coefficient $\operatorname{Exp}(\psi)$, define
\begin{equation}\label{eq:P3}
 \begin{split}
 \widehat T_\sigma&=f_{\sigma,!}
             \operatorname{Trans}_{+\psi_\sigma}q_\sigma^!,\\
 \widehat H_\sigma&=q_{\sigma,*}[-2e_\sigma]
             \operatorname{Trans}_{-\psi_\sigma}f_\sigma^!.
 \end{split}
\end{equation}
In mixed calculations the shift in the second line includes
$(-e_\sigma)$.  Filtered determinants identify the twists along
these operations.  They therefore lift to the same frame
torsors and add no new Kummer input.  The opposite sign replaces
$\psi$ by $-\psi$ everywhere.

\begin{proposition}[Finite projection diagrams]\label{deform:stages}
There is a canonical finite triangle
\[
 \widehat M_{I,m}\longrightarrow\widehat F
                  \longrightarrow\widehat F_\perp
\]
such that $\widehat M_{I,m}$ is a finite extension of images of the
$\widehat T_\sigma$ and every $\widehat H_\sigma$ kills
$\widehat F_\perp$.  After field specialization, Fourier transform,
and $!$-restriction at $t=1$, its first map is the counit of
Whittaker projection of $\pi^!F_W$.  Denote that projection by
$M_{I,m}$.  The augmented stages
\begin{equation}\label{eq:P4}
        \pi_!\widehat M_{I,m}
                    \longrightarrow F_W\boxtimes\delta_{u=0}
\end{equation}
form a geometric diagram under increases of pole bounds and
closed diagonals of marked parameters.  This entire diagram is
linear in the single initial Kummer input.
\end{proposition}

\begin{proof}
The stratum calculation of
Proposition~\ref{whittaker:strata-properties} gives the adjunction
$\widehat T_\sigma\dashv\widehat H_\sigma$ and
$\widehat H_\sigma\widehat T_\sigma=\operatorname{Id}$.
There are only finitely many relevant strata in a pole bound.
Order them from the higher open strata to the lower ones.
For each successive stratum, take the cone of its projection
counit on the current remainder.  Lower shriek extensions have
zero restriction to the higher open strata already treated.
Induction therefore kills all the $\widehat H_\sigma$ and
produces the stated triangle.  Orthogonality to the generated
subcategory characterizes the triangle, so its construction is
independent of the chosen refinement of the order and is
functorial, with coherent comparison maps.

Cartan scaling by $2\rho$ acts on the Pl\"ucker data and scales
each $\psi_\sigma$ by the square (or inverse square in the other
action convention).  Give $u$ the same scaling.  The diagram is
equivariant for this action.  On the twist line pulled from $Y$
the action is trivial; its identification with a stratum line
over $S_\sigma$ carries the induced filtered-determinant action.
Thus the required dilation equivariance persists in frame
presentations containing Kummer factors.  The Fourier
specialization rules apply.

After specialization, $\widehat T_\sigma$ becomes the corresponding
Whittaker shriek extension.  It satisfies character equivariance
on the stratum, and its holonomic extension preserves that
equivariance by the smooth-arrow tests in \eqref{eq:W1}.
Vanishing of all the specialized $\widehat H_\sigma$ is precisely
the stratum criterion for zero Whittaker projection.
Consequently the triangle specializes to the claimed projection
triangle.  The object $M_{I,m}$ is locally holonomic.  It is
supported cleanly on a bounded-zero quasi-compact open containing
the relevant strata: its $!$-restriction to the complement is
zero by the stratum criterion; its $*$-restriction is zero by
holonomic duality and the same criterion for the opposite
character.  This support conclusion is asserted after Fourier
specialization; the pre-complex need not have both kinds of
vanishing support.

Apply $\pi_!$ to the first arrow and use its adjunction with
$\pi^!$ to obtain \eqref{eq:P4}.  These are finite geometric
operations on every output chart.  The map $\pi$ in a pole
bound is relatively finite type; alternatively the first term
is computed by its finite extensions from quasi-compact
relevant strata.

For transitions use the closed inclusions that increase pole
bounds and the closed diagonals $S_I\hookrightarrow S_J$
associated to surjections $J\twoheadrightarrow I$.  Increase
the new pole bound enough to include sums at coincident marks.
The $\widehat F$'s restrict by $!$ to each other.  Closed image
preserves the subcategory generated by the
$\widehat T_\sigma$: for pole inclusions the old relevant strata
are strata at the new bound, and for diagonals coincidences
are already part of the stratification.  The earlier
$\widehat M$ therefore maps to the later $\widehat F$ through
the later $\widehat M$.  Orthogonality gives the factorization
and all its coherences.  Every step is geometric and linear
in the single $M_q(r)$; hence the logarithm lifts and the
recording presentation \eqref{eq:P2} apply to the diagram.
\end{proof}

For clarity, its colimit can be calculated without a compactness
claim about the kernel.  Let $r_{I,m}$ be the $!$-restriction
functors in the limit presentation of the Whittaker category,
and $a_{I,m}$ their left adjoints in the corresponding colimit
presentation.  For an object $A$ in that category,
\[
                A\simeq\operatorname*{colim}_{I,m}
                                  a_{I,m}r_{I,m}A.
\]
Indeed mapping this formula to $B$ gives, by adjunction, the
limit of the mapping complexes between $r_{I,m}A$ and
$r_{I,m}B$, which is the mapping complex in the limit category.
Here $M_{I,m}$ computes $r_{I,m}R_sF_W$.  At $s=0$, applying
the known left adjoint $!$-Poincar\'e to
$a_{I,m}M_{I,m}$ gives $\pi_!M_{I,m}$, again by adjunction.
Both classical functors are continuous, so their tensored
adjunction permits the spectator; the partial $!$-images on
the holonomic arguments are their actual left adjoints there.
Thus the Fourier-specialized colimit of \eqref{eq:P4} at
exponent zero is exactly the ordinary $!$-Poincar\'e counit.

We will apply Lemma~\ref{deform:colimit} to finite calculations
presenting this colimit.  Take the simplicial replacement on a
skeleton of the stage diagram, restrict to finite subdiagrams,
and then to finite skeleta.  These form a filtered system with
the same colimit.  In the skeletal filtration, simplicial
degree $l$ contributes with shift $[l]$.  Taking the augmented
cone gives a filtered system $\{\widehat C_\beta\}$ of finite diagrams,
where $\beta$ records the finite subdiagram and skeletal truncation.
These cones have all the coefficient and mixed-lift properties just
proved.  The shift $[l]$ will be included in the weight estimate in
Section~\ref{sec:plancherel}.

\subsection{Comparison through the compactified diagonal}

We next identify each finite Fourier-specialized stage with
the formula defining $J_s$.  This is a finite comparison, so
Lemma~\ref{deform:rank}, rather than the colimit lemma, suffices.
Recall the proper Vinberg correspondence from
Section~\ref{sec:pseudoidentity},
\[
 b_{\mathrm V}:\bar Y\longrightarrow Y\times Y,
 \qquad J^b=(j_0)_!l_*k_Y,
\]
where $J^b$ carries the ratio twist with exponent $-s$ in the
first slot.  For a locally holonomic Whittaker object $D$ in
a pole bound, form
\[
 \mathscr Z=\mathcal Y_m\times_Y\bar Y,
 \qquad g:\mathscr Z\longrightarrow\mathcal Y_m\times\bar Y,
 \qquad p_2:\mathscr Z\longrightarrow Y.
\]
The last map uses the second factor of the Vinberg
correspondence.  All these formulas retain spectator
variables, and initially also the marked parameter $S_I$.

Let $\mathscr Z^{\mathrm{tr}}$ be the open where the Borel of
the Pl\"ucker data and the opposite parabolic of the Vinberg
data are transverse at the generic point of $X$.  It contains
the data over $j_0$.  It is open because, in the section-matrix
description, transversality is detected by generic
nonvanishing of the Vinberg maps on the Pl\"ucker lines.
The open-orbit description of each Vinberg fibre gives the
test on fundamental lines for the omitted roots; for the
actual root datum take positive multiples of the relevant
weights.  Transversality then implies nonvanishing for every
highest-line map.

\begin{lemma}\label{deform:transverse-map}
The restriction $p_{2,\mathrm{tr}}$ is representable and
separated.  On finite output charts the unrestricted
$p_{2,*}$ has the unipotent-stabilizer finiteness required for
the finite geometric calculus.
\end{lemma}

\begin{proof}
At the generic point, an automorphism of the Pl\"ucker data
lies in the maximal unipotent.  Nonzero Vinberg matrices on
the highest lines force every residual torus scaling to be
trivial.  The remaining kernel of the Vinberg map is in the
radical of the opposite parabolic, whose intersection with
this maximal unipotent is trivial.  Automorphisms over the
output are therefore trivial, proving representability.

For separatedness use the valuative criterion for isomorphisms.
The torus scalings comparing the matrices extend as units:
their compositions on the Pl\"ucker lines are generically
nonzero on the special curve.  At the valuation given by the
generic point of that special curve, choose integral frames
putting both systems of Pl\"ucker vectors in standard form.
This is possible because the reductions are nondegenerate.
The isomorphism over the fraction field is now an element
$n$ of the maximal unipotent.  Choose an actual strictly
dominant highest-weight representation, highest vector $e$,
and an output covector whose composition with the Vinberg
matrix takes unit value on $e$.  The resulting row $\ell$
is integral, $\ell(e)$ is a unit, and $\ell n$ is integral.

These conditions force $n$ to be integral.  To verify this,
write its ordered root coordinates by increasing positive
root height.  Evaluate $\ell n$ on $f_\beta e$, where
$f_\beta$ lowers by a positive root $\beta$.  The coordinate
for $\beta$ occurs with coefficient a nonzero root-string
integer times $\ell(e)$, hence a unit over the complex DVR.
All other nonconstant terms involve coordinates of smaller
height: the total raising height cannot exceed that of
$\beta$, and a different root of the same height cannot
reach the highest weight.  Induction recovers each
coordinate integrally.  The isomorphism consequently
extends at the generic point of the special curve.
Normality and the codimension-one extension criterion for
sections of the affine isomorphism scheme extend it over
the entire curve family; its compatibility with the data
extends as well.  The argument retains marked parameters;
if they are forgotten, their own separatedness supplies
that part of the criterion.

Finally, before restricting to the transverse open, inertia
over an output chart embeds in the inertia of the
Pl\"ucker data, because the Vinberg compactification is
representable.  The latter is unipotent.  The previously
established unipotent-stabilizer reduction therefore
applies to $p_{2,*}$.
\end{proof}

Consider the two geometric arrows
\begin{equation}\label{eq:P5}
\begin{split}
 p_{2,\mathrm{tr},!}
       \bigl(g^*(D\boxtimes J^b)|_{\mathrm{tr}}\bigr)
 &\longrightarrow
 p_{2,\mathrm{tr},*}
       \bigl(g^!(D\boxtimes J^b)|_{\mathrm{tr}}\bigr)[2b]
 \\[-2pt]
 &\longleftarrow
 p_{2,*}g^!(D\boxtimes J^b)[2b].
\end{split}
\end{equation}
The first arrow uses the orientation of the smooth diagonal
of $Y$ and then the $!$-to-$*$ comparison on the transverse
open.  More explicitly, pull the diagonal class to the
fibre product by the canonical base-change map and use the
tensor-to-$!$-pullback morphism; this gives
$g^*\to g^![2b]$.  The tensor morphism also follows from
adjunction and the projection formula.  The second arrow
is restriction to the transverse open.  Mixed versions
include the Tate conversion belonging to $[2b]$.

These are finite calculations on an output chart.  Indeed,
cut $\mathcal Y_m$ to a bounded-zero quasi-compact open
containing the relevant strata.  The clean support of $D$
just proved shows that the star pull in the first term
extends by shriek from this cut, whereas the shriek pulls
in the other terms extend by star.  Thus all three images
are computed from the cut.  Properness of the Vinberg
correspondence now gives the finite calculation above an
output chart.  Enlarging the cut gives the same arrows by
restriction and transitivity of the $!$-to-$*$ comparison.
Proper base change and compatibility of orientation also
make them natural under closed pushes in the stage indices.

After integrating the proper marked parameter $S_I$, the
left term of \eqref{eq:P5} is $\pi_!D$: this is the
shriek calculation of the diagonal using the Vinberg
correspondence.  The right-source term is $J_s(a_{I,m}D)$.
For the latter identification, \eqref{eq:W2} transposes
$a_{I,m}$ to restriction.  Pairing with $D$ makes the
opposite Whittaker-projection counit an isomorphism, so
one can use the unprojected pullback of the $!$-diagonal
kernel.  Vinberg proper pushforward and $!$-base change
then give exactly the right-source expression.  The
$2\delta_N$ in $J_s$ cancels the normalization in
\eqref{eq:W2}; the remaining diagonal shift is $2b$.

\begin{proposition}[Finite Vinberg comparison]\label{deform:vinberg}
For $s\notin\mathbb Q$ and $D=M_{I,m}$, both arrows in
\eqref{eq:P5} are isomorphisms.  They are natural under the
stage transitions, smooth chart changes, and retention of
marked parameters.  The same conclusion holds for the
fixed-point, fixed-label standard Whittaker objects of
\eqref{eq:W3}.
\end{proposition}

\begin{proof}
At exponent zero the two maps are precisely Lin's
geometric comparison
\cite[Propositions~2.2.4--2.2.5, Section~2.2.6,
Section~8.1.3 and Proposition~8.1.4, Theorem~9.0.2]{Lin}.
The restriction map is his vanishing after image from
the nontransverse locus; the first arrow is the diagonal
orientation and transverse $!$-to-$*$ construction of
his transformation.  The theorem says that both are
isomorphisms for ordinary Whittaker arguments, also with
marked parameters retained.  Its diagonal is the
ordinary diagonal; thus its torus-quotient presentation
includes the central-cover push appearing in $J^b$.
Its dimension is the stack dimension $b$, with the
unshifted pair of opposite Whittaker objects used here.

The assertion permits our chart spectator.  To check
this without an assertion about arbitrary tensor
extension, first add extra marked points with zero
pole allowance.  Their pole model is the pullback of
the original one, and the smooth modification tests
preserve the Whittaker condition.  Embed the affine
spectator into affine space.  Smoothly, or \'etale
locally, that space is covered by schemes \'etale over
it which are open in powers of $X$: use coordinate
charts on the curve and translations in affine space.
For such opens the assertion follows from the retained
parameter theorem by open shriek extension to the
whole power of $X$.  Smooth-arrow character tests
preserve equivariance under that extension.  Restrict
back to the open, and then use the closed image of
the spectator.  Proper base change commutes that
closed image with every formula, since the spectator
is carried along.  Closed inclusions of pole bounds
handle changes of bounds in this reduction.

Now express the three terms and two arrows of
\eqref{eq:P5} as finite pre-axis calculations on the
chosen bounded-zero cut, using
$\widehat M_{I,m}$ and the regular constructible
frame-ratio formula for $J^b$.  The latter introduces
a Kummer factor but no exponential.  Multiply its
recording coordinate with that of
$\widehat M_{I,m}$ in \eqref{eq:P2}.  Pullback by $g$
lifts to the frame torsors, and the inner frame
descends because its two recording scalings are
opposite.  Thus all the arrows, including orientation
and the push comparison, have the finite geometric
presentation required by Lemma~\ref{deform:rank}.

Apply $C_!^u$ to the cones of the two arrows.  The
Cartan dilation preserves transversality, so the
Fourier specialization rules identify their field
specializations with the cones in \eqref{eq:P5} on
the punctured Fourier axis.  At $q=1$ these projected
cones vanish by the ordinary-level result and
dilation.  Although the pre-object itself need not
have clean support, after Fourier at every nonzero
character parameter the bounded-zero cut suffices
by the clean support of the Whittaker object.  This
justifies the use of that cut in the test at $q=1$.
Lemma~\ref{deform:rank} makes the projected cones
zero for every non-torsion $q$, in particular for
$q=\exp(2\pi i s)$ with $s\notin\mathbb Q$.
Fourier and restriction at $t=1$ prove the claim.

For a fixed-label standard replace
$\widehat M_{I,m}$ by the formula
\eqref{eq:P3} applied to $\delta_{u=0}$ and its
stratum factor on frame torsors, with its fixed
normalization shift.  The stratum exponentials
extend cleanly as in \eqref{eq:W3}, and the same
finite argument applies.  All maps used were
geometric and compatible with the indicated
base changes, which proves the stated naturality.
\end{proof}

By continuity, the colimit of the stage comparisons
identifies the Fourier-specialized augmentation
\eqref{eq:P4}, for irrational $s$, with a canonical
map
\begin{equation}\label{eq:P6}
                         J_sR_s(F_W)\longrightarrow F_W.
\end{equation}
The projection counits and the Vinberg arrows make
this a map of kernels, compatible with smooth
spectator charts.  At exponent zero the analogous
counit has only an anti-tempered cone.  The next
section applies the finite Whittaker test to that
cone, proves the uniform upper weight bound needed
in Lemma~\ref{deform:colimit}, and thereby proves
that \eqref{eq:P6} is an isomorphism at irrational
exponent.
==== END Global-quantum-geometric-Langlands-at-irrational-level//05-deformation.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//06-plancherel.tex ====
\section{The finite Whittaker test and Plancherel identity}
\label{sec:plancherel}

We complete the deformation argument of Section~\ref{sec:deformation}.
Its remaining input is a uniform upper weight bound for the finite
approximations to the cone of \eqref{eq:P6}.  A finite Whittaker test makes
that cone vanish at exponent zero and detects its vanishing at irrational
exponents.  The weight bound will then allow Lemma~\ref{deform:colimit} to
transfer the vanishing.  Throughout, $Y=Y_H$, $b=\dim Y$, and
$D_s=D(L_H^s)$, as in that section.  All the tests below act in the output
bundle variable; the spectator variable is retained.

\subsection{A test at one point}
\label{planch:finite-test-construction}

Fix $x_0\in X$ and a parameter $z$ at $x_0$.  Set
\[
 K=L^+H,\qquad K_1=\ker(K\longrightarrow H),\qquad
 I^-=\operatorname{ev}^{-1}(B^-),
\]
and define
\[
 K^d=\operatorname{Ad}_{z^{-\rho}}K,\qquad
 I^d=\operatorname{Ad}_{z^{-\rho}}I^-,\qquad
 K^d_1=\operatorname{Ad}_{z^{-\rho}}K_1.
\]
These conjugations take place in the adjoint group.  They do not require
$\rho$ to belong to the cocharacter lattice of $H$.
Write $Y(J)$ for the stack with level $J$ at $x_0$, so that $Y(K)=Y$.
One may construct these stacks by formal frames and disk gluing, or by the
corresponding smooth parahoric group schemes.  The level changes used here
are finite-type morphisms: after passage to a sufficiently deep congruence
subgroup, they are the usual finite-dimensional homogeneous-space
fibrations of jet-frame torsors.  In particular, with
\[
 V=Y(K^d_1),
\]
the map $V\to Y(K^d)$ is an $H$-torsor and $V/B^-=Y(I^d)$.

We define a functor $\mathcal T_s$ in three steps.  First pull by $!$ from
$Y$ to $Y(I^-)$.  Next apply the star intertwiner
\[
 p_{2,*}p_1^!,\qquad
 Y(I^-)\xleftarrow{\ p_1\ }Y(I^-\cap I^d)
             \xrightarrow{\ p_2\ }Y(I^d).
\]
Finally apply finite $(N,\chi_0)$ star averaging, where $\chi_0$ is a
nondegenerate residual character.  Either sign of $\chi_0$ is allowed.
The last operation is the kernel transform for the open incidence
\begin{equation}
 \{([vn],v):v\in V,\ n\in N\}
 \ \subset\ V/B^-\mathbin{\times}_{Y(K^d)}V:
 \qquad ([vn],v)\longmapsto \chi_0(n).
 \label{eq:P7}
\end{equation}
It uses $!$-pull from the first projection, tensor with the exponential of
the displayed function, and star image to $V$.  Fixed normalization shifts
do not affect any vanishing or conservativity assertion in this
subsection.

Here are the twisting lines in this construction.  On $Y(K^d)$ use the
determinant line defined by the lattice
$\operatorname{Ad}_{z^{-\rho}}\mathfrak h_{\mathcal O}$, where
$\mathcal O=\mathbb C[[z]]$.  On $Y(I^d)$ correct its pullback by the
inverse relative determinant of this lattice and
$\mathfrak h_{\mathcal O}$.  On frames the relative determinant is a fixed
vector line whose torus character is
\begin{equation}
 \nu=\sum_{\alpha>0}2\operatorname{ht}(\alpha)\alpha
       =\operatorname{Kil}(\rho,-).
 \label{planch:det-character}
\end{equation}
The formula follows by counting the shifted root-lattice lines.  Extend
this character to the Iwahori through its torus quotient and form the
associated line.  Its inverse is the required correction.  After taking
exponent $s$, the resulting line agrees with the input twisting line on
the intersection correspondence.  Characters of that intersection are
determined on its torus, so this check proves the agreement globally.
On $V$ the frame trivializes the correction line, with its specified
character law; use the same trivialization at $vn$ in \eqref{eq:P7}.
Thus $\mathcal T_s$ is a well-defined functor from $D_s$ to twisted
D-modules on $V$.

\begin{proposition}[Finite Whittaker test]
\label{planch:finite-test}
The functor $\mathcal T_0$ kills the anti-tempered subcategory of $D(Y)$.
If $s\notin\mathbb Q$, the functor $\mathcal T_s$ is conservative on all
of $D_s$.  Both assertions remain valid after tensoring with a spectator
D-module category.
\end{proposition}

\begin{proof}
For the assertion at zero we use the local Whittaker description of
anti-tempering.  In the notation of
\cite[Lemmas~4.2.0.1 and~4.3.0.1, Equation~(4.3.1)]{FaergemanRaskin},
the kernel of nondegenerate Whittaker $!$-averaging on positive-loop
invariants is exactly the anti-tempered subcategory.  That averaging
factors through the translated first adolescent Whittaker level, whose
inclusion in the full Whittaker category is fully faithful.  Its
finite-level formula is forgetful passage to the opposite Iwahori,
star intertwinement, and residual finite Whittaker averaging.  The last
$!$-averaging agrees with star averaging on these inputs up to a fixed
shift; see also \cite[Section~3.1, Lemma~3.1.4]{BogdanovaWhit}.
The resulting finite-level diagram on bundle stacks is exactly the
construction above.  The use of adjoint conjugation for a group on which
$\rho$ is not a cocharacter is allowed in this theorem; see
\cite[Remark~4.4.0.1]{FaergemanRaskin}.

For completeness, the categorical mechanism explains why this assertion
applies to all D-modules on the level tower.  An object of positive-loop
invariants determines an equivariant functor out of the Grassmannian
module category.  The Whittaker Grassmannian category is a tempered
spherical module.  Consequently the functor associated to an
anti-tempered object becomes zero after taking Whittaker coinvariants:
the tempered projector acts as the identity on that module, whereas
precomposition with the projector kills the given functor.  Whittaker
invariants and coinvariants agree in the character conventions under
consideration.  The finite test lands in the first adolescent level,
where the fully faithful inclusion detects zero.  Its operations commute
with the equivariant functor: this is clear for restriction and finite
averaging, and for the intertwiner follows from the finite unipotent
root-piece quotients of the two compact subgroups by their intersection.
This also proves the assertion with spectators, without a holonomicity
restriction.

We next prove conservativity for irrational $s$.  The first $!$-pull is
conservative because the level map is smooth and surjective.  To treat
the intertwiner, choose a minimal gallery from the alcove of $I^-$ to
that of $I^d$.  The open Bruhat convolution for the associated reduced
word identifies its product correspondence with the intersection-level
correspondence.  The star intertwiner therefore factors into the
intertwiners for adjacent alcoves.  Twists factor as well: at each
Iwahori use its own Lie-lattice determinant and the inverse relative
determinant with respect to $\mathfrak h_{\mathcal O}$.  Transitivity of
relative determinant gives the comparisons at consecutive intersections
and recovers the prescribed endpoint lines.

For adjacent alcoves, work smoothly locally over the wall-level stack.
The correspondence is the open of distinct flags in a product of two
projective lines for the rank-one reductive quotient.  The wall-lattice
determinant comes from the base; the remaining twists are the torus lines
on the two flag varieties.  Compose the star open kernel with the
reverse shriek open kernel.  To calculate this composition, fix the
first and last flags.  If they differ, the middle flag ranges over a
twice-punctured projective line, with star extension at one puncture and
shriek extension at the other.  The regular-singular rank-one
connection occurring in the tensor product has the same cohomology as
its restriction to the link of the shriek puncture; the relative
cohomology specified by shriek extension at that puncture is
therefore zero.  In coordinates with punctures $a\ne c$, this is the
fibre of the restriction map
\[
 R\Gamma(\mathbb P^1\setminus\{a,c\},\mathcal L)
       \longrightarrow R\Gamma(\operatorname{link}_c,\mathcal L),
\]
which is an isomorphism for every Kummer system $\mathcal L$, including
the trivial one.  The $!$-tensor calculation has this description by
Verdier duality and the projection formula.  If the flags coincide,
the ratio twists cancel.  The
projection formula for star image and $!$-tensor reduces the calculation
to the constant kernel on an affine line, whose integral is a nonzero
constant complex up to shift.  The calculation is smooth along the
diagonal.  The composite is thus a nonzero constant multiple, up to
shift, of the identity kernel.  Each adjacent intertwiner is
conservative.  The alcove argument uses affine Weyl representatives,
which exist for the actual root datum; it does not assert the existence
of $z^{-\rho}$ in $LH$.

It remains to treat finite Whittaker averaging.  Smoothly locally over
$Y(K^d)$, its source is the flag variety $H/B^-$ and its output is the
frame space $H$.  The flag twist has torus character $-s\nu$.
Compose its star open kernel with the reverse star open kernel, using
opposite pairing twists and the negative exponential.  By the
projection formula this is computed on the two open incidences.
The composite is equivariant for the left $H$-action on the pair of
flags.  On a Bruhat stratum indexed by $w\ne1$, the stabilizer torus
carries the Kummer character
\[
 s(w\nu-\nu)
\]
up to the harmless overall sign convention.  The weight $\nu$ is regular
on the root system.  Thus $w\nu-\nu$ is nonzero, and irrationality of
$s$ makes the character nonintegral on some cocharacter of the
stabilizer torus.  The Kummer inertia test forces the equivariant
category on that stratum to vanish.

For the diagonal calculation we use $!$-restriction, rather than
ordinary star pull.  If $j_1,j_2$ are the two open incidences in the
space of a pair of flags and an intermediate frame, the $!$-tensor
projection formula and $!$-base change identify the tensor of their
star extensions with star image from their open intersection.
For the projection $q$ forgetting the intermediate frame and the
diagonal inclusion $i_\Delta$, schematic $!$-base change gives
$i_\Delta^!q_*=q_{\Delta,*}(i'_\Delta)^!$, with $i'_\Delta$ the pulled-back
diagonal.  Thus no interchange of nonproper star image with ordinary
star restriction is being used.
On the diagonal the two $n$-coordinates agree.  Both the exponential
factors and the twisting factors cancel.  After fixing a flag and a
representative of it, the remaining integral is the de Rham integral of
the dualizing object on $N\times B^-$.  Its cohomology is nonzero and
varies constantly along the diagonal.  The composite therefore acts as
the identity tensored with a nonzero complex of vector spaces, and is
conservative.  These are ordinary D-module kernel calculations on fixed
finite-dimensional correspondences.  They apply to all D-modules and
commute with spectator factors.  This proves the proposition.
\end{proof}

\subsection{Preparing both character parameters}
\label{planch:two-axes}

Let $\{\widehat C_\beta\}$ be the finite augmented cone diagrams
constructed at the end of Section~\ref{deform:finite-stages}.  They
already carry the pre-axis $u$.  Write $\widehat{\mathcal T}$ for the
pre-axis version of the finite Whittaker test: adjoin
$\delta_{u'=0}$, perform its pullback and intertwiner, and replace the
exponential in \eqref{eq:P7} by translation of the second axis $u'$ by
$\chi_0(n)$.  Fourier specialization at its character parameter $1$
recovers $\mathcal T_s$.

The new finite diagram has the dilation property in both axes.  For
$u'$ use the right torus action on frames: it conjugates $n$, scales
$\chi_0$ and $u'$ by the same square, and leaves the first flag $[vn]$
fixed.  The correction line transforms by its character law.  The
matched determinant lines define maps between frame torsors, and
$\widehat{\mathcal T}$ transports the single coefficient input along
these maps.  At logarithm length $a$ this is one $\mathcal L_a$ input;
no tensor product over $\mathbb Q$ of separate logarithm variations is
taken.  A change of trivialization merely rescales the one frame-ratio
coordinate.  Thus these operations introduce no new monodromy input.

Define the tested cones
\begin{equation}\label{planch:tested-cones}
 \widehat K_\beta
   =C_!^u C_!^{u'}\widehat{\mathcal T}(\widehat C_\beta).
\end{equation}
On fixed output and spectator charts, including their frame torsors,
these are finite geometric calculations linear in the original Kummer
input, with the compatible logarithm lifts of
Section~\ref{sec:deformation}.  To apply Lemma~\ref{deform:colimit} we
must prove two properties of this family: its cohomology colimit is zero
at $q=1$, and its weights in each stalk degree have an upper bound
independent of $\beta$.  The next lemma proves the first property; the
rest of the section establishes the bound.

\begin{lemma}
\label{planch:zero-colimit}
For every stalk $\xi$ on a test chart and every degree $j$,
\[
 \operatorname*{colim}_\beta
       H^j\bigl((\widehat K_\beta)_{\xi,q=1}\bigr)=0.
\]
\end{lemma}

\begin{proof}
With both Fourier parameters nonzero, the exponent-zero Poincar\'e
counit has anti-tempered cone by the enhanced classical Whittaker
theorem recalled in Section~\ref{sec:deformation}.  Proposition~
\ref{planch:finite-test} kills this cone.  Continuity gives the same
statement with spectators.  Dilation spreads vanishing at unit
parameters to the product of the two punctured Fourier axes.

This argument also commutes with the colimit used here.  On smooth
charts the calculations lie in Ind-holonomic complexes.  The dilation
identifications hold stage by stage, so the colimit vanishes after
pullback to the dilation covers and hence before that pullback.
Shriek extension from the punctured axes is computed in the same
Ind-holonomic category and preserves this zero colimit.  The Fourier
description of $C_!^u C_!^{u'}$ now gives the assertion for the
tested cones $\widehat K_\beta$.
\end{proof}

We will need one boundedness property of the fixed finite test.  At
$q=1$, its pre-axis version increases upper weights and upper ordinary
cohomological degrees by fixed constants on fixed test charts.  To see
the weight assertion, factor the intertwiner into adjacent-alcove
steps and compactify each open by its proper wall-flag correspondence.
Smoothly locally over the input flag, the star extension is external
product with a fixed fibre kernel.  Its weight cost is therefore fixed;
the remaining push is proper.  The same reasoning applies to
\eqref{eq:P7}: compactify the incidence inside
$V/B^-\times_{Y(K^d)}V$, retaining $u'$.  To see the product over the
input, choose a local lift $g(f)$ of its flag $f$.  The incidence
$[vn]=f$ writes $v=g(f)b n^{-1}$, with $b\in B^-$.  In the coordinate
$g(f)^{-1}v$, its open fibre is the fixed cell $B^-N$ and its character
graph $u'=\chi_0(n)$ is independent of $f$.  An arbitrary input is
therefore external-tensored with this fixed star-extended graph kernel.
The remaining projection, from the compactification to $V$ with the
axis retained, is proper.  Ordinary cohomological dimension gives the
corresponding fixed degree bounds.  Above a fixed output chart these
level and flag correspondences require only fixed quasi-compact input
charts.  Finally each pre-axis projection uses a one-dimensional
projection with fixed shifts.  None of these costs depends on the
number of marks, the pole bound, the Satake labels, or the skeletal
degree.

The uniform estimate has three parts.  The two opposite Hecke kernels
have cancelling weights, removing dependence on the Satake labels.
A spectral calculation then gives an upper cohomological degree
independent of those labels and the number of marked points.
Configuration integration lowers this degree by the collision
codimension, and the simplicial shifts lower it further.  Only finitely
many collision codimensions and simplicial degrees can therefore
contribute to any fixed stalk degree; their weight bounds will suffice.

\subsection{Satake decomposition of the finite terms}
\label{planch:satake-decomposition}

All weights from now until the final application are computed at
$q=1$.  On frame-torsor charts the monodromy input is then constant in
mixed Hodge modules.  We use the mixed-complex convention of
Section~\ref{sec:deformation}, and work on fixed final test charts.
Smooth-chart dimension shifts are fixed constants there.

Take a term $\pi_!\widehat M_{I,m}$ of \eqref{eq:P4}, retaining its
pre-axis.  Stratify $X^I$ by coincidence patterns.  The filtration by
$!$-restrictions expresses it by star extensions from these strata.
Let $S_\alpha$ have $n$ distinct points.  Its closure is a diagonal copy
of $X^n$, denoted $\overline S_\alpha$, and
$j_\alpha:S_\alpha\hookrightarrow\overline S_\alpha$ is the open
configuration inclusion.  We first retain the marked parameters and
use the entire stack $Y$ as spectator, with
$F_Y=\Delta_!k_Y$ in output--spectator order.  All comparisons are
relative and will subsequently
be pulled to a smooth spectator chart $W$.  Comparisons made modulo
$u$-constants are used only after applying the pre-axis projections.

For a tuple $\lambda=(\lambda_1,\ldots,\lambda_n)$ of dominant
coweights within the pole bound, let $\mathsf S_\lambda$ be the relative
Satake operator.  Our convention on bounded holonomic diagrams is
\[
 \mathsf S_\lambda(D)
   =p_{2,!}\bigl(p_1^*D\otimes^* K_\lambda\bigr),
\]
where $p_1$ is the input projection and the Hecke projections are proper
on the support.  The labels are chosen so that application to the
vacuum gives the corresponding Whittaker standard objects.  Extend
the kernels over $\overline S_\alpha=X^n$ by fusion.  On the bundle
Hecke correspondence set
\[
 d_\lambda=\sum_i\langle2\rho_{\rm rt},\lambda_i\rangle,
 \qquad
 K_\lambda=\operatorname{IC}_\lambda[-(b+n)].
\]
There are no half Tate twists.  Thus $K_\lambda$ has complex weight
$d_\lambda$, and its relative dual over the smooth base of dimension
$b+n$ is
\begin{equation}
 K_\lambda^r=K_\lambda(d_\lambda),\qquad
 \operatorname{wt}(K_\lambda^r)=-d_\lambda.
 \label{planch:dual-satake}
\end{equation}
The right adjoint $\mathsf S_\lambda^r$ uses the exchanged paths and
this relative dual.  The Tate twist disappears after forgetting the
mixed structure, but its weight in \eqref{planch:dual-satake} is
essential below.

We recall the exchange properties being used.  Fusion Satake kernels
are universally locally acyclic (ULA) relative to either Hecke
projection.  The relative duality formula identifies
$K\otimes^*p_i^*(-)$ with relative Hom from the dual kernel into
$p_i^!(-)$; kernels and their relative duals commute with base change.
Together with properness, this gives the displayed adjoints, exchange
with $!$-pull, and exchange with star image under parameter changes.
These are comparisons in mixed complexes.  They may be checked after
forgetting to constructible complexes, because the comparison maps,
duals, and adjunctions are themselves mixed.  At distinct fixed points
the Hecke correspondences are locally products over both projections,
using finite jet frames.  Thus the same formulas hold for the
holonomic exponential expressions after Fourier specialization.
We use the usual fusion and Casselman--Shalika theorems with these
relative normalizations; see \cite{BravermanGaitsgory,GLCII}.

The pole-model Hecke correspondence is the base change of the bundle
Hecke correspondence along either projection: modifying the extension
at the marked points transports the Pl\"ucker data.  Its kernels are
the star pullbacks of $K_\lambda$.  All calculations in a fixed bound
use finite bounds.  In particular, a modification bounded by
$\lambda$ takes the vacuum pole bound to the bound allowing exactly
those dominant pole orders on the highest-line maps.  This is the
Schubert-bound estimate on representation lattices; at collisions
the pole orders add.  It also permits the adjoint calculation back
to the vacuum using just the indicated forward bound.

Over $S_\alpha$, denote the vacuum operations of \eqref{eq:P3} by
$\widehat T_0$ and $\widehat H_0$.  Their vacuum stratum consists of
true $N_0$-bundles, with no poles or defects.  Put
\[
 \widehat Q_\lambda=\mathsf S_\lambda\widehat T_0,
 \qquad
 \widehat H^\lambda=\widehat H_0\mathsf S_\lambda^r.
\]
The following arrows are the evaluation maps of these adjunctions and
the map induced by the Whittaker projection counit:
\begin{equation}
 \widehat M\longleftarrow
 \bigoplus_\lambda\widehat Q_\lambda
                    \widehat H^\lambda(\widehat M)
 \longrightarrow
 \bigoplus_\lambda\widehat Q_\lambda
                    \widehat H^\lambda(\widehat F).
 \label{eq:P8}
\end{equation}
Here $\widehat M$ and $\widehat F$ are $!$-restricted to $S_\alpha$,
and the finite sum ranges over its labels in the pole bound.  Both
arrows are isomorphisms modulo $u$-constants.  Indeed Fourier commutes
with the Hecke operations, whose kernels are external to the axis.
The diagram has dilation because Hecke transport commutes with the
Cartan action on Pl\"ucker data.  At a nonzero character parameter,
Casselman--Shalika identifies the vacuum translates with the standard
objects of the Whittaker summands, while $\widehat M$ is the Whittaker
projection.  This gives the assertion.  The verification may be made
on $!$-fibres over $S_\alpha$, also with spectators: properness and
the local product description give the Hecke base changes, the
Whittaker decomposition gives the projection base changes, and the
vacuum bound is constant over the parameter base.  Notice that the
restricted $\widehat F$ uses $\pi^!$ over $S_\alpha$, so includes the
base dualizing factor.

Define two objects without marked parameters:
\begin{equation}
 U_0=\pi_!\widehat T_0(\delta_{u=0}),\qquad
 U'_0=\widehat H_0
       \bigl(\pi^!(F_Y\boxtimes\delta_{u=0})\bigr).
 \label{eq:P9}
\end{equation}
They live respectively on the output and spectator copies of $Y$,
each with a pre-axis.  Equivalently one may use a fixed marked point
with pole bound zero in this definition.  Write $*_{!,u}$ for external
product followed by $!$-image under addition of the two pre-axes.

\begin{lemma}[Separation of the two bundle variables]
\label{planch:separation}
After \eqref{eq:P8}, the term indexed by $\lambda$, star extended to
$\overline S_\alpha$ and pushed by $\pi_!$ with marked parameters
retained, is, modulo $u$-constants,
\begin{equation}
 \mathsf S_\lambda^{(1)}\mathsf S_\lambda^{\prime(2)}
 \left((U_0*_{!,u}U'_0)\boxtimes
                    (j_\alpha)_*\omega_{S_\alpha}\right).
 \label{eq:P10}
\end{equation}
Both Hecke operations are over $\overline S_\alpha$.  The second is
the transpose-path version of $\mathsf S_\lambda^r$: its path is
reversed back to the forward direction, but its kernel is
$K_\lambda^r$, rather than $K_\lambda$.  Formula~\eqref{eq:P10}
includes the base dualizing factors and has no additional shifts
depending on $I$, $\alpha$, or $\lambda$.
\end{lemma}

\begin{proof}
First work over $S_\alpha$.  In
$\widehat H^\lambda(\widehat F)$, move
$\mathsf S_\lambda^r$ from the pole model to the bundle stack.
The Cartesian Hecke squares, properness, and the ULA exchange formulas
justify this move through $\pi^!$.  Only the vacuum restriction for
$\widehat H_0$ is needed, so the forward Schubert bound just described
suffices.  Acting on the first slot of the $!$-diagonal kernel is the
same as acting on the second slot by the transpose-path operator
$\mathsf S'_\lambda$.  In both expressions, star pull followed by
the diagonal $!$-image produces the same $!$-image along the Hecke
path, with the same relative dual kernel.  Now commute
$\widehat H_0$ with that spectator operation.  Its $!$-restrictions,
vacuum translation, and star image satisfy the same exchange
formulas.  This gives
$\mathsf S'_\lambda(U'_0\boxtimes\omega_{S_\alpha})$.

Next apply $\widehat T_0$ and the forward pole-model Hecke operator.
The vacuum operation commutes with the spectator Hecke calculation
by star-pull and shriek calculus.  Before either Hecke operation we
therefore have an external object with parameter factor
$\omega_{S_\alpha}$.  The ULA fusion kernels exchange with
$(j_\alpha)_*$, including the closed embeddings of pole bounds.
Consequently extension to $\overline S_\alpha$ changes only that
external factor, to $(j_\alpha)_*\omega_{S_\alpha}$.

Finally move $\pi_!$, with parameters retained, through the
pole-model Hecke operation.  The Cartesian square and the exchange
between star pull and $!$-image turn it into bundle-stack Hecke.
Before this operation, the vacuum translation formula is exactly
\[
 \widehat T_0(E)
   =\widehat T_0(\delta_0)*_{!,u}E.
\]
It gives the addition convolution in \eqref{eq:P10}.  Every step
is relative to the spectator and hence commutes with smooth pull to
$W$.  The closed diagonal embedding
$\overline S_\alpha\hookrightarrow X^I$ may be included afterwards
and introduces no further shifts.
\end{proof}

\subsection{Uniform bounds before configuration integration}
\label{planch:uniform-bounds}

Replace the two vacuum inputs by
\[
 \bar U_0=C_!^uU_0,\qquad \bar U'_0=C_!^uU'_0.
\]
This does not change \eqref{eq:P10} modulo $u$-constants: addition
convolution with a $u$-constant is again $u$-constant, by a change
of coordinates and the $!$-projection formula.  The benefit is that
both bar objects have a global upper weight bound, even though $Y$
is not quasi-compact.

\begin{lemma}
\label{planch:vacuum-weight}
The objects $\bar U_0$ and $\bar U'_0$ have fixed upper complex-weight
bounds on their entire bundle stacks.  In \eqref{eq:P10}, omit
$(j_\alpha)_*\omega_{S_\alpha}$ and instead use the unshifted constant
complex $k_{\overline S_\alpha}$, of weight zero and ordinary degree
zero.  With the bar inputs, the resulting
object has an upper complex-weight bound independent of
$I,\alpha,\lambda$ and the pole bound.
\end{lemma}

\begin{proof}
For $U_0$, the vacuum $!$-image factors through shriek extension from
a fixed quasi-compact open of $Y$.  Its source is the finite-type
stack of $N_0$-bundles with fixed Cartan bundle, and its image is
contained in such an open.  The mixed object on that open is bounded;
shriek extension preserves upper weights.  Applying $C_!^u$ has a
fixed weight cost.

For $\bar U'_0$, use Fourier only to identify support, not to assert
preservation of Hodge weights.  At nonzero Fourier parameter,
$\mathsf F U'_0$ is the $!$-Poincar\'e object of the opposite vacuum,
up to a fixed shift and line.  This is Lin's vacuum comparison
\cite[Theorem~9.0.2 and Section~8.1]{Lin}, applied to the coefficient
of the $!$-diagonal kernel as in \eqref{eq:W2} and \eqref{eq:P5}.
More explicitly that coefficient is
\[
 q_{0,*}\bigl(\operatorname{Exp}(-\psi_0)\otimes^!
                  f_0^!\pi^!F_Y\bigr)[-2\delta_N].
\]
Dilation changes the character parameter while leaving the bundle
variable fixed.  Hence over the punctured Fourier axis this object
extends by shriek from one fixed quasi-compact bundle open.
The Fourier transform of $\bar U'_0$ is its shriek extension across
the zero parameter.  Inverse Fourier, which operates only in the
axis variable, preserves the same bundle-open support property.
The shriek-extension comparison exists in mixed complexes and is an
isomorphism after forgetting the mixed structure.  It is therefore
an isomorphism in mixed complexes.  On the fixed quasi-compact open,
$\bar U'_0$ has an upper weight bound, which extends to all of $Y$.

Apply the two Hecke operations to these bar inputs.  Star pull,
star tensor, and proper image give upper weight bounds, and the
kernel contributions $d_\lambda$ and $-d_\lambda$ cancel by
\eqref{planch:dual-satake}.  Addition convolution uses $!$-image and
has a fixed cost.  Thus the upper bound is independent of every
index.  Restoring the parameter factor in \eqref{eq:P10} simply
star-tensors this object with its pullback, by projection formula.
\end{proof}

Restoring $(j_\alpha)_*\omega_{S_\alpha}$ will introduce weight bounds
that depend on collision codimension, which is unbounded as the number
of points grows.  We now prove a uniform degree bound that will limit
the codimensions contributing to a fixed cohomology degree.  This is
the one place where the full classical geometric Langlands theorem
enters the estimate.

\begin{lemma}
\label{planch:satake-degree}
Fix smooth output charts on the two copies of $Y$.  With the bar inputs
and the parameter factor $k_{\overline S_\alpha}$ used in
Lemma~\ref{planch:vacuum-weight}, the star restriction of
\eqref{eq:P10} to every fibre of $\overline S_\alpha$ has a uniform
upper ordinary cohomological degree.  The bound includes all tuples
of points, all further collisions, and all Satake labels.
\end{lemma}

\begin{proof}
On a parameter fibre, fusion identifies Hecke with fixed-point
Satake operators for tensor products of the given representations.
We first bound the Fourier transform away from zero.  For
$\bar U'_0$, the value at character parameter $1$ on a spectator
chart $W$ is a kernel-valued Hom of the form
\begin{equation}
 \operatorname{Hom}_{D(Y)}
       \bigl(\mathsf S P\operatorname{vac}_!,F_W\bigr),
 \qquad P\operatorname{vac}_!=\pi_!T_0(k).
 \label{eq:P11}
\end{equation}
Here $T_0$ is the exponential-specialized vacuum of the corresponding
sign, $\mathsf S$ is the usual fixed-point Satake operator, and Hom
means application in the first kernel variable, with values in
$D(W)$.  Formula~\eqref{eq:P11} follows from \eqref{eq:P9} by the
transpose calculation in Lemma~\ref{planch:separation}, the smooth
vacuum adjunction with its specified shift, and $!$-image adjunction.
These adjunctions tensor with the spectator.  Their fixed-point
kernel formulas hold for every representation tensor produced by
fusion, with no additional shifts depending on its labels.  Tate
factors can now be forgotten, since we are proving a degree bound.

The object $F_W$ is compact in $D(Y)\otimes D(W)$.  Indeed it is the
$!$-image of a holonomic object on the finite-type affine scheme $W$
along its schematic graph map.  The partial $!$-image has genuine
adjunction against arbitrary D-modules, and the corresponding
$!$-pullback is continuous.  Compactness follows.  The tensor-product
theorem for the ordinary D-module category of $Y$ then expresses
$F_W$ as a retract of a finite construction by extensions and shifts
from exterior products of compact objects.

Let $\mathcal L=\operatorname{LS}_{\check H}$ be the derived stack
of de Rham local systems.  Under classical geometric Langlands,
those first-factor compact objects become coherent compacts in
$\operatorname{IndCoh}_{\operatorname{Nilp}}(\mathcal L)$, and
\[
 \mathsf S P\operatorname{vac}_!
 \quad\longleftrightarrow\quad
 \Xi(\mathcal V),
\]
where $\mathcal V$ is the evaluation representation vector bundle,
or the tensor product of such bundles at the specified points.
The global shift and line are fixed.  This normalization is the
vacuum normalization of \cite[Sections~1.3--1.4]{GLCI}, with Hecke
compatibility from its Theorem~1.2.4 and Proposition~1.7.2, and the full equivalence from
\cite{GLCV}.  More explicitly,
$\mathbb L^L_{H,\operatorname{temp}}(\mathcal O)
=P\operatorname{vac}_!$ and
$\Psi\mathbb L_H=\mathbb L_{H,\operatorname{coarse}}$ by
\cite[Corollary~1.6.5]{GLCI}.  Uniqueness of left adjoints after the
equivalence gives
$\mathbb L_H\mathbb L^L_{H,\operatorname{temp}}=\Xi$.
It follows that the vacuum maps to $\Xi(\mathcal O)$ and its
Satake translates to $\Xi(\mathcal V)$.
The chosen theta characteristic transports the ordinary-D Hecke
conventions; any resulting constant label lines do not change
cohomological degrees.

For a fixed coherent compact $C$, adjunction computes the mapping
complex as
\begin{equation}
 \operatorname{Hom}\bigl(\Xi(\mathcal V),C\bigr)
   =R\Gamma\bigl(\mathcal L,
                    \mathcal V^\vee\otimes^*\Psi(C)\bigr).
 \label{planch:spectral-bound}
\end{equation}
The right adjoint $\Psi$ takes $C$ to a bounded coherent
quasi-coherent complex.  Tensoring with a vector bundle is t-exact,
irrespective of its rank, highest weight, or evaluation points.
The fixed derived QCA stack $\mathcal L$ has finite cohomological
dimension for quasi-coherent global sections in characteristic
zero; we use the QCA finiteness theorem
\cite{DrinfeldGaitsgoryFiniteness}.  Thus
\eqref{planch:spectral-bound} has an upper bound depending on $C$
and $\mathcal L$ only.  Finite extensions, shifts, retracts, and
the fixed compact $D(W)$ factors yield a uniform perverse-D upper
bound in \eqref{eq:P11}.

For $\bar U_0$, Lin's vacuum comparison identifies its Fourier
transform on the puncture with the corresponding transform for
$\bar U'_0$ of the opposite sign, up to a fixed normalization.
Applying fixed-point Satake therefore gives the same bound.  The two
bundle charts may be chosen independently.

The fixed-fibre Satake operations commute with shriek extension
across the Fourier origin, by their kernel formulas.  On the
puncture, dilation makes the preceding bound uniform in the
parameter, with only a fixed dimension shift after the dilation
cover.  The principal affine-open extension $j_{t,!}$ is t-exact
on holonomic complexes.  Fourier is t-exact with its fixed
normalization shift.  The same perverse bounds thus hold on the
whole pre-axis.  The chart dimensions are fixed, so they also give
uniform upper ordinary degrees.  Finally the addition convolution
$*_{!,u}$ has fixed relative dimension and commutes with the
fixed-fibre Hecke calculations.  It changes this upper bound by a
fixed constant.  This proves the lemma.
\end{proof}

\subsection{Configuration strata and the uniform weight estimate}
\label{planch:configuration-estimate}

We now restore $(j_\alpha)_*\omega_{S_\alpha}$ and integrate the
marked parameters.  Stratify $\overline S_\alpha=X^n$ again by
coincidence patterns.  If a stratum $S'$ has $n'$ distinct points,
write
\[
 c'=n-n'\geq0.
\]
In this second filtration we use star restrictions to strata and
shriek extensions for integration.

The star restriction of $(j_\alpha)_*\omega_{S_\alpha}$ to $S'$
has ordinary upper degree
\begin{equation}
 c'-2n.
 \label{planch:link-degree}
\end{equation}
Indeed local coordinates on the curve identify a normal slice with
a product of central braid arrangements of total rank $c'$.  The
local links have the cohomology of their arrangement complements,
whose cohomology vanishes above $c'$.  The dualizing normalization
$[2n](n)$ gives \eqref{planch:link-degree}.
The same restriction has a complex-weight upper bound depending
only on $c'$.  This follows from the mixed arrangement description,
which is \'etale local in curve coordinates.  For fixed $c'$ there
are only finitely many nontrivial collision types: every block of
size $r>1$ consumes $r-1$ units of codimension, and singleton blocks
contribute only smooth factors.  Finally $[2n](n)$ is neutral for
the complex-weight convention.  Thus neither the number of singleton
blocks nor $n$ changes this weight bound.

Let $E$ be the fixed upper degree from
Lemma~\ref{planch:satake-degree}, with the fixed costs of convolution
included.  Integrating the piece over $S'$ by $!$ adds at most
$2n'$ to ordinary cohomological degree, while preserving upper
complex weights.  Its contribution on the test charts therefore
has upper degree
\begin{equation}
 E+c'-2n+2n'=E-c',
 \label{planch:codimension-decay}
\end{equation}
and an upper complex-weight bound depending only on $c'$.
The closed embedding of $\overline S_\alpha$ back into $X^I$ adds
no further cost.

The comparison with the original terms is made after the pre-axis
projections.  Every arrow used in
Lemmas~\ref{planch:separation}--\ref{planch:satake-degree} is
defined in mixed complexes.  Where its being an isomorphism was
checked by Fourier and projection, that check was on underlying
field complexes and therefore also detects the mixed isomorphism.
The fixed finite test and both pre-axis projections add only the
fixed costs established in Section~\ref{planch:two-axes}.
The augmentation-target term has fixed bounds as well.

\begin{proposition}[Uniform weight estimate]
\label{planch:weight-estimate}
Fix a test chart, a stalk $\xi$, and a degree $j$.  The mixed Hodge
structures $H^j((\widehat K_\beta)_{\xi,q=1})$ have a common upper
weight bound, independent of $\beta$ and hence of all pole,
configuration, label, and skeletal indices.
\end{proposition}

\begin{proof}
For skeletal degree $l$, the shift $[l]$ in the finite approximation
lowers \eqref{planch:codimension-decay} to
\[
 E'-c'-l,
\]
where $E'$ includes all fixed shifts.  The complex-weight bound
depends only on $(c',l)$.  A term can contribute to degree $j$
only if
\[
 c'+l\leq E'-j.
\]
Since $c',l$ are nonnegative integers, only finitely many such
pairs occur.  Take the maximum of their degree-$j$ weight bounds.
The finite-stage filtrations use finite sums and extensions, and
an upper weight bound on cohomology in a fixed degree passes to
extensions.  The resulting bound is uniform in every remaining
index, as asserted.
\end{proof}

\subsection{Completion of the identity}

\begin{theorem}[Plancherel identity]
\label{planch:identity}
For $s\notin\mathbb Q$, the natural transformation \eqref{eq:P6}
is an isomorphism.  Consequently
\[
 J_sR_s\simeq\operatorname{Id}_{D_s},
\]
as asserted in \eqref{eq:P1}.
\end{theorem}

\begin{proof}
On every test chart the family $\{\widehat K_\beta\}$ of
\eqref{planch:tested-cones} is linear in the single frame-ratio Kummer
input and has compatible logarithm lifts.  It therefore satisfies
the finite-stage logarithm and rank hypotheses of
Lemma~\ref{deform:colimit}.  Its cohomology colimit vanishes at
$q=1$ by Lemma~\ref{planch:zero-colimit}, and
Proposition~\ref{planch:weight-estimate} supplies the required
degreewise uniform upper weights.  The colimit lemma and stalk
detection in Lemma~\ref{deform:stalks} therefore give
$\operatorname*{colim}_\beta\widehat K_\beta=0$ at every parameter
that is not a root of unity, in particular at the monodromy of every
irrational $s$.

Fourier transform in the two pre-axes and take $!$-fibres at the
two character parameters $1$.  The pre-axis diagrams are regular
holonomic; Fourier and specialization are computed in the holonomic
Ind-categories on charts.  Away from zero character parameters, the pre-axis
projections do not change them.  We obtain a zero cone after
applying $\mathcal T_s$ to \eqref{eq:P6}.  Its conservativity in
Proposition~\ref{planch:finite-test} proves \eqref{eq:P6} on each
spectator chart.  All arrows used were natural stack morphisms,
so this is the required isomorphism of kernels.  The
pseudo-identity equivalence of Section~\ref{sec:pseudoidentity}
then gives \eqref{eq:P1}.
\end{proof}
==== END Global-quantum-geometric-Langlands-at-irrational-level//06-plancherel.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//07-loop-categories.tex ====
\section{Loop categories over moving divisors}\label{sec:loop}

The local comparison will involve marked points that move and collide.  We
therefore construct its categories and pairings over a parameter scheme,
including their infinitesimal transport.  The essential output is the
pairing \eqref{eq:L3} and its continuous, parameter-linear right adjoints.
The construction also identifies the strong equivariance used in that
pairing with the Harish--Chandra models used for the representation-theoretic
calculations in the next sections.

\subsection{Divisors, determinant lines, and parameter directions}

Let $S$ be a smooth affine scheme with a finite tuple of maps $x_l:S\to X$.
Additional parameters are allowed.  Complete along the \emph{sum} of their
graphs.  Repeating a graph changes its multiplicity but not the resulting
formal completion or meromorphic algebra.  After an \'etale localization,
separate disjoint clusters and choose a curve coordinate $z$ and a frame
of the theta characteristic.  On one cluster the divisor has equation
\[
                 u=P(z)=\prod_l(z-x_l).
\]
Only the formal neighborhood of the marked points in this coordinate
chart is used.  Write $LG$ and $H=L^+G$ for the resulting relative loop and
arc groups.  The groups are twisted by $P_0$; the displayed coordinates
include a trivialization of $P_0$.  In particular, the intrinsic unipotent
group is $N_0$.

The relative determinant extension is the multiplicative line
\[
     \mathcal L(g)=\det(g\mathfrak g_{\mathcal O}:\mathfrak g_{\mathcal O}),
\]
with first lattice over second lattice and the adjoint action on
$\mathfrak g$.  The lattice indices vanish, so this is an ordinary
multiplicative line.  It is the local line in the global Hecke
correspondence.  Its Lie cocycle is
\[
                       (x,y)\longmapsto
                       \operatorname{Res}\operatorname{Kil}(dx,y).
\]
Indeed, the Tate trace cocycle is the trace of the two off-diagonal
blocks relative to the regular lattice; on modes $az^m,bz^{-m}$, with
$m>0$, it is $m\operatorname{Tr}(ab)$.  This proves the formula at distinct
points.  Both sides are polynomial identities in each bounded collection
of Laurent coefficients, and the relative residue formula gives the same
identity on the collision locus.

Put $\ell=\widehat{\mathfrak g}_{\mathcal K}$ for this Tate Lie algebra
with its central extension.  Its central element acts by
\[
                         -s=\frac{k_a}{2h^\vee}.
\]
Thus the basic-form level is $k_a$ and the shifted level is $a$.  The
sign comes from the frame convention: a frame for a modification glues
from the inside to the outside, and sections transform by its inverse
power.  Left convolution consequently uses the twist $\mathcal L^{-s}$.

For a module $M$, a loop $g$, and a determinant lift $\widetilde g$, the
family of conjugated modules, paired with this convolution twist, is
given in left-module normalization by
\begin{equation}\label{eq:L1}
 \begin{split}
 x&\longmapsto\operatorname{act}_M(\operatorname{Ad}_{g^{-1}}x),\\
 \partial_v&\longmapsto\partial_v+
       \operatorname{act}_M(\widetilde g^{-1}\partial_v\widetilde g).
 \end{split}
\end{equation}
Here $\partial_v$ differentiates in the parameter direction while holding
the curve coordinate $z$ fixed.  Changing the determinant lift adds
$-s$ times its logarithmic derivative, so this family has twist $s$.
The second line is essential: conjugation alone does not specify the
connection when the marked points move.  We construct the categorical
meaning of \eqref{eq:L1} below.

We use crystals of presentable categories in the sense of
\cite[Section~2 and Appendix~B, especially Sections~B.3.8--B.3.11
and~B.14]{GLCII}.  By
\cite[Theorem~2.6.3]{Gaitsgory1Affine}, a crystal over $S$ is equivalently
a $D(S)$-module category.  Its underlying quasi-coherent category is
\[
 \mathcal E^0=\operatorname{QCoh}(S)
                   \otimes_{D(S)}\mathcal E(S),
 \qquad b:\mathcal E(S)\longrightarrow\mathcal E^0.
\]
The functor in this tensor product is the left-crystal forgetful functor:
it sends $F^{\langle S\rangle}$ to its underlying quasi-coherent module.
After applying $b$, equivariant operations on smooth schemes over $S$
are relative operations.  This follows either from crystal base change
or from the relative differential-operator algebra.

We will check crystal compatibilities on the formal diagonal groupoid
of $S$.  This means all finite infinitesimal neighborhoods of every
multiple diagonal, with their coherent pullback maps.  Since $S$ is
smooth, the projections may be computed using cofinal neighborhoods
finite and flat over $S$.  The use of this groupoid will permit moving
congruence subgroups; it does not declare a fixed finite jet group to
be horizontally constant.

\subsection{Relative convolution and equivariance}

Let
\[
                     \mathcal A=D_{*,\mathcal L^{-s}}(LG).
\]
The subscript specifies star convolution and its pro convention.
In the ind direction we use closed direct images.  In the pro direction
we use systems compatible under star direct image.  The pro projections
have smooth unipotent affine-space fibers; their star-compatible limit
is also the colimit under normalized smooth pulls, the left adjoints of
those direct images.

Concretely, on a bounded part of $LG$ stable on the right under $H$, take
quotients by sufficiently deep congruence subgroups $K\subset H$.  Left
quotients give the same category: bounded conjugation sends sufficiently
deep congruences into any specified congruence, so both presentations
have common higher-jet refinements.  A bounded part may be the preimage
of a closed bound in the affine Grassmannian.  Multiplication is star
direct image with the multiplicative line.  All products are relative:
for the finite-type schemes used in these presentations,
\[
                 D(U)\otimes_{D(S)}D(V)
                       \simeq D(U\times_S V).
\]
This is the D-module tensor-product formula, followed by base change
along the base diagonal; it also holds with the indicated twists.
For any finite string of products and any output jet level, sufficiently
deep input levels make the calculation finite-dimensional.  Composition
of star direct images then proves associativity, including its higher
coherences.  The delta system at the identity is the unit.

We need an equivariance calculation over $S$, where jet groups can vary
with the divisor.  A \emph{strong action} in the following statement
means an action of the convolution category of D-modules.  This is
stronger than an algebraic action on the underlying category.

\begin{proposition}[Relative invariant and coinvariant calculation]
\label{loop:relative-equivariance}
Let $J\to S$ be one of the smooth affine finite jet groups of the arc,
congruence, or unipotent groups used here.  Put
$A_J=D_*(J)$ and let $\mathbf1_J=D(S)$ be its module given by relative
de Rham direct image.  For every $D(S)$-linear strong $J$-category
$\mathcal E$, the norm calculation identifies
$\mathcal E_J$ with $\mathcal E^J$, naturally also in additional
coefficient categories.  Under this duality:
\begin{enumerate}
\item geometric relative tensor products over $A_J$ are D-modules on
contracted products, with star integration in a free group variable;
\item $\operatorname{End}_{A_J}(\mathbf1_J)=D(B_SJ)$, with ! tensor;
\item the unbalanced evaluation
\[
 \operatorname{ev}_J:
 {}_{A_J}\mathbf1_J\otimes_{D(S)}\mathbf1_J{}_{A_J}
       \longrightarrow A_J
\]
has a continuous right adjoint, linear for the left and right
$A_J$-actions.  Both actions on $\mathbf1_J$ are given by the
de Rham augmentation.
\end{enumerate}
The assertions hold for the pro-smooth groups in use, whose kernels
over a finite jet stage are pro-unipotent, and commute with change of
smooth parameter base.  Character versions use the corresponding
exponential and its inverse on opposite sides.
\end{proposition}

\begin{proof}
For a constant finite-dimensional group, this is strong-group-action
duality; see the distinction between group schemes and loop indgroups
in \cite[Section~1.5.4]{RaskinW}.  We reduce the relative case to this
one without assuming that $J$ is constant.

Locally on $S$, embed $J$ in a constant general linear group $D_0$ over
$S$, with schematic quotient $W=D_0/J$.  The embeddings needed here
can be constructed as follows.  An arc jet group acts faithfully on a
finite free module over the finite divisor algebra.  Restriction of
scalars embeds it in a general linear group over $S$.  The intermediate
quotient parametrizes module structures over the divisor algebra for which it is
locally free of the specified rank over the divisor algebra.  This is
an open locus in an affine representation scheme.  The remaining
quotient is affine, since the original reductive group has affine
quotient in a general linear group and restriction of scalars along a
finite flat map preserves affines.  Split unipotent jet groups admit
upper-triangular embeddings using their root and congruence filtrations.
The same construction applies after restricting to a base chart.

The relative D-module category of a finite-type schematic map $U\to S$
is dualizable over $D(S)$: its coevaluation is the relative diagonal
kernel $\Delta_*\omega_U$, and evaluation is ! tensor followed by star
integration.  The triangle identities are base change on the relative
square.  Transposing an action using this duality gives its !-coaction
and hence the usual descent diagram for strong invariants.

Consider
\[
            \mathcal B=A_{D_0}\otimes_{A_J}\mathcal E.
\]
We first show, by a free-action computation, that
\[
       \mathcal B\simeq
          \bigl(D(D_0)\otimes_{D(S)}\mathcal E\bigr)^J.
\]
Here $J$ acts by right translation on $D_0$, and inversion identifies
the required right and left conventions.  If $m\mapsto m^h$ denotes
the transposed coaction, the comparison is star averaging: on a
trivialized right torsor $U\to W$, its ! family at a section $x$ is
the star integral in $h$ of
$K(xh)\otimes^!m^h$.  The coaction identity makes this an equivariant
family.  Shearing two group variables shows that convolution on $K$
is the action on $m$, so the comparison is balanced; the same
calculation for strings supplies the coherent balance.

For a trivial torsor, pulling the family to a section splits its
descent bar.  The extra degeneracy identifies its invariants with
coefficients on $W$, and the ! fiber of the displayed averaging
formula is exactly the original action.  Thus the comparison is an
equivalence on a trivial torsor.  A smooth torsor has sections \'etale
locally.  Restriction to such a cover commutes with the descent limits,
by the relative dualizability just proved, and with the tensor bars.
Descent for $D(W)$-module categories follows from the 1-affineness of
$W_{\mathrm{dR}}$.  It therefore gives the claimed equivalence over
$W$, and over $S$.  The construction is $A_{D_0}$-linear.

Associativity of relative tensors, constant-group duality, and this
free-action formula now give
\[
        \mathcal E_J\simeq\mathcal B_{D_0}
                    \simeq\mathcal B^{D_0}
                    \simeq\mathcal E^J.
\]
For the last equivalence, interchange the two commuting descent
diagrams and apply translation descent on $D_0$.  These computations
also work with coefficient categories and hence identify the claimed
module duality, rather than just its value on one category.

On the regular category, translation descent identifies the two
trivial modules with $D(S)$.  Evaluation is consequently the smooth
pull kernel on $J$, up to an invertible normalization from the base.
Its right adjoint is star integration with the corresponding smooth
shift.  The projection formula and the shear map identify the
canonical adjunction-linearity maps for the two convolutions with
isomorphisms.  The contracted-product formula and the description of
the endomorphism category follow from the same descent calculation.

Finally, for a pro-unipotent kernel, normalized smooth pull of the
unit is a convolution idempotent at every finite quotient.  Its image
is strong equivariance, and its continuous right adjoint is star
averaging.  The delta unit in the pro-convolution convention is the
filtered colimit of these idempotents for decreasing kernels: this
is checked by projection to every finite level.  This reduces the
assertions to a finite quotient.  In particular $D(B_SJ)$ may be
computed at any finite stage with pro-unipotent kernel.  All maps in
the argument are relative pull, star integration, or tensor bars,
so the construction commutes with the asserted base changes.
\end{proof}

This proposition is used only for finite-dimensional or compact
pro-groups.  Coinvariants for the indgroup $LN$ will instead be formed
as a colimit over compact subgroups.

\subsection{Renormalized Harish--Chandra categories}

We next describe the underlying quasi-coherent categories, after applying
$b$.  Every compact-generation assertion in this subsection is made
in these categories.  It does not assert generation by horizontal
objects; in particular, multipoint Kazhdan--Lusztig categories need
not be ULA-generated \cite[Remark~B.14.25]{GLCII}.  Put
$R=\mathcal O(S)$.  Modules for $\ell$ are smooth: every vector is
annihilated by a sufficiently deep Lie lattice.  Let $J$ be an arc
group, a congruence subgroup, or a pro-unipotent compact open of the
kind above, with Lie algebra $j$ and a specified splitting of the
extension.  A character $\chi_J$ is incorporated by the augmentation
\[
       K\longmapsto p_{J,*}
                      \bigl(K\otimes^!\operatorname{Exp}(\chi_J)\bigr).
\]
The corresponding Harish--Chandra condition identifies the differential
of the $J$-action with the $\ell$-action \emph{minus} $\chi_J$.
This sign agrees with \eqref{eq:L1}.

Define the renormalized category for $(\ell,J)$ as the Ind category
of the idempotent-complete stable category generated by
\begin{equation}\label{eq:L2}
                       \operatorname{Ind}_{j}^{\ell}(V),
\end{equation}
where $V$ ranges over algebraic $J$-representations that are vector
bundles over $S$.  Impose the specified central value and character
correction.  The mapping complexes on these generators are the
ordinary derived Harish--Chandra mapping complexes.  By induction
adjunction, they are the derived algebraic $J$-maps from $V$ to the
restricted target.  Renormalized pro-group representations are defined
in the same way, omitting Lie induction.

\begin{proposition}\label{loop:hc-renormalization}
The underlying quasi-coherent categories have the following properties.
\begin{enumerate}
\item The comparison with the ordinary derived category of smooth
Harish--Chandra modules is t-exact and is an equivalence on complexes
bounded below in cohomological degrees.
\item The category for $(\ell,J)$ is monadic over renormalized
$J$-representations, with the PBW induction monad.  Its t-structure is
created by the forgetful functor.
\item The compact-generator mapping calculations commute with smooth
affine base change, restriction to smooth closed collision and graph
charts, residue fields, and the finite flat scalar
extensions used on infinitesimal diagonals.  In the last case the
generators are perfect relative to the scalar extension.
\item Restriction to a smaller compact open with smooth finite-dimensional
quotient preserves compact objects.  Between pro-unipotent opens it
is fully faithful and has a continuous right adjoint.
\end{enumerate}
\end{proposition}

\begin{proof}
The pointwise renormalization and bounded comparison are developed in
\cite[Sections~9.11--9.13 and~10.8]{RaskinHomological}; we give the
relative calculation needed here.  First consider finite group stages.
The quotient presentations in
the proof of Proposition~\ref{loop:relative-equivariance} give the
resolution property.  Indeed, quasi-coherent representations are
generated by vector bundles: use generation on the quasi-affine
homogeneous space, and then finite algebraic subrepresentations of
its sections.  General linear group invariants are exact in
characteristic zero, and a quasi-affine scheme of finite type has
bounded cohomological dimension.  On the smooth base, bounded coherent
equivariant complexes are therefore perfect and compact.  Ordinary
equivariant descent supplies the derived representation category at
these stages.  Restricted perfects generate after the stated base
changes, either by the same quotient presentation or by testing after
the affine pushforward.

Every algebraic representation of a pro-group is exhausted by the
subobjects fixed by deeper kernels, as follows from its coaction.
Mappings between bounded coherent objects are computed by the group
cobar resolution.  If the target complexes are bounded below in a
fixed range, maps from a finite vector bundle commute with filtered
colimits: degree by degree the cobar computation uses only a finite
part of the resulting first-quadrant total complex.  This proves the
bounded-below comparison for renormalized group representations.
More explicitly, the hearts are generated by the finite-stage
representations; bounded complexes lift using the mapping calculation;
and bounded-below complexes are obtained as colimits of their good
upper truncations.  The same calculation shows that the renormalized
pro-category is the colimit of the finite-stage renormalized
categories.  The finite-stage transition functors need not be fully
faithful.

For later use, in the pro-unipotent case the compact representation
category is thickly generated over $R$ by the trivial representation.
At a finite stage, the successive vector-group quotients act by
commuting locally nilpotent operators.  On a coherent representation
they supply a finite filtration by coherent subobjects with trivial
successive action; repeat with the quotient group and resolve over
the smooth affine base.  This proves the assertion at each stage and
hence in the pro-category.

The Lie induction monad is the PBW monad.  The quotient of the Tate
Lie algebra by an open lattice is a union of finite locally free
summands, so PBW makes induction exact.  The adjoint $J$-action is
algebraic: sufficiently high congruences fix each specified bounded
Laurent expression and each vector fixed at a sufficiently deep
level.  The monad thus acts on the representation categories just
constructed.  Its extension to their Ind categories agrees on
bounded-below objects with the ordinary induction monad.  Its free
modules have exactly the mapping complexes prescribed in
\eqref{eq:L2}, proving the monadic assertion.  The ordinary induction
and forgetful functors are exact, the latter is conservative and
commutes with colimits, and all monad iterates agree in the
bounded-below comparison.  This proves the first two assertions for
Harish--Chandra modules.

For base change, compute the generator mappings with the same relative
cobar complexes and PBW modules.  Their terms are $R$-flat, and the
asserted changes have finite Tor dimension; for a regular closed
chart this is given by its finite Koszul resolution.
The bounded-below calculation therefore commutes with the derived
tensor product.  A finite flat algebra from an infinitesimal diagonal
is treated as an algebra object in these categories.  Its compact
generators are obtained from perfect scalar extensions, not by
declaring every coherent module over that possibly singular algebra
compact.  The free-generator cobar calculation gives the same
base-change and bounded-below assertions there.

Finally, restriction of \eqref{eq:L2} to a smaller open is resolved
by the finite-dimensional relative Chevalley complex for the
difference of the Lie lattices.  Retain the smaller group's
equivariance throughout.  This finite resolution proves compact
preservation.  The opens may first be refined through a common
deeper congruence with smooth quotients.  If both opens are
pro-unipotent and the smaller is normal, the assertion of full
faithfulness is the equality of the relative Chevalley calculation
and algebraic group cohomology for the finite-dimensional unipotent
quotient.  For a nonnormal inclusion use a still smaller normal
subgroup.

On bounded-below modules the essential image consists of those
objects whose corrected Lie action integrates to the larger
pro-unipotent group.  Such integration is unique and is preserved by
extensions.  Smoothness reduces this assertion, vector by vector,
to local nilpotence for a finite-dimensional unipotent quotient.
The fully faithful assertion then extends from the compact
presentations to the Ind categories.  Compact preservation implies
that the right adjoint to restriction is continuous.
\end{proof}

We define the renormalized Kac--Moody category $\mathcal M^0$ as the
colimit of these categories over decreasing untwisted pro-unipotent
congruence levels.  The maps into this category will be called
forgetful maps.  They are distinct from passage to the ordinary
abelian module model.  In particular, Proposition~\ref{loop:hc-renormalization}
does not identify the two unbounded categories; this restriction is
essential for Whittaker objects, as explained in
\cite[Sections~1.5.7--1.5.9]{RaskinW}.

\subsection{Infinitesimal transport and the strong loop action}

The preceding categories must now be supplied with their connection
in $S$.  The issue is that a fixed congruence level moves when a graph
moves.  The full completed arc algebra does not have this defect.

\begin{proposition}\label{loop:crystal}
The categories just constructed and their restriction and induction
functors descend along the formal diagonal groupoid of $S$.
Their sections are $D(S)$-linear categories, with underlying categories
as above.  The construction commutes with smooth parameter change and
with separation into disjoint clusters, including after restriction
to a collision locus.
\end{proposition}

\begin{proof}
Two infinitesimally close lifts of the marked graphs give cofinal
divisor ideals.  Explicitly, if their monic equations $P,Q$ agree
modulo an ideal $I$ with $I^r=0$, the binomial expansion gives
\[
               Q^{n+r-1}\in(P^n),\qquad
               P^{n+r-1}\in(Q^n).
\]
Every surviving term in the first expansion contains at least $n$
factors of $P$, and the other inclusion is symmetric.  They therefore
define the same completions,
full arc groups, and Laurent algebras.  The determinant extension
has the same canonical identification.

Choose a congruence deep enough for one lift to lie in the two
levels being compared.  It may be chosen normal in the full arc
group.  The two relative quotients are smooth: the quotient polynomial
in each such divisibility is monic, so the finite difference of the
corresponding lattices is locally free even over the nonreduced base.
Restriction to this common deeper level is fully faithful and
compact-preserving by Proposition~\ref{loop:hc-renormalization}, also
after the finite flat scalar extensions under consideration.

The two essential images there agree.  To see this on bounded-below
modules, filter by the powers of the nilpotent base ideal.  On every
successive quotient the two integrability conditions coincide.
Integrability is stable under extensions, so it coincides on the
whole module.  Equivalently, a change with coefficients in a
nilpotent ideal integrates by the finite exponential.  Restriction
is t-exact, by the group-representation comparison followed by the
induction monad.  Thus the same argument applies to the scalar
extensions of the compact generators.  Their images are compact in
the common-depth category; when they lie in the other fully faithful
subcategory they remain compact there.  The two compact subcategories
therefore agree, and so do their Ind-completions.

These identifications are independent of the common depth, because
passing to a deeper one is fully faithful.  On multiple
infinitesimal diagonals choose a single common deeper level for
all lifts.  All identifications are then restrictions of the same
full Laurent algebra, and their compositions agree there.  This
supplies the higher cocycle compatibilities, not just transport on
first-order tangent vectors.  Smooth descent and the formal
diagonal description of crystals give the claim.  The base-change
mapping calculations in Proposition~\ref{loop:hc-renormalization}
identify the underlying categories on a new smooth parameter
scheme with its corresponding Harish--Chandra categories.

The derivative at fixed $z$ preserves the full regular arc
algebra.  It need not preserve a chosen finite-depth subgroup;
the comparison through deeper levels is exactly what makes that
derivative act on the crystal.  Disjoint-cluster products follow
from products of the completed algebras.  The cofinal-ideal
argument applies equally at collisions.
\end{proof}

Write $\mathcal M$ for the resulting crystal category of Kac--Moody
modules.  We next verify that its geometric loop action is the strong
action whose invariants were used above.

\begin{proposition}\label{loop:strong-action}
Formula~\eqref{eq:L1}, paired by ! tensor and star integration, defines
a unital associative $\mathcal A$-action on $\mathcal M$ in
$D(S)$-module categories.  For each compact open $J$ used above, its
strong $(J,\chi_J)$-equivariant category is the renormalized
Harish--Chandra category for $(\ell,J)$ with differential
$\operatorname{act}_\ell-\chi_J$.  These identifications respect
the forgetful maps into $\mathcal M$.
\end{proposition}

\begin{proof}
Suppose an input belongs to a congruence level $K$, and take a bounded
right-stable part $Z$ of the loop group.  Project a convolution kernel
to $Z/K$.  On this quotient it is paired with $M^g$ from
\eqref{eq:L1}.  A sufficiently deep output level $K'$ satisfies
$g^{-1}K'g\subset K$ uniformly on the bound; thus $M^g$ has values in
the smaller Harish--Chandra category, and the pairing and integration
are ordinary relative operations with that category as coefficients.

One can compute $M^g$ after ! pull to any smooth affine plot of $Z/K$.
Choose a local loop lift and determinant frame.  At $b$, the first
line of \eqref{eq:L1} is conjugation restricted to $K'$.  The
positive-depth splittings agree with conjugation after increasing
the depth; this follows directly from the relative determinant
calculation.  Across an infinitesimal diagonal, two lifts differ by
a formal group element.  Exponentiating its Lie action, including
the prescribed scalar on the center, gives the comparison.  Each
exponential terminates on a nilpotent test ideal.  Changing the loop
lift by $K$ has precisely the Harish--Chandra equivariance already
imposed.  First-order differentiation yields the second line of
\eqref{eq:L1}, while products of formal elements give all higher
compatibilities.  By Proposition~\ref{loop:crystal} these are
calculations in crystal categories.

Here plots need not be smooth over a Schubert bound.  We record why
! plots still suffice.  For a finite-dimensional scheme $V$, a proper
map $\widetilde V\to V$ that is an isomorphism off a closed subset
$D$ gives a descent square with $D$ and
$\widetilde D=\widetilde V\times_VD$.  The open/closed triangles
prove this: open objects extended by star and closed objects have
the indicated pullbacks, and their extension maps agree by proper
direct-image/!-pull adjunction and Kashiwara's theorem.  These
identities remain valid with coefficient categories.  Resolution
of singularities and induction on the dimension of the closed
complement therefore reduce descent to smooth plots.  To compare
with any other smooth plot, resolve the closure of its lift over
the open locus and apply the same square; plots supported entirely
over the closed complement are handled by the induction.  This
also proves descent for morphisms and coherent data.  Twists and
restriction to affine plots do not change the argument.

The kernel formula is compatible with changes of level by the
projection formula.  For multiplication choose an intermediate
depth so that its left action on the second bounded right
quotient is trivial.  Project both inputs to these finite
quotients.  Pulling $M^g$ along the multiplication map gives the
iterated formula: on product plots conjugation composes, the
Maurer--Cartan term obeys its product rule, and the determinant
line multiplies.  Relative base change now identifies convolution
followed by the action with the two successive actions.
Coefficient functors commute with this integration because they
are continuous and parameter-linear.  The calculation with
strings proves coherence, and the delta kernel acts identically.

It remains to identify strong equivariance, including its derived
structure.  Take a deeper open $K_1$ normal in $J$, on which
$\chi_J$ vanishes.  The pro-unipotent forgetful identifications
reduce the calculation to the finite-dimensional smooth quotient
$Q=J/K_1$.  On a fixed input level, averaging and the bar maps
factor through such quotients by the kernel formula; further
output levels give the same calculation.

First replace $\ell$ by $j$ and work at a finite group stage of
$J$.  Harish--Chandra modules for $(j,K_1)$ are equivalent to
weakly right $J$-equivariant relative D-modules on $K_1\backslash J$.
Here weak equivariance is algebraic equivariance, without
identifying its differential with the D-module differential.
Indeed, translation descent of the underlying quasi-coherent
module identifies it with a representation of the stabilizer
$K_1$.  Differential-operator induction adds the monad
\[
                  U(j)\otimes_{U(\operatorname{Lie}K_1)}(-).
\]
Invariant first-order differentiation gives the Lie operators;
the stabilizer derivatives are already prescribed.  Brackets
and relative PBW identify the entire monad.  This is a derived
identification, using equivariant descent and its flat filtered
induction monad.

In this model the $Q$-action is strong translation, and its
conjugation kernel is exactly \eqref{eq:L1}.  Strong translation
invariants descend the D-module to the base while retaining the
weak $J$-action.  They therefore give algebraic representations
of $J$, as required.  A character twists this same computation.
Passing to pro $J$ is legitimate on induced generators: only a
finite-dimensional Lie quotient is added, and a generator
trivial on an open normal stage remains at that stage under
induction.  The representation cobar calculation proves the
colimit assertion, and taking $Q$-invariants commutes with it
by Proposition~\ref{loop:relative-equivariance}.

Finally add the remaining Lie operators from $\ell$ by their
induction monad.  Induction and its continuous forgetful right
adjoint are $Q$-linear.  For restriction this is immediate from
\eqref{eq:L1}; for induction the adjoint change on the
enveloping factor gives the same derivative rule.  Forgetful
is conservative by testing the induced generators.  Equivariant
bars thus give the Harish--Chandra induction monad for $(\ell,J)$.

There is no assumption here that infinite-level equivariance has
already been identified.  For a pro-unipotent congruence $K$, its
convolution idempotent is the identity on a level already
$K$-equivariant.  On a smaller level pass to a normal smaller
one; the finite calculation shows that the image of the
idempotent is exactly the modules equivariant for $K$.  This
identifies its image in the colimit defining $\mathcal M$.
All other compact opens are then handled by the same finite
quotient calculation.  This proves the asserted identification
and its compatibility with the forgetful functors.
\end{proof}

The spherical category constructed from the inner $P_0$ presentation
is the ordinary Kazhdan--Lusztig category.  Trivializing $P_0$ on the
disks gives the identification; changes of trivialization descend
because the modules are strongly positive-loop equivariant and the
extension splits there.  The same change of frames identifies the
localization constructions.

\subsection{The spherical--Whittaker pairing and its adjoints}

Set
\[
 \begin{split}
 C_S&=\mathcal M^H,\qquad \mathbf1_S=D(S),\\
 a_S&=\mathcal A\otimes_{D_*(H)}\mathbf1_S
           =D_{\mathcal L^{-s}}(LG/H),\\
 d_S&=(a_S)_{LN,\chi},\qquad
 Q_S=\mathcal M_{LN,\chi}.
 \end{split}
\]
The Grassmannian category uses closed ind-proper bounds.  Since $LN$
is an increasing union of compact pro-unipotent groups, its
coinvariants are the colimit of compact-subgroup coinvariants.
After the norm identification at each compact stage, the transition
functors are star averaging with character.  Denote the image of a
Grassmannian object $K$ in $d_S$ by $\mathbf d(K)$.

These conventions give
\begin{equation}\label{loop:whittaker-coinvariants}
                     d_S=\mathcal W_{-s}^-(S).
\end{equation}
The sign can be
checked without a choice of local coordinates.  Transpose the
Grassmannian pairing, which is star integration of ! tensor.
Coinvariants at phase $\chi$ become ! equivariance at phase $+\chi$
on the opposite, exponent-$s$ Grassmannian.  The model comparison
\eqref{eq:W2} therefore identifies $\mathbf d(K)$ with the functional
\begin{equation}\label{eq:loop-whittaker-functional}
                 W\longmapsto
                  \Gamma_{\mathrm{dR}}(K\otimes^!\mu^!W).
\end{equation}
The pairing is relative over $S$ when parameters are present.

For clarity, this duality also permits coefficient categories.
Grassmannian duality is computed by closed bounds and ordinary
D-module duality.  Compact-subgroup equivariant categories are
retracts by pro-unipotent averaging.  To pass from their colimit to
the tensor-compatible dual category, one may use the compact opens
of Proposition~\ref{walg:adolescent}: its presentation and transition
calculation apply to $a_S$, since deep-open-equivariant categories
generate it.
For a chosen compact open, use a cofinal system of closed bounds
stable under that group.  Such a system exists: on a fixed bounded
subscheme a sufficiently deep normal congruence acts trivially,
so its saturation factors through a finite-type quotient of the
compact open and remains bounded; take its stable closed closure.
On these bounds the open-equivariant stage uses finite-type
quotients with unipotent stabilizers.  The closed-bound transitions,
and then the compact-open transitions in that calculation, are
fully faithful with continuous $D(S)$-linear right adjoints.
Their stages are dualizable and their colimit remains dualizable.
Consequently its dual is the limit of the !-equivariance diagrams,
also after tensoring with coefficients.  This is the Whittaker
category in \eqref{eq:W2}.  Alternatively, the same conclusion
follows from the linear parametrized Whittaker comparison.

The loop action on a spherical module factors through the
Grassmannian.  Passing to $LN$-coinvariants gives the pairing
\begin{equation}\label{eq:L3}
                     F_S:C_S\otimes_{D(S)}d_S\longrightarrow Q_S.
\end{equation}
It is balanced over the spherical monoidal category
$\mathcal S_S=\operatorname{End}_{\mathcal A}(a_S)$, with factor
orientations chosen for the left loop action.  All its categories
and maps have the disjoint-cluster product property.  For $d_S$
this property comes from coinvariants, or equivalently from the
Whittaker model comparison with its dual pairing; this specifies
the normalization even when stack dimensions contribute shifts.

\begin{proposition}\label{loop:pairing-adjoints}
At the irrational levels under consideration, the balanced functor
\[
               \overline F_S:
               C_S\otimes_{\mathcal S_S}d_S\longrightarrow Q_S
\]
is fully faithful and has a continuous $D(S)$-linear right adjoint.
The projection
$C_S\otimes_{D(S)}d_S\to C_S\otimes_{\mathcal S_S}d_S$
also has a continuous $D(S)$-linear right adjoint.
These assertions hold over the full parameter scheme, including
its collision loci.
\end{proposition}

\begin{proof}
First identify the spherical category.  On any nonunit Schubert
orbit the twisting character of a stabilizer is nonintegral on
a one-dimensional torus.  At a nonzero coweight it is the
Killing pairing with that coweight, up to the fixed sign, times
the irrational exponent.  The corresponding equivariant
D-module category vanishes.  On a bounded Schubert chart this
argument applies to each distinct-point stratum; ! restrictions
to the collision strata give the same calculation for their
merged coweights.  After removing the unit locus there is no
remaining stratum.  Thus
\[
                         \mathcal S_S\simeq D(B_SH),
\]
with ! tensor, the convolution product on the unit orbit.

The multiplication of $D(B_SH)$ is ! pull along its diagonal.
Compute it using a finite smooth quotient of $H$ with
pro-unipotent kernel.  The diagonal is then smooth,
representable, and affine; its right adjoint is star direct
image with the smooth shift.  The ! projection formula makes
this adjoint bimodule-linear and continuous.  This uses the diagonal,
not the map $B_SH\to S$, and makes no compactness assertion about
the unit of $D(B_SH)$.  The balanced
projection for two $\mathcal S_S$-modules is obtained by
tensoring this multiplication and its adjunction with those
modules.  Hence its right adjoint has the same properties.
It is also conservative: the terms coming from the unbalanced
tensor generate the balanced tensor, as its tensor bar shows.

Put
\[
        a^r=\mathbf1_S\otimes_{D_*(H)}\mathcal A,
        \qquad
        \mathcal S_S=a^r\otimes_{\mathcal A}a_S.
\]
Consider the evaluation
\[
             e:a_S\otimes_{\mathcal S_S}a^r\longrightarrow\mathcal A.
\]
Before balancing, its geometric calculation is the evaluation
kernel for $H$ from Proposition~\ref{loop:relative-equivariance},
followed by
\[
             \mathcal A\otimes_{D_*(H)}\mathcal A
                            \longrightarrow\mathcal A.
\]
The latter functor is multiplication on $LG\times^H LG$.
Separating out the product coordinate identifies its fibers
with the ind-proper affine Grassmannian.  The right adjoint is
! pull and is continuous and bimodule-linear by base change.
Precisely, at each congruence quotient of the product coordinate
use closed proper Grassmannian bounds and then their compatible
system.  This constructs the adjoint in the pro convention;
no finite-type adjunction for the whole loop group is being
assumed.

Consequently the pre-balance evaluation has a continuous
bimodule-linear right adjoint.  The balanced evaluation $e$
has those same properties.  To check this formal passage,
write $p$ for balancing.  The right adjoint of $ep$ is
$p^Re^R$.  Since $p^R$ is continuous, conservative, and
bimodule-linear, it detects both the colimit comparison maps
for $e^R$ and its canonical linearity maps.  They are
isomorphisms because they are so for $(ep)^R$.

The functor $e$ is fully faithful.  Put
$E=a_S\otimes_{\mathcal S_S}a^r$.  Proposition~\ref{loop:relative-equivariance}
expresses $H$-invariants as a relative tensor operation and gives
\[
 E^{H\times H}\simeq
    a_S^H\otimes_{\mathcal S_S}(a^r)^H
       \simeq\mathcal S_S\otimes_{\mathcal S_S}\mathcal S_S,
 \qquad \mathcal A^{H\times H}\simeq\mathcal S_S.
\]
On these two-sided invariants evaluation becomes
$\mathcal S_S\otimes_{\mathcal S_S}\mathcal S_S\to\mathcal S_S$,
which is the identity.  Both adjoints are bimodule-linear,
so the unit of the adjunction is an isomorphism on these
invariants.  They generate the domain under the two loop
actions.  Indeed, $a_S$ and $a^r$ are generated by the loop
actions on their equivariant Grassmannian delta units, from
their relative tensor presentations; the tensor bar then
gives the assertion for the balanced product.  Continuity
and bimodule-linearity extend the unit isomorphism to the
whole domain.

Tensor this fully faithful adjunction with $\mathcal M$ over
$\mathcal A$ and take $LN$-coinvariants on the other side.
Both operations preserve the adjunction and its unit
isomorphism because both functors in the adjunction are
continuous and linear.  The resulting functor is precisely
$\overline F_S$.  This proves the first assertion, and the
earlier balancing calculation proves the second.  Every
calculation took place over $S$ and in its finite/pro
presentations, so the argument applies across collisions.
\end{proof}

\begin{corollary}[Parameter operations for the pairing]
\label{loop:parameter-operations}
Let $f:T\to S$ be a change between the smooth affine parameter charts
used here, including a principal open or a smooth closed collision or
graph chart.  For each of the families
$\mathcal E=\mathcal M,C,a,d,Q$, there is an identification
\[
             \mathcal E_T\simeq
                   D(T)\otimes_{D(S)}\mathcal E_S.
\]
The pairing and its continuous linear right adjoint base change under
these identifications.  For a smooth closed inclusion $i:Z\to S$ and a
principal open inclusion $j:U\to S$, they commute with
$i_*,i^!,j_*,j^!$, including the units and counits of the adjunctions.
\end{corollary}

\begin{proof}
For the Harish--Chandra categories, first apply the underlying
quasi-coherent functor $b$.  The monic divisor algebras and lattice
presentations commute with the indicated base changes.
Proposition~\ref{loop:hc-renormalization} identifies the mappings
between the base-changed compact generators, which still generate.
All the maps in question have finite Tor dimension since the source
and target charts are smooth.  The underlying category of
$D(T)\otimes_{D(S)}\mathcal E_S$ is
\[
 \operatorname{QCoh}(T)\otimes_{D(S)}\mathcal E_S
 \simeq\operatorname{QCoh}(T)\otimes_{\operatorname{QCoh}(S)}
                   \mathcal E_S^0,
\]
which is the category just computed.  The formal-diagonal transport
of Proposition~\ref{loop:crystal} restricts along $f$ and agrees on
every multiple infinitesimal diagonal with the transport on that
category.  This identifies the crystals.  The equivalence between
crystals and $D(S)$-module categories recalled at the start of this
section identifies their sections as asserted; see
\cite[Theorem~2.6.3 and Lemma~3.2.4]{Gaitsgory1Affine}.
For $a$ the assertion is the relative D-module tensor-product formula;
for $d$ and $Q$ it follows by base change in their coinvariant bars.

Tensoring both functors of a continuous $D(S)$-linear adjunction with
$D(T)$ tensors its unit and counit as well.  Thus its right adjoint
over $T$ is the base-changed right adjoint.  Finally, on any of these
module categories, the open and closed localization projectors
$j_*j^!$ and $i_*i^!$ are the actions of $j_*\omega_U$ and
$i_*\omega_Z$, respectively.  Linearity makes both adjoints commute with these
projectors.  Identifying their images with the restricted categories
gives the claimed commutation with the pushes and pulls, and the
tensor construction preserves their adjunction maps.
\end{proof}
==== END Global-quantum-geometric-Langlands-at-irrational-level//07-loop-categories.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//08-w-algebras.tex ====
\section{A common target through principal \texorpdfstring{\(W\)}{W}-algebras}\label{sec:walg}

The pairings of \eqref{eq:L3} take values in Whittaker coinvariants of
Kac--Moody modules.  We identify these targets for the two inverse shifted
levels.  At a single point this is categorical Feigin--Frenkel duality.
The assertion needed here is stronger: the identification must respect
moving points, collisions, and the crystal structure on the parameter
scheme.  We construct it on compact generators and prove the necessary
base-change assertion for their mapping complexes.

Throughout this section \(S\) is a smooth affine coordinate chart for a
finite tuple of points of the curve, \(R=\mathcal O(S)\), and \(u=P(z)\)
is the monic equation of their effective divisor in the coordinate
\(z\).  Write \(\mathcal O\) for its completed regular ring and
\(\mathcal K=\mathcal O[u^{-1}]\).  All residues are total residues with
respect to \(dz\).  We use the Kac--Moody category \(\mathcal M\), its
Whittaker coinvariants \(Q_S\), and the passage \(b\) to the underlying
\(\operatorname{QCoh}(S)\)-linear category from Section~\ref{sec:loop}.
A superscript \(0\) denotes this last passage, as in \(Q_S^0=bQ_S\).
The definitions are local on \(S\); the coordinate and crystal comparisons
below will make the resulting equivalence independent of this chart.

\subsection{Compact open approximations}

For a positive root \(\alpha\), let \(h_\alpha\) be its height.  The
adolescent Whittaker subgroup \(I_i\), for \(i\geq1\), has the following
root and Cartan orders:
\begin{equation}\label{eq:L4}
\begin{array}{c|ccc}
 &e_\alpha&\mathfrak t&f_\alpha\\ \hline
 I_i&-i h_\alpha&i&i(h_\alpha+1).
\end{array}
\end{equation}
Here an order \(m\) means coefficients in \(u^m\mathcal O\).
Write \(N_i\) for the positive-root part and \(B_i\) for the
Cartan and negative-root part.  These are pro-unipotent groups, with
product coordinates, and \(I_i=N_iB_i\).  The character \(\chi\)
extends to \(I_i\) by zero on \(B_i\).

The multiplicative twisting splits over these groups with the Lie
splitting specified by the standard cocycle.  Indeed the order bounds
make that cocycle zero.  At integral depth the splitting is the
conjugated arc splitting.  Its possible Cartan correction vanishes on
positive-order Cartan coefficients.  We will also use intermediate
rational orders.  To see the splitting there without assuming that
\(\rho\) is an integral cocharacter, give the root piece of signed height
\(l\) the order \(m+\lambda l\).  The group preserves the corresponding
rationally filtered arc lattice in the adjoint representation.  Its
order-zero action is upper unipotent in root height, and all other
pieces increase order.  The determinant of its action on every finite
lattice difference is therefore canonically trivial.  The differential
of this trivialization has zero root correction by weights and zero
Cartan correction because positive-order Cartan modes have zero trace
on the successive order quotients.  Passing through the finite
nilpotent quotients gives the asserted splitting.

\begin{proposition}[Adolescent presentation]\label{walg:adolescent}
There is a canonical presentation
\begin{equation}\label{eq:L5}
 Q_S\simeq\underset{i\geq1}{\operatorname{colim}}P_i,
 \qquad P_i=\mathcal M^{I_i,\chi},
\end{equation}
whose transition functors are star averaging with character.  They are
fully faithful and have continuous \(D(S)\)-linear right adjoints.
After passage to \(b\), they preserve compact objects.  These
assertions hold over collision loci and after the base changes of Section~\ref{sec:loop}.
The same presentation and transition assertions apply with
\(a_S\) in place of \(\mathcal M\).
\end{proposition}

\begin{proof}
First fix \(N_i\).  Its invariant category is generated by objects with
additional \(B_j\)-equivariance for sufficiently large \(j\geq i\).
In fact, the underlying category is generated by objects with deep
congruence equivariance.  For such an object choose \(j\) so that
\(B_j\) lies in that congruence subgroup.  Product coordinates identify
its \(N_i\)-averaging with its \(N_iB_j\)-averaging.  Forgetting the
additional equivariance is fully faithful and has continuous right
adjoint by Proposition~\ref{loop:relative-equivariance}.  Thus these
subcategories exhaust the \(N_i\)-invariant category.  Increasing
\(i\) commutes with these exhaustions after increasing \(j\), by the
same averaging formula.  The diagonal depths are cofinal.  This proves
\eqref{eq:L5}; bounded loop supports give the identical argument for
\(a_S\).

It remains to prove the transition assertion relatively.  We give the
Fourier argument underlying the adolescent construction
\cite[Section~2]{RaskinW}, retaining the divisor coefficients to account
for collisions.  Interpolate the orders using rational slopes
\(\lambda>1\).  At a break the new orders are
\[
 a_l=-\lfloor\lambda l\rfloor\quad(l>0),\qquad
 b_r=\lfloor\lambda(r+1)\rfloor\quad(r\geq0),
\]
where \(r=0\) denotes the Cartan part.  The old orders immediately
before this break are
\[
 a_l^0=a_l+\mathbf1_{\lambda l\in\mathbb Z},\qquad
 b_r^0=b_r-\mathbf1_{\lambda(r+1)\in\mathbb Z}.
\]
Let \(J\) use the new positive orders and the old nonpositive orders,
and let \(K\subset J\) use the old positive orders and the new
nonpositive orders.  Then \(K\) is normal, and its character is
\(J\)-invariant.  Here are the order checks.  In positive roots they
follow from the floor inequality
\(\lfloor x\rfloor+\lfloor y\rfloor\leq\lfloor x+y\rfloor\);
when the sum is an integral break, either both summands are integral
or their fractional parts add to one.  A sum of two old nonpositive
orders reaches the new order, since \(\lambda>1\).  For a mixed
bracket of positive height \(l\) and nonpositive height \(-r\),
\(a_l+b_r^0\) reaches the old nonpositive order if \(l\leq r\),
and can fail to reach the new order only when both participating
breaks are integral.  If \(l>r\), it already reaches the old positive
order of height \(l-r\).  A simple-root residue can occur only when
\(l-r=1\), both breaks are integral, and both coefficients are among
those changed at the break.  This verifies normality and invariance
of the character on \(K\).  In the Cartan case the orders also exclude
a residue in the central Lie cocycle.  Exponentiating the finite
nilpotent quotients gives the group assertions.

Modulo \(K\), denote the new/old positive difference by \(U\) and the
old/new nonpositive difference by \(V\).  The group \(U\) is unipotent,
and \(V\) is a normal vector group.  Relative Fourier transform for
\(V\), applied to the category of \((K,\chi)\)-equivariant objects,
identifies the old invariant category with the fiber over zero in
\(V^*\).  The new condition is \(U\)-equivariance.  Its action on
\(V^*\) is affine: exchanging the two factors in multiplication
introduces both conjugation on \(V\) and the residual character of
\(K\); Fourier transform turns the latter into translation.  Thus
this description also accounts for the character twisting of the
quotient action.

The orbit of zero is free and closed over \(S\).  To check this,
order coordinates by root height.  The linear term from a height
\(l\) coordinate of \(U\) to the height \(l-1\) component of
\(V^*\) is the pairing
\[
 (x,y)\longmapsto\chi([x,y]).
\]
It is injective in \(x\), with locally split image: the Lie-algebra
factor is the injective principal \(\mathfrak{sl}_2\) operator
\([f_{\mathrm{prin}},-]\) on positive grades, and the coefficient
factor is the perfect residue pairing
\[
 (\overline f,\overline g)\longmapsto
       \operatorname{Res}\frac{f(z)g(z)}{P(z)}\,dz
       \quad\text{on }R[z]/(P).
\]
This pairing is perfect for every monic \(P\), including repeated
roots.  The higher terms in the orbit map at height \(l-1\) involve
only lower-height coordinates.  Induction in height therefore identifies
the orbit with a closed graph over the locally split linear image,
and identifies its parameter space with \(U\).

Closed pushforward from this orbit is fully faithful with continuous
right adjoint, also for categories with coefficients.  On the orbit,
translation descent for the free \(U\)-action identifies equivariant
objects with the fiber over zero.  Under this identification, forgetting
from that fiber and star averaging is exactly the transition at the
break, with the fixed smooth/Fourier normalization.  It is therefore
fully faithful with continuous linear right adjoint.  Composing the
finitely many breaks between successive integral slopes gives the
claimed transition.  After passage to \(b\), such a left adjoint
preserves compact objects because its right adjoint is continuous.  All operations used in the
argument are relative Fourier transform, closed pushforward, smooth
descent, and the base-change operations of Section~\ref{sec:loop};
this proves the remaining assertions.
\end{proof}

\subsection{The point input and its module convention}

We specify the representation-theoretic input before extending it over
\(S\).  For a split root Lie algebra over a characteristic-zero field,
Raskin's affine Skryabin theorem identifies the Whittaker coinvariants
of the renormalized Kac--Moody category with the renormalized category
of principal \(W\)-modules \cite[Theorem~5.1.1]{RaskinW}.
Here the latter category means
\[
 \operatorname{Ind}\bigl(\operatorname{thick}
       \{V_i^W:i\geq0\}\subset D^+(W\text{-}\mathrm{sm})\bigr).
\]
In particular the mapping complexes in its specified compact category
are the ordinary derived mapping complexes between the generalized
vacua.  This is the convention in \cite[Section~5.1 and the proof of
Theorem~5.1.1]{RaskinW}; it does not assert that these objects are
compact in the unrenormalized unbounded derived category.

The composite with the underlying-complex functor is principal
Drinfeld--Sokolov reduction.  It sends the Whittaker averaging
counits to isomorphisms.  On the induced \((I_i,\chi)\)-generator,
using the \(N_i\)-lattice, it gives the generalized vacuum \(V_i^W\)
in degree zero.  Changing the lattice introduces its relative
cohomological shift and determinant line.  The vacuum calculation,
its nonnegative PBW filtration, and the strict surjections
\(V_j^W\twoheadrightarrow V_i^W\) for \(j\geq i\) are
\cite[Theorems~4.2.1, 4.5.1, and 4.6.1]{RaskinW}.
The ordinary vacuum is the case \(i=0\).

We use unshifted vertex modes \(w_{(m)}\).  For homogeneous principal
PBW generators \(w\) of conformal degree \(d_w\), the vacuum is
annihilated by \(w_{(m)}\) for \(m\geq i d_w\), and ordered
monomials in the remaining modes give its PBW basis.  One can recover
the exact, rather than merely associated-graded, annihilation from
the free-field realization \cite[Theorem~6.1.1]{RaskinW}.  The vacuum
embeds in the depth-\(i\) Heisenberg vacuum.  A homogeneous
normally ordered monomial of weight \(d\) has \(k\) currents and
\(r\) derivatives with \(k+r=d\); its mode of index \(m\) has
current indices with sum \(m-(k-1)-r\).  If \(m\geq id\) and
\(i\geq1\), this sum is greater than \(k(i-1)\), so one of the
annihilating current modes occurs on the right after normal ordering.
The creation axiom gives the case \(i=0\).  The associated-graded
calculation then proves the stated PBW basis.

We also use Feigin--Frenkel duality of principal conformal vertex
algebras at the inverse shifted forms.  Our levels are irrational;
thus they lie in the proved range of the duality, including its
vacuum-preserving form \cite[Section~7.3, especially Warning~7.3.1
and Lemma~7.3.3]{RaskinW}.  The input is a Lie-algebra theorem and
is unaffected by the isogeny type of the group.  The principal BRST
construction provides closed fields acting on reduction, with the
usual conformal coordinate action.  One may use its nonnegative
ghost-degree subalgebra: in the principal integer grading it has
cohomology only in degree zero, and its degree-zero kernel realizes
the \(W\)-algebra.  This is the subcomplex \(\mathcal C^-\) of
\cite[Section~4, Theorem~4.1 and Remark~4.1]{KacWakimoto}, with
homological charge reversed to cohomological degree.  Its absence
of negative cohomological degrees means that no degree-zero boundary
has to be quotiented out.  Thus its degree-zero cycles form an actual
vertex subalgebra of the BRST algebra, acting on every module complex.
This supplies the strict action needed here; the corresponding
factorization action is discussed in \cite[Section~4.4]{RaskinW}.

For completeness, the coordinate convention agrees with our
\(P_0\)-twist.  The improved Virasoro term is
\(L_n^{\mathrm{Sug}}-(n+1)\rho_n\), together with its ghost terms.
On a current mode \(x_m\) of height \(l\) its noncentral commutator is
\[
 [L_n,x_m]=-(m+(n+1)l)x_{m+n}.
\]
This is the coordinate action on the root coefficient with density
\((dz)^l\).  The central term is the affine gauge correction for
\((dz)^\rho\); differentiating \(z\mapsto z+\epsilon z^{n+1}\)
gives precisely the correction contributed by \(-(n+1)\rho_n\).
The charged ghosts transform by the dual densities.  Corrected
currents and their normally ordered ghost bilinears preserve the
closed-field model, with possible scalar corrections from reordering.
Consequently the BRST action and the conformal vertex-algebra action
use the same changes of coordinate.

\subsection{Reduction over a moving divisor}

Let \(\mathcal H_R^W\) be the abelian category of smooth modules for
these \(W\)-modes over \(R\).  Concretely, fields are smeared by
coefficients in \(\mathcal K\), with total residue and the translation
and operator-product relations; equivalently one uses the chiral
module construction of \cite[Section~3.6]{BDChiral}.  Smoothness means
that, on any fixed vector, sufficiently high-order coefficients of each generating field
act by zero.  The identity field acts by total residue against \(dz\).
For coefficients \(f,g\), commutators use the vertex products with
coefficients \((\partial^n f/n!)g\).  Composite fields act by their
normal-product formulas.  Write \(D(\mathcal H_R^W)\) for the ordinary
derived category.

These formulas make sense also when the divisor has multiple points.
For example, the two expansions of a pole \((z-w)^{-j}\), \(j>0\),
are obtained by writing
\[
 (z-w)^{-j}
 =\left(\frac{P(z)-P(w)}{z-w}\right)^j(P(z)-P(w))^{-j}
\]
and expanding the last factor in the two orders in the nested
Laurent rings.  In each resulting normally ordered series, the
coefficients of the modes applied first tend to arbitrarily high
order.  Thus each expression acts on a smooth vector by a well-defined
sum.  The construction is unchanged upon replacing \(P\) by an
equation giving a cofinal completion, commutes with scalar extension,
and uses products of mode algebras on disjoint formal divisors.

Give a generating field its principal degree \(d_w\).  A commutator
of two generating modes has strictly smaller total PBW degree: a
nonnegative vertex product has conformal weight smaller than the sum
of the two weights, and expressing it through products and derivatives
does not increase that bound.  We will repeatedly use this fact to
reorder words.  Every such calculation is vectorwise; it requires no
single smoothness bound valid for an entire module.

The reduction functor on compact Kac--Moody modules is the complex
\begin{equation}\label{eq:L6}
 \operatorname{DS}_R(M)=
 C^{\infty/2,h}\bigl(\operatorname{Lie}(LN),M\otimes R_{-\chi}\bigr),
 \qquad h=\operatorname{Lie}N_0(\mathcal O).
\end{equation}
We use cochains on the compact lattice.  If \(h'\supset h\),
changing from \(h'\) to \(h\) tensors the complex by
\(\det(h'/h)[\dim(h'/h)]\).  The critical bracket correction is zero
on the unipotent Lie algebra.  The complex uses the algebraic smooth
module and the Fock ghost module.  On a bounded complex of inputs,
quasi-isomorphisms remain quasi-isomorphisms: the input direction is
bounded and the ghost factors are free.  The compact calculations in
Proposition~\ref{loop:hc-renormalization} therefore define
\eqref{eq:L6} on compact objects, and then define its continuous
extension to \(\mathcal M^0\).

The BRST differential and the closed \(W\)-fields are given by the
same residue formulas on every chart.  Their identities can be
checked in the affine-plus-Clifford enveloping algebra.  The preceding
expansions guarantee convergence in the smooth-module topology;
induced modules for sufficiently deep lattices have free PBW terms,
and the Clifford factor has its free Fock presentation.  These
modules detect the identities.  Equivalently, on them one may check
the identities after every residue-field extension, where they are
the ordinary BRST identities in uniformizers.  This also verifies
closure of the degree-zero fields acting on the complex.

The full regular lattice \(h\), its Fock module, and its vacuum are
canonically identified across nilpotently different lifts of the
parameter scheme.  Indeed the corresponding regular and Laurent
completions are canonically equal.  Thus \eqref{eq:L6}, after forgetting
the \(W\)-action if necessary, defines a crystal functor.  In
particular the ordinary vacuum cycle, consisting of the inducing
vector times the full-lattice Fock vacuum, is horizontal.  We do not
assert that a prescribed positive-depth cutoff is itself horizontally
constant.

\begin{lemma}[Relative depth vacua]\label{walg:vacua}
The reduction of the induced \((I_i,\chi)\)-generator, calculated
with the \(N_i\)-lattice, is a flat heart module \(V_{i,R}^W\).
It has a distinguished vector \(\mathbf v_i\) with presentation
\begin{equation}\label{eq:L7}
 w(u^{i d_w}\mathcal O)\mathbf v_i=0,
 \qquad
 V_{i,R}^W\text{ has PBW basis in modes of orders }<i d_w.
\end{equation}
It corepresents vectors satisfying the displayed annihilations.
These constructions commute with the scalar extensions used here,
including the finite flat infinitesimal tests for the crystal.
The ordinary vacuum satisfies the same assertion for \(i=0\).
\end{lemma}

\begin{proof}
Coefficients of each fixed \(u\)-order have basis
\(z^a\), \(0\leq a<\deg P\).  Apply ordered words in the closed
fields to the inducing vector times the lattice Fock vacuum.
This defines a map from the free \(R\)-module on the proposed PBW
words to the BRST complex.  Its terms and its construction commute
with ordinary scalar extension.  Over every residue field it is a
quasi-isomorphism by the point vacuum calculation: at a point of
multiplicity \(m\), the equation \(u\) has order \(m\), so that
its depth is \(mi\); separated points give the product calculation.

The BRST terms are flat over \(R\).  Since \(R\) is smooth of finite
dimension, the scalar extensions to residue fields have finite Tor
dimension, and ordinary tensor with these complexes computes derived
tensor.  The cone of the displayed map therefore has zero derived
fiber at every prime.  Residue fields at all primes detect zero
objects of \(D(R)\), so the map is a quasi-isomorphism.  Its
cohomology is the indicated free module.  The annihilations follow
from the point annihilations, since an operator between free
\(R\)-modules that is zero at every prime is zero.  The same free
presentation remains valid on the finite flat infinitesimal tests.

Finally let \(v\) be a smooth vector satisfying \eqref{eq:L7}.
Reorder a word by decreasing total PBW degree, moving annihilating
modes to the right; each commutator lowers degree.  The same operation
on \(\mathbf v_i\) gives its PBW expansion.  At each comparison the
normal-product tails terminate by the smoothness bounds for the two
vectors.  Hence the substitution \(\mathbf v_i\mapsto v\) respects
every mode action.  Its uniqueness follows from cyclicity, proving
the universal property.
\end{proof}

The modules \(V_{i,R}^W\) generate \(\mathcal H_R^W\), because every
vector satisfies a sufficiently deep cutoff.  Kernels, cokernels,
and filtered colimits are computed on the underlying modules, with
the induced smooth action.  Consequently \(\mathcal H_R^W\) is a
Grothendieck category with exact filtered colimits.

\subsection{Mapping complexes and scalar extension}

The passage from the point theorem to families needs a finiteness
statement for derived maps.  Smoothness alone would not justify
commuting an infinite collection of mode conditions with tensor
product.  The finite differences between the depth vacua provide
the required replacement.

\begin{lemma}[Ext continuity]\label{walg:ext}
For every depth \(i\geq0\) and every \(p\geq0\), the functor
\(\operatorname{Ext}^p_{\mathcal H_R^W}(V_{i,R}^W,-)\) commutes
with filtered colimits.  If \(R\to T\) is a scalar extension occurring
in our affine or infinitesimal charts, then, for a smooth \(W\)-module
\(M\) over \(T\), restriction gives
\begin{equation}\label{walg:restriction-ext}
 \operatorname{Ext}^p_{\mathcal H_T^W}(V_{i,T}^W,M)
 \simeq
 \operatorname{Ext}^p_{\mathcal H_R^W}
             (V_{i,R}^W,\operatorname{oblv}_R M).
\end{equation}
For smooth affine or residue-field extensions, and for the finite
flat infinitesimal tests, the ordinary derived mapping complexes
between depth vacua consequently satisfy
\begin{equation}\label{walg:mapping-base-change}
 \operatorname{RHom}_{\mathcal H_R^W}(V_{i,R}^W,V_{j,R}^W)
       \otimes_R^{\mathbf L} T
 \simeq
 \operatorname{RHom}_{\mathcal H_T^W}(V_{i,T}^W,V_{j,T}^W).
\end{equation}
\end{lemma}

\begin{proof}
\emph{Resolving the difference between two depths.}
Put \(K_{ji}=\ker(V_{j,R}^W\twoheadrightarrow V_{i,R}^W)\) for
\(j>i\).  We first construct a resolution of \(K_{ji}\), possibly
infinite to the left, whose terms are finite sums of depth vacua.
The PBW filtration on its defining surjection is strict; in its
associated graded the kernel is generated by the finitely many mode
coefficients between the two cutoffs.  Choose homogeneous generators
and lift them to \(K_{ji}\).  Their smoothness supplies a common
sufficiently deep vacuum mapping to each chosen lift, with the
appropriate filtration shift.  These maps give a strict surjection
onto \(K_{ji}\).

The construction can be iterated.  Indeed, at each step choose a
common upper depth \(m\).  All the associated-graded modules involved
are finitely presented over the polynomial ring in the mode variables
below depth \(m\).  Modes above that depth act as zero in associated
graded, because commutators lower degree; they need not annihilate the
filtered modules.  Increasing from one finite depth to another adds
only finitely many variables.  The polynomial ring in all the
remaining, countably many, variables over the Noetherian ring \(R\)
is coherent.  To verify precisely the fact used here, a matrix between
finite free modules and its finitely many coefficients descend to a
polynomial ring in finitely many variables; its kernel is finitely
generated there, and extension to the full polynomial ring is flat.
Thus kernels between finitely presented modules remain finitely
presented.  The next graded kernel is therefore finitely generated,
and the lifting argument repeats.  This proves the resolution
assertion.

Every vacuum is \(R\)-flat.  The first kernel \(K_{ji}\) is flat
because it is the kernel of a surjection between flat modules with
flat quotient.  Inductively, every subsequent kernel in the resolution
is flat for the same reason.  Hence these exact sequences remain
exact after arbitrary scalar extension in the charts.  The PBW
presentations of Lemma~\ref{walg:vacua} identify the extended sequences
with the corresponding sequences over \(T\).  This argument applies
also over the finite flat infinitesimal charts.

\smallskip
\noindent\emph{Continuity of Ext under filtered colimits.}
For \(p=0\), a map from a vacuum is determined by its generating
vector.  If that vector is given in a filtered colimit, lift it to
one member.  The lift is annihilated above some depth \(j\).
Only finitely many coefficients between depth \(i\) and depth \(j\)
remain to be tested.  Their vanishing holds in a common later member,
which proves continuity of \(\operatorname{Hom}(V_{i,R}^W,-)\).
The universal property also gives \eqref{walg:restriction-ext} for
\(p=0\).

For the induction in \(p\), the exact underlying-module functor has
the presentation
\[
 \operatorname{oblv}(M)=
 \underset{j}{\operatorname{colim}}
       \operatorname{Hom}(V_{j,R}^W,M).
\]
Applying this formula to an injective resolution and using exactness
of filtered colimits gives
\[
 \underset{j}{\operatorname{colim}}
       \operatorname{Ext}^p(V_{j,R}^W,M)=0\qquad(p>0).
\]
The long exact sequences for \(K_{ji}\), followed by this colimit,
therefore give, for every \(p>0\), the useful formula
\begin{equation}\label{walg:ext-bootstrap}
 \operatorname{Ext}^p(V_{i,R}^W,M)
 \simeq\operatorname{coker}\left(
 \underset{j>i}{\operatorname{colim}}
       \operatorname{Ext}^{p-1}(V_{j,R}^W,M)
 \longrightarrow
 \underset{j>i}{\operatorname{colim}}
       \operatorname{Ext}^{p-1}(K_{ji},M)\right).
\end{equation}
For fixed \(j,i\), the resolution just constructed calculates
\(\operatorname{Ext}^{p-1}(K_{ji},M)\) by the first-quadrant
resolution spectral sequence.  Its terms are finite sums of
\(\operatorname{Ext}^q(V_{m,R}^W,M)\), and its diagonals in any
fixed total degree are finite.  Only \(q\leq p-1\) is needed.
The induction hypothesis therefore gives filtered-colimit continuity
for this Ext group.  Formula~\eqref{walg:ext-bootstrap} proves the
claim in degree \(p\).

\smallskip
\noindent\emph{Restriction under scalar extension.}
The same induction proves \eqref{walg:restriction-ext}.  Its degree
zero assertion was proved above; use the base-extended resolutions
of \(K_{ji}\), which remain exact, in the spectral sequences on the
two bases, and then use \eqref{walg:ext-bootstrap}.  All comparisons
are the natural maps obtained by restriction and the canonical
\(V_{i,R}^W\to V_{i,T}^W\), so this is an identification of the
natural derived mapping complexes, not just a coincidence of their
cohomology dimensions.

\smallskip
\noindent\emph{Derived tensor base change.}
It remains to justify tensoring the target in
\eqref{walg:mapping-base-change}.  First tensor by a finite projective
\(R\)-module; the assertion follows from finite additivity and
retracts.  A flat module is a filtered colimit of finite free modules,
so the Ext-continuity statement proves the assertion for a flat
module as well.  The extensions from a smooth affine \(R\) under
consideration have finite Tor dimension; resolve \(T\) by a bounded
complex of flat \(R\)-modules.  The preceding comparisons totalize
over this finite range and prove the required derived tensor
identity.  Formula~\eqref{walg:restriction-ext} then changes the
ambient module category from \(R\) to \(T\).  Finite flat
infinitesimal tests are already covered by the flat case.
\end{proof}

\subsection{The relative Skryabin comparison}

The stages in \eqref{eq:L5} have positive depth, so we first use
only their vacuum images.  Let \(\mathcal V_R^W\) denote the thick,
idempotent-complete subcategory of \(D(\mathcal H_R^W)\) generated
by \(V_{i,R}^W\), \(i\geq1\).  Keeping the ordinary mapping
complexes in this definition is essential.  After constructing the
relative comparison, Lemma~\ref{walg:ordinary-vacuum} will prove
geometrically that adjoining \(V_{0,R}^W\) does not change this
category.  Define the relative renormalized category by
\[
 \mathcal W_R^{\mathrm{ren}}=
       \operatorname{Ind}(\mathcal V_R^W).
\]
The next proposition constructs this category intrinsically as the
Whittaker target and then checks its dependence on the formal curve.

\begin{proposition}[Relative Skryabin comparison]\label{walg:relative-skryabin}
Reduction induces an equivalence
\[
 Q_S^0\simeq\mathcal W_R^{\mathrm{ren}}.
\]
On compact objects its composite with the ordinary comparison to
\(D(\mathcal H_R^W)\) is \eqref{eq:L6}.  This comparison is fully
faithful on compact objects and has essential image
\(\mathcal V_R^W\).  It commutes with the scalar extensions in
Lemma~\ref{walg:ext} and with the crystal transport of
Section~\ref{sec:loop}.
\end{proposition}

\begin{proof}
We first descend the continuous extension of \eqref{eq:L6} through
the Whittaker projection.  Apply it to the counit of star averaging
by a compact subgroup \(N_i\).  After forgetting the \(W\)-action,
this comparison commutes with tensor to every residue field:
on congruence compact generators this follows from the BRST formula,
and on their continuous extensions it follows by colimits.  The
point Skryabin theorem says that every such fiber of the counit is
an isomorphism.  Residue-field detection in \(D(R)\) therefore makes
the original counit an isomorphism.  The isomorphisms are the images
of the averaging counits themselves and hence carry their composition
coherences.  They define a single functor
\[
 Q_S^0\longrightarrow D(\mathcal H_R^W)
\]
using \eqref{eq:L5}, whose composite with the canonical projection
from \(\mathcal M^0\) is reduction.  In particular this is a
comparison on functors and maps, as well as on objects.

By Proposition~\ref{walg:adolescent}, every compact object of
\(Q_S^0\) is a retract of an object generated at a finite stage,
and the stage transitions preserve compactness.  Each stage is
thickly generated over \(R\) by the induced trivial representation,
with its character correction.  This is the pro-unipotent
Harish--Chandra calculation of
Proposition~\ref{loop:hc-renormalization}.  Its reduction is the
vacuum of Lemma~\ref{walg:vacua}, with the prescribed relative lattice
line and shift.

Take two such generators.  Move them to a common later stage of
\eqref{eq:L5}.  Their source mapping complex commutes with the allowed
scalar extensions by Proposition~\ref{loop:hc-renormalization}
and the continuous linear averaging adjunctions.  Their target
mapping complex commutes with these extensions by
Lemma~\ref{walg:ext}.  On each residue field the resulting mapping
comparison is the point Skryabin equivalence.  Apply residue-field
detection to its cone.  The comparison is consequently an
isomorphism over \(R\), proving full faithfulness on the generators
and hence on their thick closure.  Its essential image is exactly
\(\mathcal V_R^W\); taking Ind proves the equivalence.

For crystal transport, work on a finite infinitesimal neighborhood
of a diagonal in the parameter scheme.  The two lifts have the same
absolute completed curve, full regular lattice, and Laurent algebra.
The ordinary BRST comparison on Kac--Moody compact objects is
therefore the same for both lifts, including its Fock vacuum.
Its mapping comparisons agree by the finite flat case of
Lemma~\ref{walg:ext}.  To pass to \(Q\), refine to common deep
congruence subgroups as in Proposition~\ref{loop:crystal}; their
systems are cofinal.  On an equivariant stage the projection to
coinvariants is precisely its map into the colimit, so its ordinary
comparison is calculated by first forgetting to the Kac--Moody
category and applying the same BRST functor.  Thus the comparisons
agree on the stages and their transition maps, and hence on the
colimit.  The identifications are induced by the identity of the
absolute formal curve.  They compose identically on higher
infinitesimal diagonals, which supplies the crystal coherence.
\end{proof}

\begin{lemma}[The ordinary vacuum is in the compact image]
\label{walg:ordinary-vacuum}
The image in \(Q_S^0\) of the spherical Kac--Moody vacuum is compact.
Its ordinary comparison is \(V_{0,R}^W\).  Consequently adjoining
\(V_{0,R}^W\) to the generators of \(\mathcal V_R^W\) does not
change that category.
\end{lemma}

\begin{proof}
Let \(\mathrm{Vac}_H\in C_S\) be the module induced from the
trivial arc-group representation, and let \(\delta_1\in a_S\)
be the convolution unit supported at the unit section of the
Grassmannian.  By \eqref{eq:L2}, \(b\mathrm{Vac}_H\) is compact
in \(C_S^0\).  We first verify that
\(b\mathbf d(\delta_1)\) is compact in \(d_S^0\).

In the global pole model of Section~\ref{sec:whittaker}, the zero-pole
bound has just one relevant label, namely zero.  The stratum tests
remove all defective strata, leaving the fixed stack
\(\operatorname{Bun}_{N_0}\times S\).  Exponential pullback identifies
its Whittaker category with \(D(S)\), with the fixed determinant line
and normalization from that section.  This assertion is uniform in
the marks: a zero divisor does not change when marks collide.
The inclusion of this Whittaker pole bound into the full pole model
has continuous \(D(S)\)-linear right adjoint given by \(!\)-restriction,
by Propositions~\ref{whittaker:strata-properties} and
\ref{whittaker:duality}.  Tensoring this adjunction with
\(\operatorname{QCoh}(S)\) shows that the image of the compact
module \(R\) is compact after passage to \(b\).

Under \eqref{eq:W2}, this opposite zero-label object represents
\(\mathbf d(\delta_1)\), up to the fixed invertible line and shift
specified by the pairing.  To verify the identification, its
functional on a Whittaker object \(W\) is the normalized zero-label
coefficient.  Indeed restriction to the zero-pole bound extracts that
coefficient by exponential cancellation.  The functional defining
\(\mathbf d(\delta_1)\) in Section~\ref{sec:loop} is
\[
 W\longmapsto
 \Gamma_{\mathrm{dR}}(\delta_1\otimes^!\mu^!W).
\]
It is the same coefficient: the Grassmannian unit maps to the trivial
\(N_0\)-torsor, a section of the zero-label stratum over \(S\),
and normalized \(!\)-pullback to that section inverts exponential
pullback of the coefficient.  More explicitly, if \(q_0\) is the
zero-label projection and \(\sigma\) the trivial-torsor section,
then \(q_0\sigma=\operatorname{id}_S\) and
\(\psi_0\sigma=0\).  On the pairing side,
\(q_{0,*}q_0^!\) contributes \([2\delta_N]\), exactly canceled
by the \([-2\delta_N]\) in \eqref{eq:W2}.
Proposition~\ref{whittaker:duality}
identifies closed inclusion with the transpose of \(!\)-restriction,
so this identifies the two functionals on every object, not just
on standard generators.  Invertible lines and shifts preserve
compactness.  This proves the assertion about
\(\mathbf d(\delta_1)\).

By Proposition~\ref{loop:pairing-adjoints}, the pairing \(F_S\) has a
continuous \(D(S)\)-linear right adjoint, as the composite of the
adjoints to balancing and to the balanced pairing.  After passage to
\(b\) its adjunction remains continuous and linear.  Thus it
preserves compact objects.  The exterior product of the two compact
inputs above is compact, and the convolution unit gives
\[
 F_S(\mathrm{Vac}_H,\mathbf d(\delta_1))
      =q(\mathrm{Vac}_H),
\]
where \(q:\mathcal M\to Q_S\) is the coinvariant projection.
Hence this particular coinvariant image is compact; no compactness
claim about \(q\) on arbitrary inputs is being used.

Proposition~\ref{walg:relative-skryabin} identifies its ordinary
comparison with \(\operatorname{DS}_R(\mathrm{Vac}_H)\), which is
\(V_{0,R}^W\) by Lemma~\ref{walg:vacua}.  The already proved
full faithfulness on compact objects has essential image
\(\mathcal V_R^W\), generated by positive depths.  Therefore
\(V_{0,R}^W\) belongs to that image, as required.
\end{proof}

The relative renormalization thus agrees with the convention that
includes the ordinary vacuum.  The order of the argument matters:
its compactness follows from the geometric pairing, not from the
possibly infinite resolution in Lemma~\ref{walg:ext}.

\subsection{Duality and compatibility}

For either Lie algebra, compose the ordinary comparison of
Proposition~\ref{walg:relative-skryabin} with the underlying-complex
functor:
\[
 Q_S^0\longrightarrow D(\mathcal H_R^W)\longrightarrow D(R).
\]
Its crystal compatibility descends this composite to a continuous
\(D(S)\)-linear functor
\begin{equation}\label{eq:walg-forgetful}
 \operatorname{oblv}_W:Q_S\longrightarrow D(S).
\end{equation}
After passage to \(b\), its composite with the coinvariant projection
is the continuous Drinfeld--Sokolov functor computed on compact
inputs by \eqref{eq:L6}.  The scalar-extension comparisons used in
this construction are those of Lemma~\ref{walg:ext}: finite-Tor
smooth-affine and residue-field extensions, and finite flat
infinitesimal tests for the crystal.

\begin{theorem}[Common Whittaker target]\label{walg:common-target}
For the inverse shifted levels on the two Langlands-dual Lie
algebras, there is a canonical identification of the crystal
categories \(Q_S\).  It respects pullback in \(S\), collisions of
marks, changes of curve coordinate, and exterior products for
disjoint clusters.  The underlying-complex functors from these
categories to \(D(S)\) are identified.  The ordinary vacuum and its
distinguished horizontal generating cycle are identified as well.
\end{theorem}

\begin{proof}
The conformal Feigin--Frenkel isomorphism identifies the two
\(W\)-algebras \cite{FeiginFrenkel}.  At a point it identifies depth
vacua with their cyclic vectors, by \cite[Lemma~7.3.3]{RaskinW}.  In the relative
presentation \eqref{eq:L7} this identification is equally explicit.
The principal degrees agree.  Replacing homogeneous principal PBW
generators by the images of the dual generators does not change the
presentation.  We verify the needed cutoff assertion before applying
its universal property.  Suppose \(i\geq1\), and a vector \(v\) obeys
the generating-field cutoffs.  Induction on conformal weight proves
that every homogeneous field of weight \(d\) has modes of index
\(m\geq id\) annihilating \(v\).  Derivatives preserve the assertion:
\((\partial^r A)_{(m)}\) is a scalar multiple of \(A_{(m-r)}\),
and \(m\geq i(a+r)\) implies \(m-r\geq ia\) for a field of
weight \(a\).

For a normal product of fields of positive weights \(a,b\), use
\[
 (:AB:)_{(m)}=
 \sum_{k<0} A_{(k)}B_{(m-1-k)}+
 \sum_{k\geq0}B_{(m-1-k)}A_{(k)}.
\]
If \(m\geq i(a+b)\), the first sum kills \(v\), as do the terms
\(k\geq ia\) in the second.  In the remaining finitely many terms,
commute \(B_{(s)}\) past \(A_{(k)}\), where \(s=m-1-k\geq ib\).
The reordered term kills \(v\).  Each commutator term is a scalar
multiple of \((B_{(n)}A)_{(m-1-n)}\); this field has weight
\(c=a+b-n-1<a+b\), and
\[
 m-1-n-ic=(m-i(a+b))+(i-1)(n+1)\geq0.
\]
It therefore kills \(v\) by induction.  Weight-zero fields contribute
only the identity mode of index \(-1\), which cannot occur in this
range.  Strong generation now proves the assertion.  The case
\(i=0\) is the ordinary creation axiom.

Over the moving divisor, specialize the transported generator action
to every residue field.  The calculation just made applies at each
support point, with its multiplicity multiplying the depth.  Thus
these modes annihilate the cyclic vector on every fiber.  The module
\(V_{i,R}^W\) is \(R\)-free and \(R\) is reduced, so the same
annihilation holds over \(R\); it persists under the flat
infinitesimal base changes by its presentation.  Hence the images of
the dual generators satisfy \eqref{eq:L7}.  Applying the inverse change of
generators proves equality of the two annihilation conditions.
The common cyclic vector and Lemma~\ref{walg:vacua} determine mutually
inverse maps of the relative vacua.

It follows that Feigin--Frenkel identifies the two thick subcategories
\(\mathcal V_R^W\), with their ordinary mapping complexes, and hence
their Ind completions.  Proposition~\ref{walg:relative-skryabin}
identifies both with \(Q_S^0\).  Its crystal compatibility identifies
this equivalence across the formal diagonals, so it descends to the
claimed equivalence of crystal categories \(Q_S\).

The coordinate compatibility follows from the conformal coordinate
action on the strict closed fields and the BRST calculation above.
For disjoint divisors the modes, Fock modules, and semi-infinite
complexes are tensor products.  The determinant lines and shifts use
the graded tensor symmetry, so the comparisons also respect products
and their permutation coherences.  Finally, the Feigin--Frenkel
identification changes the algebra acting on a module and preserves
its underlying complex.  The two ordinary comparison functors to
\(D(S)\) consequently agree.  At depth zero
Lemma~\ref{walg:ordinary-vacuum} places both vacuum images in the compact subcategories where the ordinary comparison is
fully faithful.  The vacuum-preserving module isomorphism therefore
lifts to their unique isomorphism in \(Q_S^0\), and its crystal
compatibility follows from the same comparison on infinitesimal
charts.  It preserves the actual inducing/Fock cycle, which was
shown to be horizontal.
\end{proof}

Theorem~\ref{walg:common-target} supplies the common target for the
pairings of \eqref{eq:L3}, with the relative structure needed to
compare their right adjoints.  Its proof has used the point
representation theorems together with explicit relative mode and
mapping calculations; no factorization form of a fundamental local
equivalence enters this construction.
==== END Global-quantum-geometric-Langlands-at-irrational-level//08-w-algebras.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//09-local-comparison.tex ====
\section{Local comparison in families}\label{sec:local}

The common category $Q_S$ of Section~\ref{sec:walg} receives the two
semi-infinite pairings, one for $G$ and one for $\check G$.  We now compare
these pairings over the entire space of marked points, including its
collision diagonals.  The point calculation supplies a gap in low
cohomological degrees.  That gap determines the equivalence of hearts
across a collision; derived realization then supplies the equivalence of
DG categories.  A further argument with translation of the marked points
will make the comparison compatible with insertion in both slots.

Throughout this section $a$ is irrational and $d$ is its dual shifted
level.  The spherical category $C_S$, the test category $d_S$, and their
pairing $F_S$ are those of Section~\ref{sec:loop}, with
$d_S=\mathcal W_{-s}^-(S)$ as in
\eqref{loop:whittaker-coinvariants}.  In particular, $d_S$ is a category of tests,
whereas $d$ without a subscript is a level.  Checked symbols refer to the
dual group.  We work on smooth affine coordinate charts for tuples of
points.  Extra auxiliary coordinates are allowed.  When adding a
collision diagonal, we retain the complements of any other diagonals
already deleted; thus the new diagonal is a smooth divisor and its
complement is principal affine locally.

\subsection{The heart and its derived realization}

We use the $t$-structure on $C_S$ whose heart has the perverse
$D$-module normalization in the parameter $S$.  In the underlying
quasi-coherent description $b$, its test includes the shift $[\dim S]$.
This convention is relevant to both insertion and the open--closed
arguments below.

\begin{proposition}\label{local:heart}
On the coordinate charts just described, the $t$-structure on $C_S$ is
separated and compatible with filtered colimits.  Its heart is a
Grothendieck abelian category, and there is a continuous equivalence
\begin{equation}\label{eq:L8}
                     D(C_S^\heartsuit)\simeq C_S.
\end{equation}
The functors $j^!,j_*,i_*$ are $t$-exact when $j$ is a principal open and
$i$ is a smooth closed collision diagonal.  For a smooth divisor
$i$, the functor $i^!$ has amplitude $[0,1]$.  These assertions also hold
for products of the group categories, with separate subsets of the
marks assigned to the different factors.
\end{proposition}

\begin{proof}
We first identify the connection that must be added to the
Harish--Chandra description of Section~\ref{sec:loop}.  On an affine
smooth $S$, forgetting the connection has the continuous left adjoint
of $D$-induction.  Its projection formula allows this adjunction to be
tensored with a crystal of categories.  Forgetting is conservative:
if the underlying object of $M$ vanishes, the action of the
$D$-induced structure sheaf on $M$ vanishes, and this induced object
generates $D(S)$.  The adjunction is therefore monadic.

The connection monad is the differential-operator monad with crystal
transport.  Filter it by the order of distributions on the formal
diagonal of $S$.  At every finite order the transport is the one over
that infinitesimal diagonal; the associated graded is tensoring with
$\operatorname{Sym}(T_S)$.  Combine this filtration with the
Harish--Chandra induction monad of Proposition~\ref{loop:hc-renormalization}.
In coordinates, the combined induction adds the Laurent-current
action and the derivatives on $S$, the latter taken at fixed curve
coordinate $z$.  Regular currents are stable under this derivative.
For a distribution of fixed order, the order of a vanishing current
can decrease only by a bounded amount.  Thus the formula respects
algebraic arc equivariance and smoothness.  Each associated graded
piece for distribution order is the induction in \eqref{eq:L2}
tensored with a vector bundle.  It is $t$-exact.  The increasing
filtration and passage to Ind extend this conclusion from bounded
below arc representations to the renormalized category in both
$t$-directions.

Let $\mathcal H_S$ denote the ordinary abelian category of smooth
connection Harish--Chandra modules obtained in this way, with the
specified parameter shift.  The two consecutive forgetful functors
are conservative and continuous, so their monadic descriptions give
a $t$-exact comparison
\[
                       u_C:C_S\longrightarrow D(\mathcal H_S)
\]
which is an equivalence on bounded below objects.  The heart
$\mathcal H_S$ is Grothendieck.  It is generated by the modules induced,
including their connection, from the arc representations that are
locally free over the base.  The same induction adjunctions give
bounded compact generators of $C_S$.  Forgetting further to underlying
$D$-modules, and removing the parameter shift when using left crystal
normalization, commutes with the parameter operations.

We verify conservativity also without a boundedness condition.
At a point the category $C$ is DG semisimple, with basis the Weyl
modules $M_\eta$ for actual dominant weights of $G$.  Here is the
irrational-level argument.  The Sugawara operator $L_0$ on $M_\eta$
has bottom energy equal to the finite Casimir divided by $2a$, and
its spectrum consists of this number plus nonnegative integers.
A submodule with a higher bottom energy has, at its bottom, a finite
$G$-type killed by the positive currents.  Its energy difference
would be a nonzero rational number divided by $2a$ and a positive
integer.  Irrationality excludes this.  A submodule reaching the
bottom contains the generating irreducible finite $G$-module.
Hence each $M_\eta$ is simple.  Extensions split by energy separation;
at a common bottom energy, a splitting of finite $G$-modules lifts
the bottom vectors, which the positive currents kill, and induction
splits the extension.  The statements use the algebraic energy
decomposition from PBW, and generalized eigenspaces for an extension.
The unipotent filtration of a finite arc representation now shows
that its induction is a sum of these Weyl modules.  These inductions
generate the heart.  Bounded comparison and generation by bounded
compacts give the asserted DG semisimplicity.

On a cell of fixed coincidence pattern the coordinate description
of $C_S$ is the tensor product of the corresponding point categories
with $D(S)$.  This description respects the crystal: infinitesimal
translation of centered coordinates is inner, through Sugawara
translation, since $a\ne0$.  Between nilpotent lifts its exponential
is a finite expression, acting on modules and their mappings.
Consequently $u_C$ detects objects on each cell.  The ! restrictions
to the cells are jointly conservative by the open--closed triangles,
so $u_C$ detects objects on the chart.  This proves separation.
The claimed exactness and amplitude of the parameter operations
can therefore be checked in ordinary $D$-modules.  In particular,
principal-open star extension is exact; the localization triangle
then gives the amplitude of $i^!$.  Compatibility with filtered
colimits follows from the monadic formulas.

It remains to justify unbounded realization.  We describe the
construction, since bounded comparison alone would not suffice.
An exact, sum-preserving functor from a Grothendieck heart to the
heart of a separated DG category with filtered-colimit-compatible
$t$-structure can be realized on complexes as follows.  For a
bounded above complex, shift its upper bound to degree zero and
use the additive Dold--Kan construction, followed by geometric
realization in the target.  Equivalently, use the filtered colimit
of the finite totalizations of its lower stupid truncations.
Changing the upper bound gives the same construction by the
normalized-chain suspension identification.  For an arbitrary
complex, take the filtered colimit over its good upper truncations.

The inclusion of the heart has discrete mapping spaces: negative
cohomology of a mapping complex between heart objects vanishes.
Thus the diagrams and their coherences used here are actual diagrams
in the target.  Short exact sequences of heart objects are cofiber
sequences there.  Finite totalizations compute the usual cohomology;
filtered-colimit compatibility gives the same assertion for the
successive infinite totalizations.  In particular, quasi-isomorphisms
are inverted.  The cone calculation is compatible with changing the
truncation bounds, since the resulting comparison is an isomorphism
on cohomology and the $t$-structure is separated.  This constructs an
exact continuous DG realization functor.

Apply this construction to the inverse of $u_C$ on hearts.  Its
composition with $u_C$ is the identity by the same realization formula
in the ordinary derived category.  The opposite composition is the
identity on bounded objects by bounded comparison, and hence on $C_S$
by its bounded compact generators and continuity.  This proves
\eqref{eq:L8}.  The construction is linear: tensoring with a complex
of vector spaces is computed by the same finite operations, sums,
and realizations.
\end{proof}

We will also use the corresponding statement about natural
transformations.  Suppose two continuous functors out of
$D(\mathcal H_S)$ have images of heart objects in a common subcategory
whose mapping spaces are discrete.  A natural transformation on the
heart extends uniquely by the realization formulas above.  Indeed,
apply it term by term to Dold--Kan diagrams and their truncation
colimits.  An existing transformation must equal this extension by
naturality with the colimit cocones.  This argument identifies the
mapping spaces as well as the sets of transformations.  It applies
separately in each variable of a multilinear functor, including
auxiliary scheme $D$-module variables.

\subsection{Insertion of marked points}

Let $T\subset V$ be two assignments of marks over the same base $S$.
All collisions remain allowed.  Subscripts $T$ and $V$ indicate the
corresponding categories, and $T=\varnothing$ means the unit input
$D(S)$.

\begin{lemma}\label{local:insertion}
Insertion of the unit defines $t$-exact functors
$C_T\to C_V$ and $d_T\to d_V$.  They have continuous $D(S)$-linear
right adjoints, commute with parameter operations and disjoint
products, and use the vacuum when $T=\varnothing$.
On arc-induced modules the first functor is
\begin{equation}\label{eq:L9}
 \operatorname{Ind}_{j_T}^{\ell_T}W
       \longmapsto
 \operatorname{Ind}_{j_V}^{\ell_V}(W|_{H_V}).
\end{equation}
Here restriction is along the restriction map on regular arcs.
\end{lemma}

\begin{proof}
The Lie-algebra formula factors through currents over $V$ whose
poles are allowed only at $T$.  Modulo regular currents, this
intermediate algebra has the same principal parts as the algebra
for $T$.  Inducing further gives \eqref{eq:L9}, the usual insertion
formula for localization; compare \cite{GLCII}.  The formula is
horizontal for the fixed-$z$ connection of
Section~\ref{sec:loop}.  It preserves compact generators.  Its
right adjoint is therefore continuous, and the quasi-coherent-linear
adjunction descends to the crystal.  Compact generation and the
mapping base-change statement show that the adjoint commutes with
base change as well.

We check exactness even when points collide.  Add one new point $x$
and write $w=z-x$.  If $f$ is the old divisor equation, the quotient
of Laurent currents by the intermediate algebra is exhausted by
principal parts in $w$ after inverting $f$.  At finite order the
underlying module is built from
$\mathcal O/(w^n)[f^{-1}]$, where $\mathcal O$ is the completion
for all the marks.  The quotient $\mathcal O/(w^n)$ is the finite
jet ring at $x$.  Inverting $f$ in it is equivalent to inverting
$f(x)$ in the base: their difference is nilpotent modulo $w^n$.
These modules, and hence the quotient of current algebras, are flat
over $S$.

For completeness, flatness suffices for PBW induction even if the
quotient is not projective.  Modulo the common regular lattice, the
sequence of principal-part modules is pure exact with flat quotient.
In characteristic zero the symmetric-power filtration for this
sequence is obtained from the tensor-power filtration by the
symmetric idempotent.  The associated graded induction therefore
adds a flat symmetric-algebra factor.  Alternatively, resolve
induction by the common-lattice free bar and filter by PBW degree.
For a heart input its associated graded is precisely this
symmetric-algebra bar, concentrated in heart degree zero.
Induction on the filtration and exhaustion prove exactness of the
filtered induction.  Proposition~\ref{local:heart} transfers this
calculation to $C_S$.  Iterating adds any finite set of marks;
for the empty input the assertion is ordinary PBW.

For $d_T\to d_V$, use the opposite-character Whittaker model
\eqref{loop:whittaker-coinvariants}.  On each pole bound the functor is the closed push
of the locus with no added poles.  Its continuous extension is
$t$-exact, and its right adjoint is ! restriction.  Both preserve
the Whittaker conditions: on the relevant strata the new distinct
points have zero pole contribution, with the same $q_D$-pull and
exponential; negative defect has zero restriction.  Equivalently,
one may use the projection characterization of these conditions
after \eqref{eq:W1}.  Empty insertion is the zero-pole vacuum
locus, whose Whittaker category is $D(S)$ and whose restriction
adjoint is continuous.  By ! base change these closed pushes also
agree with insertion of the Grassmannian kernels in the
$\mathbf d$ description of \eqref{eq:loop-whittaker-functional}.  All the formulas
respect disjoint products and parameter operations.
\end{proof}

\subsection{The point calculation and its cohomological gap}

At one point, write $M_\eta$ for the Weyl module with dominant weight
$\eta$, and $D_\lambda$ for the opposite-character standard in
\eqref{eq:W3}, labeled by a dominant coweight $\lambda$.  The labels
are those of the actual root datum; no passage to a simply connected
or adjoint replacement is made.

\begin{lemma}\label{local:point-gap}
The point pairing satisfies
\begin{equation}\label{eq:L10}
 R\operatorname{Hom}_{Q}
 \bigl(F(M_\eta,D_\lambda),F(M_{\eta'},D_{\lambda'})\bigr)
 \simeq
 \begin{cases}
 H^\bullet_{\mathrm{dR}}(G),&
                    (\eta,\lambda)=(\eta',\lambda'),\\
 0,&\text{otherwise}.
 \end{cases}
\end{equation}
The map induced from the input categories identifies cohomology
in degrees at most two.  Under the common $W$-module comparison,
$F(M_\eta,D_\lambda)$ is the irrational-level simple reduction
$T^\kappa_{\eta,\lambda}$, tensored with a one-dimensional vector
space.  Its essential image agrees with the image for the dual
pairing with the two labels exchanged.
\end{lemma}

\begin{proof}
Let $q$ be the balanced-tensor projection in \eqref{eq:L3}.
The monad $q^Rq$ is obtained by applying the right adjoint of
multiplication in the spherical category $\mathcal S$ to its unit,
then acting on the two inputs.  At a finite quotient $H_f$ of the
arc group by a pro-unipotent kernel this right adjoint uses
\[
              \Delta_{BH_f,*}\omega_{BH_f}[-2\dim H_f].
\]
Pulling to the two trivial $H_f$-bundles gives
$H^\bullet_{\mathrm{dR}}(H_f)=H^\bullet_{\mathrm{dR}}(G)$.
On $C$, the spherical action supported at the unit orbit tensors
the underlying Lie module by its ! fiber at the base point of
$BH$.  The semisimplicity proved in Proposition~\ref{local:heart}
therefore computes this action on distinct Weyl labels.

For the test slot, put
$a_\lambda=\langle2\rho_{\mathrm{rt}},\lambda\rangle$ and let
$\mathbf k_g$ be the kernel at the section $g=t^\lambda$.
Equations~\eqref{eq:W2} and \eqref{eq:W3}, including integration
in dimension $\delta_N+a_\lambda$, give, after retaining the
constant one-dimensional line,
\[
                   \mathbf d(\mathbf k_g)=D_\lambda[-a_\lambda].
\]
Unit-orbit convolution on this section also contributes the fiber
at the base point of $BH$: the section trivializes its arc torsor.
Combining the two actions proves \eqref{eq:L10}.  The unit of the
adjunction is the constants map.  Since $G$ is semisimple,
$H^1_{\mathrm{dR}}(G)=H^2_{\mathrm{dR}}(G)=0$, and the positive
cohomology starts in degree at least three.

We now use the irrational-level reduction theorem of
Arakawa--Frenkel~\cite[Theorems~2.1--2.3]{ArakawaFrenkel}.
Principal reduction of the
Weyl module labeled by $\eta$, with Drinfeld--Sokolov twist by the
dominant coweight $\lambda$, is concentrated in degree zero and
is the simple highest module $T^\kappa_{\eta,\lambda}$.  Under
Feigin--Frenkel duality it identifies with
$T^{\kappa^\vee}_{\lambda,\eta}$.  In our modes the character on
a simple-root current is at mode
$-1-\langle\alpha,\lambda\rangle$, the lattice is the regular
lattice, and the $W$-action is the spectral-flow action.  Simultaneous
changes of character sign conjugate the $W$-fields and leave the
statement unchanged.

To match these conventions, apply \eqref{eq:L1} to $\mathbf k_g$.
Spectral flow changes a simple-root mode $m$ to
$m-\langle\alpha,\lambda\rangle$.  Its conjugated Fock lattice
differs from the regular lattice by the determinant line and the
shift $[-a_\lambda]$ in \eqref{eq:L6}.  This shift cancels the
shift in the displayed formula for $\mathbf d(\mathbf k_g)$.
The resulting ordinary compared module is exactly the reduction
in the preceding paragraph.  These objects are compact by the
adjunctions of \eqref{eq:L3}; full faithfulness of the ordinary
$W$ comparison on compacts promotes this calculation to $Q$.

We will need its generating vector at a collision.  It is the
highest Weyl vector times the semi-infinite vacuum for the specified
lattice.  The character and highest-vector formulas make it closed.
In the DS calculation, coinvariants for the strictly negative
$N$-modes evaluate the PBW-free negative unipotent factor at the
character, so the finite-type generating vector survives.  The
remaining nonnegative cochain calculation has no degree-zero
boundaries.  Compare also the highest-weight and grading calculations in
\cite[Sections~4.1 and~4.3]{ArakawaFrenkel}.  Thus its class is nonzero and
generates the reduction.  No rigidification of its one-dimensional
line is needed.
\end{proof}

On a cell of fixed coincidence pattern, the same calculation works
with parameters.  Indeed $Q_S$ is locally the point category tensored
with $D(S)$.  Its horizontal identification is supplied by $W$
Virasoro translation: between formal lifts, exponentiation of the
translation generator in a nilpotent ideal identifies centered
coordinates.  This is an infinitesimal crystal calculation, and
requires no exponentiation on a punctured parameter space.
The image of a pair of labels is consequently its fixed compact
point object tensored by a line with connection.  To see that only
a line can occur, use mapping base change and \eqref{eq:L10}:
negative self-Ext vanishes and the degree-zero endomorphisms are
scalars.  The vacuum line is canonically the same for the two
pairings.  In \eqref{eq:L6} its generating vector is the Weyl
vacuum times the $h$-Fock vacuum, horizontal for the absolute
currents; the vacuum presentation in \eqref{eq:L7} preserves it
over families and their collision diagonals.

The same description computes right adjoints.  On coordinate
pieces of a cell the pairing is tensoring with fixed compact point
objects and invertible line connections.  Therefore the cone of
its unit on a heart input lies in degrees at least three, and
every input is a retract of the result of applying the resulting
right-adjoint monad.  Products of points give the same assertions.
Insertions simply prescribe that some labels be the vacuum.

\subsection{Extending the equivalence across collisions}

Insert the vacuum in the second slot of $F_S$ and in the first
slot of $\check F_S$, and write
\[
 F_0:C_S\longrightarrow Q_S,\qquad
 F_1:\check d_S\longrightarrow Q_S.
\]
Their right adjoints $R_0,R_1$ are continuous and $D(S)$-linear,
by Proposition~\ref{loop:pairing-adjoints} and
Lemma~\ref{local:insertion}.  Corollary~\ref{loop:parameter-operations}
identifies these adjunctions under restriction to the smooth parameter
charts.  In particular they commute with $i_*,i^!,j_*,j^!$ for the
closed collision divisor and its principal open complement, including
the units and counits.  The same assertion applies to closed graph
maps in auxiliary parameter factors.

\begin{proposition}\label{local:one-slot}
There is a canonical continuous $t$-exact equivalence
\begin{equation}\label{eq:L11}
 \alpha_S:C_S\xrightarrow{\ \sim\ }\check d_S,
                 \qquad F_1\alpha_S\simeq F_0.
\end{equation}
It is $D(S)$-linear and compatible with the parameter operations,
disjoint products, and coordinate changes.  Exchanging the two
groups gives $\check\alpha_S:\check C_S\simeq d_S$.
The same statements apply to the simultaneous opposite levels.
\end{proposition}

\begin{proof}
The proof has three stages.  We propagate two low-degree properties
across a collision divisor, use them to construct the equivalence of
hearts, and then realize that equivalence and its compatibilities.
The final passage to DG categories will also use the cell retract
property established after Lemma~\ref{local:point-gap}.

On a cell, that lemma gives the following properties for heart inputs:
\begin{enumerate}
\item \emph{Unit gap.}  The cone of $M\to R_rF_rM$ lies in degrees
      $\ge3$, for $r=0,1$ in its respective source category.
\item \emph{Cross-pairing gap.}  For $U=R_1F_0$, the object $UM$ lies in degrees $\ge0$,
      has $H^1=H^2=0$, and $H^0U$ is an exact equivalence of
      hearts.  The adjunction map
      $F_1H^0UM\to F_0M$ is an isomorphism.
\end{enumerate}
The analogous assertions hold with the subscripts reversed.
The invertible base lines in the cell calculation are included
in these equivalences.

\smallskip
\noindent\emph{Propagating the unit gap.}
We prove these properties on a larger piece of the arrangement by
adding a smooth divisor $i$, with principal complement $j$.
Assume them on both smaller pieces.  For $M$ in the heart put
$B=j^!M$ and $E=i^!M$.  Then $B$ is in the heart and $E$ has
amplitude $[0,1]$, by Proposition~\ref{local:heart} and
Proposition~\ref{whittaker:t-structure} on the two respective sides.
Apply the cone of the unit to
\[
                  i_*E\longrightarrow M\longrightarrow j_*B.
\]
The exact pushes and the known bound on the two cohomology terms
of $E$ give the unit gap on the larger piece.  It implies, by
adjunction, that each $F_r$ identifies degree-zero maps between heart inputs with
those between their images, whose negative Hom also vanishes.

\smallskip
\noindent\emph{The boundary map and the equivalence of hearts.}
On the closed piece, the induction hypothesis gives $UE$ in
degrees $\ge0$, with $H^0UE,H^1UE$ obtained from $H^0E,H^1E$
by the heart equivalence, and $H^2UE=0$.  Moreover,
\[
                        F_1\tau_{\le1}UE\simeq F_0E.
\]
Indeed this holds on the two heart terms; the gap makes their
truncations a triangle, so it holds on $E$.

Apply $U$ to the localization triangle.  Its only possible
obstruction to the cross-pairing gap is $H^1UM$.  In the original heart,
the boundary gives an epimorphism
\[
                    j_*B\longrightarrow i_*H^1E.
\]
The corresponding boundary after applying $U$ is
\[
                j_*H^0UB\longrightarrow i_*H^1UE.
\]
To identify these maps, let $D$ be a heart object on the closed
piece of the $C$ family.  Here and below $U$ on either smaller
piece denotes its corresponding adjoint composite.  There are
natural identifications of degree-zero maps
\[
\begin{aligned}
 \operatorname{Hom}_{C_S}(j_*B,i_*D)
 &\simeq \operatorname{Hom}_{Q_S}(F_0j_*B,F_0i_*D)\\
 &\simeq \operatorname{Hom}_{Q_S}
       (F_1j_*H^0UB,F_1i_*H^0UD)\\
 &\simeq \operatorname{Hom}_{\check d_S}
       (j_*H^0UB,i_*H^0UD).
\end{aligned}
\]
The outer identifications use the unit gaps just proved on the
larger piece; the middle one uses the smaller-piece comparisons
and their compatibility with the exact pushes.  Apply this chain
with $D=H^1E$.  The comparison
$F_1\tau_{\le1}UE\simeq F_0E$, followed by the truncation map
$E[1]\to H^1E$, identifies the two boundary maps.  The gaps
ensure that the maps to the indicated heart objects are unique.
The cokernel of the second boundary is supported on $i$, because
$j^!$ is exact.  By the equivalence of hearts on the closed piece,
write it as $i_*H^0UD$ for a closed-heart object $D$.  The quotient
map from $i_*H^1UE$ then corresponds to a map
$i_*H^1E\to i_*D$.  Naturality of the displayed identifications
makes its composite with the first boundary zero.  That boundary
is an epimorphism, so this map, and hence the quotient map, is zero.
The cokernel therefore vanishes.  Hence
$H^1UM=0$; the other degree bounds and $H^2UM=0$ follow from
the same triangle.

Finally, test the adjunction map $F_1H^0UM\to F_0M$ by !
restriction to both pieces.  On the open it is the known map.
On the closed piece the gap and the amplitude of $i^!$ give
\[
                         i^!H^0UM=\tau_{\le1}UE,
\]
so it is the preceding isomorphism for $E$.  Conservativity of
the two restrictions proves the assertion.  The long exact
sequence and the gap show that $H^0U$ is exact on the heart.
Repeating with the two sides reversed gives inverse heart
functors, because the unit gap identifies their degree-zero
maps.  This completes the induction through the arrangement.

\smallskip
\noindent\emph{Realization and equivalence of DG categories.}
Realize $H^0U$ using \eqref{eq:L8}.  The heart formula commutes
with sums and filtered colimits, so its realization $\alpha_S$
is continuous and $t$-exact.  The comparison of functors in
\eqref{eq:L11} extends by the natural-transformation statement
after Proposition~\ref{local:heart}: the relevant images of
heart objects have discrete mapping spaces.  This also proves
commutation with the exact operations $j^!,j_*,i_*$, first on
hearts and then on their realizations.

We verify that $\alpha_S$ is an equivalence without presupposing
a derived-realization theorem for $\check d_S$.  The mate
\[
                            \alpha_SR_0\longrightarrow R_1
\]
is an isomorphism on the cells, by their explicit description.
It remains so after successive $i_*$ and $j_*$, by their linear
adjunctions and the comparisons just established.  Open--closed
triangles show that it is an isomorphism on all objects.
Adjunction and \eqref{eq:L11} now give full faithfulness on maps
into objects in the image of $R_0$.  On every cell, every input
is a retract of such an object by the cell monad calculation.
Applying the exact pushes and the localization triangles yields
full faithfulness everywhere.  The corresponding retracts on
the other side generate $\check d_S$ under colimits and finite
extensions.  Thus the continuous fully faithful functor is also
essentially surjective.

\smallskip
\noindent\emph{Linearity and coherence.}
Carry out all the
constructions with a list of auxiliary scheme factors.  Exterior
products of heart objects are heart objects in the ordinary
module description; for the Whittaker side this follows on the
pole bounds and then by colimits.  The same gaps therefore give
the canonical comparisons for exterior products.  The comparisons
also commute with closed graph or diagonal pushes in these
auxiliary factors, by the exact heart constructions.  Their
coherences are unique there, since the relevant mapping spaces
are discrete, and realization preserves these coherences.
Taking mates gives the coherent squares for ! pulls.  Exterior
product followed by diagonal ! pull computes the $D(S)$ action;
hence these squares give $D(S)$-linearity, including its unit
and iterated-action coherences.  The original $F_i,R_i$ commute
with the units and counits of these operations, so the comparison
$F_1\alpha_S\simeq F_0$ has the same linearity.

Disjoint products give the cell comparison used to construct
$\alpha_S$, and hence the same comparison everywhere.  To check
the $t$-structures even when each disjoint block has internal
collisions, characterize nonnegative objects by their !
restrictions to the arrangement pieces, using the localization
triangle and left exactness of ! restriction and the pushes.
The product equivalence and its inverse preserve this criterion.
Uniqueness on hearts and realization give its coherence.
The same criterion makes the $t$-structures local in the curve,
including those defined by global pole models for $d_S$.
Consequently the constructions agree on étale coordinate charts
and glue.  This proves all the assertions.
\end{proof}

\subsection{Comparison with both slots inserted}

The preceding equivalence will be used inside integrated pairings.
For this purpose it must respect insertion simultaneously in both
slots.  Let $T,U$ be two colors of independent marks over a parameter
chart $S$, let $Z$ be extra neutral marks, and put
$V=T\cup U\cup Z$.  Collisions within and between all three sets
are allowed.  The common input is
$C_T\otimes_{D(S)}\check C_U$, with $D(S)$ used for an empty color.

\begin{proposition}\label{local:two-slot}
There is a canonical comparison $G_0\simeq G_1$, where
\begin{equation}\label{eq:L12}
 \begin{aligned}
 G_0(A,B)&=F_V\bigl(\operatorname{ins}A,
                   \operatorname{ins}(\check\alpha_U B)\bigr),\\
 G_1(A,B)&=\check F_V\bigl(\operatorname{ins}B,
                   \operatorname{ins}(\alpha_T A)\bigr).
 \end{aligned}
\end{equation}
It agrees with the disjoint comparison and respects $D(S)$-linearity,
coordinate changes, disjoint products, symmetries, and all diagonal
operations for merge and repetition of marks.  Vacua give the empty
inputs.  In particular, \eqref{eq:L11} is compatible with ordinary
insertion of marks.
\end{proposition}

\begin{proof}
First the collision argument constructs an autoequivalence of the
common input that intertwines $G_0$ and $G_1$.  Translation will
identify this autoequivalence with the identity.  We will then check
that the resulting comparison specializes to the prescribed vacuum
insertion map when a neutral point meets another mark.

Both functors have continuous linear right adjoints by
Proposition~\ref{loop:pairing-adjoints} and
Lemma~\ref{local:insertion}; their parameter adjunctions are those of
Corollary~\ref{loop:parameter-operations}.  To apply the proof of
Proposition~\ref{local:one-slot}, take both source families to be
$C_T\otimes_{D(S)}\check C_U$ and replace $F_0,F_1$ by $G_0,G_1$.
Proposition~\ref{local:heart} supplies the product KL $t$-structure,
its open--closed amplitudes, and derived realization.  On each cell
the point label calculation supplies the unit and cross-pairing gaps,
as well as the retracts of right-adjoint images used for the derived
equivalence.  This holds also at cross-color collisions: the point
calculation exchanges the matched labels, while the neutral
insertions designate vacua.  The divisor induction, realization,
and auxiliary-parameter argument therefore give a $t$-exact
$D(S)$-linear autoequivalence $\beta_S$, whose heart functor is
$H^0(G_1^R G_0)$, and an isomorphism
\begin{equation}\label{eq:L13}
                           G_1\beta_S\simeq G_0.
\end{equation}
On the open where different colors do not meet, neutral marks meet
neither color, and neutral marks are distinct, this autoequivalence
is canonically the identity.  Use disjoint factorization,
\eqref{eq:L11}, and the common horizontal $W$ vacuum.  This comparison
retains the line connections; it does not choose flat frames for
conformal-weight lines.  We show that the identity comparison extends
through the deleted crossings.

\smallskip
\noindent\emph{Translation of an entire color.}
We construct the comparison first on the universal affine-line chart
with independent coordinates for the marks and any auxiliary
parameters.  The parameter comparisons then pull it to the smooth
charts under consideration; curve-coordinate descent is checked
below.  If $T$ is nonempty, select one of its marks as origin for
the relative coordinates and write
\[
                           S=\mathbb A^1_y\times S_0,
\]
where $y$ translates all $T$-marks together.  In this step remove
only those collision loci not involving $T$, so their equations
are independent of $y$.  The locus of forbidden crossings involving
$T$ is the zero set of a polynomial $f$ monic in $y$.
The common input admits a $t$-compatible presentation
\[
           D(\mathbb A^1)\otimes\mathcal E(S_0).
\]
Here $\mathcal E(S_0)$ is the product KL category for the relative
positions and the other color; it has the heart and realization
properties of Proposition~\ref{local:heart}.

We justify the horizontal assertion in this presentation also at
internal collisions of $T$.  Simultaneous translation is implemented
by $L_{-1}$, the total smeared Sugawara operator.  Its bracket with
currents is the ordinary translation derivation.  Use total residue
over the marked divisor to define it.  The mode expansion and the
bracket identity hold over the parameter ring, as can be checked on
PBW generators before specialization.  In particular this operator
is horizontal in the remaining parameters at fixed curve coordinate
$z$.  One can see this directly by splitting into the regular
lattice and the isotropic Laurent complement of rational functions
with poles on the divisor and vanishing at infinity.  Infinitesimal
variation of the poles at fixed $z$ preserves both halves.  The
quadratic expression from their residue pairing, with regular modes
on the right, has the translation bracket on all the fibers and
over the parameter ring.  Normal-order sums act on smooth modules.
Subtracting this inner translation from the parameter derivative
gives the commuting derivative in the displayed product category.
The nilpotent exponential identifies the centered and absolute
crystal transports, including on formal collision diagonals.
The same constructions allow the fixed origin among the relative
coordinates of $S_0$.

\smallskip
\noindent\emph{Unique extension on constant objects.}
For $E\in\mathcal E(S_0)^\heartsuit$ set
\[
                        M=\omega_{\mathbb A^1}[-1]\boxtimes E.
\]
This is a heart object with no nonzero subobject or quotient
supported on $f=0$.  For subobjects this follows from the ordinary
module presentation: the polynomial $f$ is monic, so multiplication
by it is injective on the module underlying $M$.  For quotients,
a generator constant along $y$ cannot have nonzero image in an
$f$-torsion module.  If $f^n e=0$ and the commuting derivative
satisfies $\partial_y e=0$, differentiation
$n\deg_y(f)$ times gives a nonzero constant multiple of $e=0$.
The argument applies generator by generator, without coherence
or a uniform torsion exponent.  Vanishing on the open $f\ne0$
is exactly $f$-torsion in this module description.

The same condition holds for $\beta_S M$.  Indeed $\beta_S$ is
an exact equivalence, commutes with open restriction along with
its inverse, and hence preserves the supported subcategory.
The identity comparison over $f\ne0$ now extends uniquely to
$\beta_S M\simeq M$.  To prove this explicitly, embed both
objects in the star push of their common open restriction; the
embeddings are monomorphisms because neither object has a
supported subobject.  The quotient of either image by its
intersection with the other is supported on $f=0$, hence zero.
Their images are equal.  The same argument proves uniqueness of
the extension.

This comparison is natural in $E$.  Extend it by realization in
$\mathcal E(S_0)$, and then by $D(\mathbb A^1)$-linearity.  The latter
step is valid because the displayed product is a free
$D(\mathbb A^1)$ factor and $\omega_{\mathbb A^1}[-1]$ is its shifted
unit.  To retain $D(S_0)$-linearity, do the construction with any
list of auxiliary parameters.  Uniqueness on the external heart
objects makes it commute with exterior products and with exact
graph and diagonal pushes in those parameters.  Realization and
the free $D(\mathbb A^1)$ factor extend these identities.  Passing
to mates supplies the coherent !-pull and linearity squares,
as in Proposition~\ref{local:one-slot}.

This allows all crossings involving $T$.  Next translate $U$ to
allow its remaining crossings.  Finally translate one neutral
point at a time to allow crossings between neutral points; the
input then has a free $D(\mathbb A^1)$ factor with no color on it.
At each step the deleted equation is monic in the translating
coordinate, while the other remaining deletions lie in the base.
We have constructed an extension of $\beta_S\simeq\operatorname{Id}$
over every collision.

\smallskip
\noindent\emph{Uniqueness and changes of chart.}
Natural transformations after \eqref{eq:L13} are controlled on
hearts by Proposition~\ref{local:heart}.  An endomorphism of the
identity of a KL input is determined on the dense open.  Indeed,
its heart is epi-generated by connection-induced arc
representations locally free over the base.  PBW makes these
inducing generators torsion-free for localization to the dense
open.  A map on such a generator that vanishes there is zero;
naturality with the generating epimorphisms then detects the
transformation on every heart object.  The same argument with
auxiliary parameters and diagonal pushes proves coherent
uniqueness.  Thus the extensions just constructed are independent
of the order of translation, agree on smaller affine charts, and
glue.  Over $X$, use étale coordinates separating disjoint
clusters.  The locality in the curve proved in
Proposition~\ref{local:one-slot} identifies these constructions
with the affine-line calculation.  This gives \eqref{eq:L12},
including its symmetries and disjoint-product compatibility.

\smallskip
\noindent\emph{Diagonal operations involving a neutral point.}
The comparison now extends over the independent marking chart.  To
identify its pullback along a neutral-point diagonal with the
specified insertion comparison, we must still determine its value
on the vacuum.  The following two-point calculation does this.

Compatibility with a same-color merge follows from uniqueness:
on the resulting scheme use its dense open with separated
colors and neutrals, where the comparison is \eqref{eq:L11}.
For a neutral point merging into another mark the same uniqueness
reduces to the locus where all other resulting marks are separated.
These reductions are legitimate inside the comparison of functors:
closed diagonal push is fully faithful and $t$-exact, and the gap
and adjunction constructions commute with it.  Thus one can push
a resulting heart object to the original chart, compare there,
and apply the same heart and realization test on the diagonal.

Disjoint product reduces the remaining calculation to two points:
a neutral point $w$ merging with a neutral point or a labeled
point $x$.  Two neutral points use the common horizontal vacuum
already described.  Now let the label at $x$ have color $T$.
The first pairing uses its Weyl module and the inserted test vacuum;
the second uses the dual Weyl vacuum and the matched test label.
Write $\lambda$ for the coweight of that dual test label; the $U$
case exchanges the groups.  Work in a coordinate neighborhood and
fix $x$ by base change if convenient.  On the
Weyl side, insertion is \eqref{eq:L9} from evaluation at $x$.
On the flow side, use the Grassmannian section
$g=(z-x)^\lambda$ and the inserted Weyl vacuum.  Retain the shifts
and determinant line from Lemma~\ref{local:point-gap}.  The line
comparison in \eqref{eq:L11} pulls back from the label at $x$.

In the compact ordinary comparison \eqref{eq:L6}, both complexes
have the closed cycles given by their highest vectors and Fock
vacua.  The mode formulas \eqref{eq:L7} define them also at $w=x$.
On the flow side the change of lattice has a finite locally free
quotient at $x$, even when $w=x$, so its determinant also extends.
The positive-current and character equations proving closure
are unchanged.  If $w\ne x$, the section $g$ is regular at $w$;
its standard central splitting acts trivially on the vacuum
generator there.  The Fock vacuum and determinant formula use
the same splitting.  Thus these cycles are exactly the external
highest cycles with the vacuum at $w$, without a factor depending
on $w$.  Their classes generate also when $w=x$, by the point
reduction theorem.

Here is the specialization argument that identifies the actual
comparison map on the diagonal.  Pass to a parameter DVR transverse
to $w=x$, with uniformizer $\pi$, and denote either shifted BRST
complex by $K$.  Its terms are flat; its generic and closed fibers
have cohomology only in degree zero.  The closed degree-zero
classes lift to $K$, because the displayed cycles and their
$W$-modes generate them.  The long exact sequence of
\[
                      K\xrightarrow{\pi}K\longrightarrow K/\pi
\]
therefore shows that multiplication by $\pi$ is injective on
$H^0(K)$ and on every $H^q(K)$.  For $q=1$ use precisely the
surjectivity of the lifting of $H^0(K/\pi)$; in other degrees
injectivity follows directly from the vanishing of the adjacent
fiber group.  The generic calculation makes $H^q(K)$ torsion
for $q\ne0$, hence zero by this injectivity.  The module $H^0(K)$
is torsion-free and consequently flat over the DVR.

On the generic fiber the comparison is the fixed line comparison
at $x$ and the identity on the vacuum.  Since the generating cycles
extend and the degree-zero modules are torsion-free, the same
formula holds before specialization and at $w=x$.  This proves
the comparison on the extremal-label vector.  Degree-zero maps
between the matched label objects are just maps of the
corresponding lines with connection, so it proves the comparison
on the full two-point test locus.  Auxiliary parameters can be
included throughout.  All elementary diagonal comparisons
therefore agree; their repeated composites are coherent by the
closed-push and discrete-mapping realization argument.  A crossing
between the two colors requires no operation identifying their
colors: it is simply parameter pull of the comparison on the
independent marked-point chart.

\smallskip
\noindent\emph{Insertion for the one-slot equivalence.}
Compare $\alpha_V\operatorname{ins}$ with
$\operatorname{ins}\alpha_T$ using \eqref{eq:L11} and
\eqref{eq:L12} with the other color empty.  After applying $F_1$
at $V$, the comparison is the one already constructed.  On heart
objects the gap makes $F_1$ fully faithful in degree zero and
negative degrees, so this lifts uniquely before $F_1$; insertion
is $t$-exact by Lemma~\ref{local:insertion}.  Realization extends
the lifted comparison.  For transitivity, test on the dense
separated locus.  In the test category insertion by a closed
pole image is fully faithful, since $i^!i_*\simeq\operatorname{Id}$
on each closed bound and hence after passage to the colimit.
Thus the same uniqueness using KL generators proves transitivity.
It also proves repetition and the auxiliary linearity coherences.
All unit conventions, determinant lines, and coordinate transports
are retained in the comparisons, as required.
\end{proof}
==== END Global-quantum-geometric-Langlands-at-irrational-level//09-local-comparison.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//10-localization.tex ====
\section{Localization and finite-level periods}\label{sec:localization}

We now connect the local pairing of Section~\ref{sec:local} with a
Whittaker coefficient of global localization.  The comparison has two
steps.  This section expresses that coefficient as an exponential period
on a stack of unipotent bundles.  Sections~\ref{sec:unipotent}
and~\ref{sec:ran} identify the period with semi-infinite cohomology after
integration over additional markings.  The integration is essential:
semi-infinite cohomology at one fixed finite set is not itself the global
period.

Continue with the notation $G,A,s,C_T,d_T$ of
Sections~\ref{sec:loop}--\ref{sec:local}.  Set
\[
 \delta_N=-\chi(X,\mathfrak n_0),\qquad
 l_N=\det R\Gamma(X,\mathfrak n_0).
\]
We compose determinant lines and their dimension shifts using graded
multiplicativity, including the relative-lattice identifications in
\eqref{eq:L6}.  For a tuple $T$ over a smooth parameter scheme $S$, write
\[
 \operatorname{Loc}_T:C_T\longrightarrow
       D(L_G^s\boxtimes\mathcal O_S),\qquad
 \Lambda_T=\operatorname{Loc}_T\otimes l_N^{-1}[\delta_N].
\]
The target is the category on $Y_G\times S$; the parameter is retained.
We construct $\operatorname{Loc}_T$ below, specifying its left-module
normalization.  The functor $\operatorname{oblv}_W:Q_S\to D(S)$ is the
common underlying-complex functor of \eqref{eq:walg-forgetful} and
Theorem~\ref{walg:common-target};
its semi-infinite computation uses \eqref{eq:L6} with the same
left-crystal normalization.  The coefficient $R_T$ retains the marked
parameters, and the pairing is that of \eqref{eq:W2}.

Write $\int_{V\supseteq T}$ for relative Ran integration with dualizing
normalization.  Its charts are all tuples enlarging $T$, with their
incidence maps, including coincidences.  It can be computed by direct
images from these tuple and incidence schemes followed by the colimit
of their $!$ systems; equivalently, one uses the colimit presentation
with closed direct images.  These presentations are pseudo-proper,
so the direct images in this calculation agree for $*$ and $!$.
Consequently the integral is continuous, commutes with $!$ base change,
and satisfies Fubini for two successive enlargements.

\begin{theorem}[Period comparison]\label{loc:period-comparison}
Fix $G$ and its level as in Section~\ref{sec:loop}, and let $T$ be a
nonempty tuple over a smooth parameter scheme $S$.  There is a natural
isomorphism of continuous $D(S)$-bilinear pairings
\begin{equation}\label{eq:T1}
 \left\langle R_T\Lambda_T(M),W'\right\rangle_S
 \simeq
 \int_{V\supseteq T}
 \operatorname{oblv}_W\!\left(
 F_V(\operatorname{ins}_{T,V}M,
             \operatorname{ins}_{T,V}W')\right),
 \qquad M\in C_T,\quad W'\in d_T.
\end{equation}
It respects the ordinary and crystalline parameter comparisons,
including permutations, repetitions, and diagonal mergers of marks.
\end{theorem}

The proof occupies the remainder of this section and
Sections~\ref{sec:unipotent}--\ref{sec:ran}.

We will also use the universal homological contractibility of the Ran
space of a connected curve \cite{GaitsgoryContractibility}, in its
relative form with a prescribed tuple contained in the set.  Indeed,
union with that tuple maps ordinary Ran to the containing-set Ran;
forgetting the containment and then taking union is the identity on the
latter.  Its augmentation is thus a retract of the ordinary Ran
augmentation, also after base change.  The augmentation therefore
integrates a $!$-pullback from the base to that same object.  Several
prescribed tuples are handled by their union.  Our prescribed tuples
are nonempty.

\subsection{Level structures with moving markings}

Fix a smooth affine chart $S$ of a tuple space.  Let $D_{\mathrm{lv}}$
be either zero or an effective thickening of the entire tuple $T$;
positive multiples of the sum of its graphs form sufficient choices.
Consider the smooth group schemes over $X\times S$
\[
 \mathfrak G=(G^{P_0})^{D_{\mathrm{lv}}},\qquad
 \mathfrak N=N_0^{D_{\mathrm{lv}}}.
\]
The superscript means dilation imposing the identity along the level
divisor.  For $N_0$, root coordinates are the corresponding root lines
twisted by $-D_{\mathrm{lv}}$; their multiplication is defined because
products have at least the required vanishing.  For $G^{P_0}$,
smoothness follows in \'etale coordinates about the identity by
dividing the coordinates by a local equation of the divisor.  Away from
that divisor the group scheme is unchanged.

Let $B_G^{\mathrm{lv}}$ and $B_N^{\mathrm{lv}}$ be their bundle stacks.
They classify bundles with the indicated level framing in the inner
presentation.  In particular $B_G^0=Y_G\times S$, by contraction with
$P_0$.  Curve deformation theory makes both stacks smooth over $S$.
Write $\mathfrak g^{\mathrm{lv}}$ and
$\mathfrak n^{\mathrm{lv}}=\mathfrak n_0(-D_{\mathrm{lv}})$ for the Lie
bundles.  The twist on $B_G^{\mathrm{lv}}$ is pulled back from $Y_G$.
For the natural map
\[
 p:B_N^{\mathrm{lv}}\longrightarrow B_G^{\mathrm{lv}},
\]
its pullback is canonically trivial: the root filtration with fixed
Cartan bundle $T_0$ identifies its determinant ratio with the ratio at
$P_0$.

For a larger tuple $E\supseteq T$, let
$H_{G,E}^{\mathrm{lv}}$ and $H_{N,E}^{\mathrm{lv}}$ be the regular arc
groups of these level group schemes.  The meromorphic Lie algebras do
not change, since the level divisor is supported on $T\subseteq E$.
We use the Harish--Chandra module categories of
Proposition~\ref{loop:hc-renormalization}; on the $N$ side no character condition is imposed here.
The affine extension on the $G$ side has central scalar $-s$ and splits
canonically on $N$.

\begin{lemma}\label{loc:levels}
The level Harish--Chandra categories have the compact presentations,
continuous functors, and ordinary and crystalline parameter comparisons
of Propositions~\ref{loop:hc-renormalization} and~\ref{loop:crystal}.  Restriction between two nonzero levels supported on
the whole old tuple is fully faithful when compared through a common
deeper level.  It preserves compact objects.  These comparisons remain
valid in the presence of extra markings, including collisions with the
old ones.
\end{lemma}

\begin{proof}
First take the underlying quasi-coherent calculation, denoted by $b$ in
Section~\ref{sec:loop}.  The level subgroup is the kernel of evaluation
from regular arcs to the level divisor.  At sufficiently deep finite
jet stages this evaluation is a smooth quotient.  Its differential is
surjective by restriction between finite divisors, whose modules are
finite locally free.  Surjectivity can be checked on geometric points:
a jet of a smooth group extends over local Artin thickenings; at an
extra distinct point choose the identity.  The resolution-property
argument in Proposition~\ref{loop:hc-renormalization} applies to this kernel.  Namely, the ambient
jet group has an embedding with quasi-affine homogeneous space, and
the additional quotient imposed here is affine.  The compact
presentation and bounded-object arguments therefore carry over.

One must distinguish two uses of unipotence.  At an added point away
from $T$, the level subgroup includes full arcs and need not itself be
pro-unipotent.  The \emph{difference between deeper levels on the old
tuple}, however, is smooth unipotent.  In the coordinates
$u=P(z)$ of Section~\ref{sec:loop}, use positive powers of $u$; successive
kernels are additive groups from square-zero thickenings.  Thus
forgetting between these levels forgets strong equivariance for a
unipotent quotient.  Normalized smooth pull and affine-space de Rham
acyclicity give full faithfulness, and the finite relative Chevalley
resolution gives preservation of compactness.  Full-arc factors at the
added points do not affect this argument.

For crystal comparisons, take two lifts of the parameter tuple across
a nilpotent ideal.  Their full formal disks and meromorphic objects,
including the affine data of \eqref{eq:L1}, agree.  Pass to common
thickenings of their level divisors.  Such thickenings are cofinal by
nilpotence, and the relevant divisor quotients are finite locally free;
locally this follows by division of their monic equations.  Differences
on the old levels again have unipotent quotients, since their defining
ideals in the larger Artin jet rings are nilpotent.

To include the full-arc factors, first divide by a common deeper normal
congruence on the entire tuple $E$.  The finite relative D-module
equivariant diagrams now compare by their relative de Rham groupoids,
together with the formal Lie transport in \eqref{eq:L1}.  In particular,
this comparison uses the specified identification of the actions; an
isomorphism of reduced spaces alone would not specify a comparison of
Lie modules.  Restriction to the common level realizes the same
comparison.  The finite equivariance calculation of
Proposition~\ref{loop:strong-action}, or
the relative Chevalley resolution for a unipotent difference, identifies
the corresponding module functors.  Zero level compares directly by
the full-disk identification.  Refinement to a third common level
proves the cocycle identities, including on multiple infinitesimal
diagonals.  This gives precisely the crystal parameter operations.
\end{proof}

\subsection{The localization algebroid}

We next construct localization on smooth bundle charts.  This also
identifies the maps that will be needed when a Grassmannian kernel
moves in its parameters.

Let $U$ be a smooth scheme over the parameter base with a map to one
of the bundle stacks and with full disk framings along $E$, compatible
with its level.  Write $\underline\ell_U$ for the meromorphic Lie
algebra, with its affine extension on the $G$ side.  Infinitesimal
replacement of the framed gluing data gives the transitive algebroid
\[
 \mathcal T_U
   =T_{U/\mathrm{base}}
       \mathop{\times}_{T_{\mathrm{Framed}/\mathrm{base}}}
                   \underline\ell_U.
\]
An element is a pair $(v,x)$ for which the replacement by $x$ induces
the deformation $v$ of the framed bundle.  The bracket is the bracket
of the right action, including differentiation of Lie sections.
Its anchor kernel is the algebra of outside currents
\[
 \mathfrak g^{\mathrm{lv}}_{P,\mathrm{out}}
       =\Gamma(X-E,\mathfrak g^{\mathrm{lv}}_P),
\]
together with the center when present; the analogous assertion holds
for $N$.

Here is why this remains an ordinary algebroid calculation despite the
Tate notation.  Every infinitesimal framed deformation can be
trivialized away from $E$: that complement is affine, so its relative
$H^1$ vanishes locally on the chart.  Such a trivialization supplies
the Laurent replacement and proves transitivity.  Its ambiguity is
exactly an outside current.  The latter is a union of locally free
finite-pole pieces with locally free successive quotients, after
starting at a sufficiently large pole bound with vanishing $H^1$.
On an affine chart, split the successive quotients: the outside
currents are then a countable direct sum of finite projective modules.
The projective form of PBW therefore applies.  Full framings can be chosen by first choosing finite jets and then lifting
through smooth jet projections with unipotent kernels on affine test
schemes.  An automorphism preserving a full framing is trivial.

The determinant line splits the affine extension on outside currents:
an outside replacement identifies the bundle back with itself, and
there is a unique lift acting trivially on its determinant.  The
relative determinant of lattices gives this action.  Differentiation
on the determinant torsor gives a map from $\mathcal T_U$ to the
relative twisted differential operators $\mathscr D_U^s$, taking the
center to $-s$.

\begin{proposition}\label{loc:construction}
Localization with level is the continuous functor whose calculation
on such a chart is
\begin{equation}\label{eq:T2}
 \operatorname{Loc}_{G,E}^{\mathrm{lv}}(M)|_U
   =\mathscr D_U^s\otimes^L_{U(\mathcal T_U)}M_U,
 \qquad (v,x)m=v(m)+xm.
\end{equation}
Here $M_U$ is the module pulled to the framed chart and the enveloping
algebra uses the specified central quotient.  Omitting the center and
twist gives $\operatorname{Loc}_{N,E}^{\mathrm{lv}}$.
These functors commute with the parameter operations of
Proposition~\ref{loop:crystal}, with pullback between smooth bundle charts, and with
the deeper-level comparisons of Lemma~\ref{loc:levels}.
At zero level divisor, $\operatorname{Loc}_{G,T}^0$ is
$\operatorname{Loc}_T$.
\end{proposition}

\begin{proof}
Formula~\eqref{eq:T2} is a formula for left modules in the
angle-bracket convention.  Its underlying $\mathcal O$ calculation is
the Chevalley chain complex of the outside Lie algebra with
coefficients in $M_U$, without an additional shift inside the
angle brackets.  Indeed, as a right $U(\mathcal T_U)$-module the TDO is
the quotient by the outside kernel, using its determinant splitting.
Resolve that kernel and apply PBW; equivalently use the Spencer
resolution and the anchor exact sequence.  Projective outside
algebras of this form cause no completion: the associated graded Koszul
calculation is the direct union of the finite calculations.

For a map of smooth charts $f:U'\to U$, not necessarily a smooth map,
the algebroid on $U'$ is the inverse-image algebroid, formed by taking
the fiber product with $T_{U'/\mathrm{base}}$.  The ordinary TDO
transfer defines the pullback comparison for \eqref{eq:T2}.  On its
generating section it sends module generators to their pulls; the
relations for $(v,x)$ become differentiation and the inverse-image
algebroid action.  Derived tensor or Spencer resolutions give the
map on complexes.  On forgetting differential operators it is exactly
derived pullback of the outside chain complex, hence an isomorphism.
Composition of transfers proves coherence.

Change of a disk frame by an arc element acts on $M_U$ through its
Harish--Chandra equivariance.  On the algebroid it acts by conjugation
and the logarithmic derivative.  The strong differential equality,
with the affine splitting on frames, identifies \eqref{eq:T2} in the
two frames.  This descends the chart formulas.  The same chart
calculation proves compatibility with forgetting additional level,
where the bundle-side operation is $!$-pullback.

We give the parameter verification since it will be used for arbitrary
kernels.  Work with relative differential operators on smooth bundle
charts.  At zero level, the full frame prestack and the Lie formula
identify across infinitesimal lifts of the marked tuple.  At nonzero
level, pull both formulas to a common deeper level as in
Lemma~\ref{loc:levels}.  The two forgetful maps agree on reduction and
supply the same crystalline comparisons; their normalized pulls are
fully faithful because the level differences are unipotent torsors.
The chart formula already agrees on those pulls.  Descent and further
common refinements give all multiple-diagonal cocycles.  For twisted
differential operators, a change of determinant frame by a nil-unit
uses Kummer equivariance, that is, its logarithmic derivative with
scalar $-s$.

Ordinary changes of tuple schemes are checked by derived base change
in the same formulas.  On the inducing compact presentations, PBW and
Chevalley terms are locally flat.  Base changes between smooth schemes
have finite Tor dimension; the cofinal infinitesimal-diagonal
projections used for the crystal test are flat.  Thus these resolutions
compute the asserted comparisons.  This argument works on all smooth
charts of a bundle-stack exhaustion and their common-level pulls; it
does not require a tensor-product assertion for the limit over
non-quasi-compact charts.

Finally take the continuous extension from the compact presentations
of Proposition~\ref{loop:hc-renormalization}.  The formulas already use colimits and derived
tensors.  Their agreement with bounded-module calculations follows by
filtered colimits in bounded ranges and the displayed resolutions.
At zero level the construction is the usual Beilinson--Drinfeld
localization: twist on the frame torsor, descend by positive-loop
splitting, and induce along the uniformization algebroid.  Pullback to
a chart gives \eqref{eq:T2}, or equivalently the outside coinvariants.
This also fixes its normalization, including the ratio rigidification
at $P_0$.  The inner presentation changes only the chosen frames.
\end{proof}

The arbitrary-level localization functor and its crystalline
transformation are constructed in \cite[Section~12.1, especially
Equation~(12.4)]{GLCII}.  Its comparison with the quasi-coherent
construction after insertion and integration is
\cite[Theorems~12.1.5 and~12.1.8]{GLCII}.  We use the standard
localization construction here.  The moving-parameter and
arbitrary-kernel comparisons required below are proved by the
algebroid calculation, and the period identity \eqref{eq:T1} will be
proved in the following sections.

\subsection{Propagation and restriction}

Adding a marked point enlarges both the outside algebra and the
principal parts.  Their quotient records the same global cohomology;
this is the reason localization is unchanged by insertion.

Fix $T\subseteq U'\subseteq E$ and the level divisor supported on $T$.
Insertion in the level categories is denoted by
$\operatorname{ins}_G$ or $\operatorname{ins}_N$.  At $b$ it is
\eqref{eq:L9} with the arcs of the level group scheme.  More explicitly,
use the intermediate Laurent algebra completed along $E$ but allowing
poles only on $U'$, restrict the old module to it, and then induce to
the Laurent algebra at $E$.  The cocycle identifies under this map by
the relative-residue formula, equivalently by the relative determinant.
For each pole bound, principal parts of the intermediate algebra
modulo its regular level lattice are the same finite
divisor-supported module as at $U'$.  This statement uses the sum of
graphs and remains true at collisions.

The induction bar with the common arc subgroup gives the bounded
module formula and its maps.  Its pole quotient is flat by the jet
localization calculation following \eqref{eq:L9}, since the level Lie
bundle is locally free on the curve.  The formula on compact
presentations therefore extends continuously and is transitive.
Arc representations are pulled by the map from arcs at $E$ to those
at $U'$.  For an infinitesimal parameter comparison, the intermediate
algebra compares by inverting the old divisor equation.  Deepening a
level on $T$ leaves this Lie induction unchanged and only restricts
its Harish--Chandra equivariance.  Thus insertion has all the parameter
and level comparisons just constructed.

\begin{proposition}\label{loc:propagation}
With parameters retained, there are natural isomorphisms
\begin{equation}\label{eq:T3}
 \operatorname{Loc}_{G,U'}^{\mathrm{lv}}(M)
 \simeq
 \operatorname{Loc}_{G,E}^{\mathrm{lv}}(
                      \operatorname{ins}_G M),
\end{equation}
and the analogous isomorphisms for $N$.  They respect transitivity,
ordinary and crystalline parameter changes, and deeper levels.
\end{proposition}

\begin{proof}
On an $E$-framed chart, restriction of an outside trivialization from
$X-U'$ to $X-E$ maps the old localization algebroid to the new one.
Its Laurent vector lies in the intermediate algebra: in vector
representations, clear powers of the old divisor equation before
completing along $E$.  This also proves the assertion when points
collide.  The central splittings and the maps to TDOs agree, since
both use the determinant of the same replacement.  Sending module
generators into the induction therefore defines the comparison.

Test the map on the inducing generators of
Proposition~\ref{loop:hc-renormalization}.  Filter the
outside-chain calculation by total PBW degree, counting the number
of chain factors as well.  For an inducing representation $W$, the
associated graded is $W$ pulled to the chart, tensored with
\[
 \operatorname{Sym}\!\left(
  \operatorname{Cone}\bigl(
   \mathfrak g^{\mathrm{lv}}_{P,\mathrm{out}}
     \longrightarrow\operatorname{PP}(
                              \mathfrak g^{\mathrm{lv}}_P)
                     \bigr)\right).
\]
The outside-current term in this cone is in shift $[1]$.
The meromorphic resolution identifies the cone with
$R\Gamma(X,\mathfrak g^{\mathrm{lv}}_P)[1]$ for either tuple,
and the comparison induces the identity on this object.  The
exhaustive increasing PBW filtration gives the isomorphism.  The
argument for $N$ is identical, and all its maps are the parameter
maps of Proposition~\ref{loc:construction}.
\end{proof}

Restriction of Lie modules from $G$ to $N$, denoted
$\operatorname{Res}$, is ordinary bounded restriction on compact
inputs at $b$, with values in the renormalized $N$ category by bounded
comparison, and is then extended continuously.  One may construct it
by the map of Harish--Chandra induction monads: restrict the group
representation and use the enveloping-algebra map.  On finite
representation generators this is the ordinary restriction map;
PBW commutes with filtered colimits in bounded ranges.  The
forgetful functors of Proposition~\ref{loop:hc-renormalization}
show that it is $t$-exact at $b$.
These descriptions also prove the common-level crystal comparisons.

We record more precisely the filtration used later to control this
restriction.  For a $G$-inducing generator with representation $W$,
count only the PBW factors outside the $N$ algebra.  Its $i$th graded
piece, as an $N$ module, is
\begin{equation}\label{eq:T4}
 \operatorname{Ind}_{\operatorname{Lie}H_N^{\mathrm{lv}}}
                   ^{\ell_N^{\mathrm{loop}}}
 \left(W\otimes
  \operatorname{Sym}^i\operatorname{PP}(
                     \mathfrak g^{\mathrm{lv}}/
                     \mathfrak n^{\mathrm{lv}})\right).
\end{equation}
Here $\ell_N^{\mathrm{loop}}$ has no center and $W$ is restricted to
$H_N^{\mathrm{lv}}$.  Order $N$ operators first in PBW.  Commuting an
arc $N$ vector past an outside operator gives its adjoint action
modulo $N$ and the regular lattice; all other terms have smaller
outside count.  Commutators of outside operators, including central
terms, also lower that count.  This proves the intrinsic filtration.
The inducing representations in \eqref{eq:T4} are exhausted by
locally free pieces at finite pole and representation bounds:
finite principal-part bounds are arc-stable, and one takes their
symmetric and tensor powers with $W$.  Consequently the filtration
is valid in the renormalized comparison and commutes with scalar
extension and the crystal maps.

The enveloping formula gives a natural map
\[
 \operatorname{ins}_N\operatorname{Res}(M)
       \longrightarrow
       \operatorname{Res}(\operatorname{ins}_G M).
\]
The map of localization algebroids on $B_N^{\mathrm{lv}}$, together
with the determinant trivialization there, gives
\begin{equation}\label{eq:T5}
 \operatorname{Loc}_{N,E}^{\mathrm{lv}}(
                              \operatorname{Res}M)
       \longrightarrow
 p^!\operatorname{Loc}_{G,E}^{\mathrm{lv}}(M).
\end{equation}
These maps commute with propagation \eqref{eq:T3}, with the displayed
insertion map for restriction, and with the ordinary, crystalline,
and deeper-level comparisons.  They are maps induced by the same
derived tensor construction \eqref{eq:T2}.

\subsection{Localization of an arbitrary Grassmannian kernel}

Let $M\in C_T$ and let
$K\in D_{\mathcal L^{-s}}(\operatorname{Gr}_T)$ be supported on a
finite closed bound $Z$, which may be taken proper over $S$.  Choose a
level divisor on $T$ sufficiently deep that its congruence subgroup
acts trivially on $Z$ and that all determinant data factor through
the resulting finite jets.  Such a choice exists: for Laurent elements
in a fixed bound, conjugation carries sufficiently deep congruences
into positive arcs.  Then $K\star M$ is a module for
$H_{G,T}^{\mathrm{lv}}$.  This follows directly from \eqref{eq:L1},
using $g^{-1}H_{G,T}^{\mathrm{lv}}g\subseteq H_{G,T}^0$.
The convolution uses the positive-loop descended kernel with $!$
pairing and star integration; its unmodified-section value is $M$.

The level framing defines the modification map
\[
 t_Z:Z\times_S B_G^{\mathrm{lv}}\longrightarrow Y_G\times S.
\]
The determinant formula for this map has the factor $\mathcal L^s$
on $Z$, which cancels the twist of $K$.

\begin{proposition}\label{loc:kernel}
There is a natural isomorphism
\begin{equation}\label{eq:T6}
 \operatorname{Loc}_{G,T}^{\mathrm{lv}}(K\star M)
 \simeq
 (\operatorname{pr}_{B_G^{\mathrm{lv}}})_*
 \left(\operatorname{pr}_Z^!K\otimes^!
                         t_Z^!\operatorname{Loc}_T(M)\right).
\end{equation}
It is compatible with ordinary and crystalline parameter operations,
with deeper levels, and with insertion into $E\supseteq T$.  In the
last comparison, $K$ is taken by closed direct image to the
Grassmannian at $E$.
\end{proposition}

\begin{proof}
\emph{1. Relative gauge comparison.}
We first compare the kernels before integrating against $K$.  On a
smooth affine plot of $Z$, choose a Laurent lift $g$ and a determinant
lift $\widetilde g$.  If $f$ is a disk frame on a localization chart,
the frames $f$ and $fg$ identify their vector/Lie pairs by
\begin{equation}\label{loc:gauge}
 (v,x)\longmapsto
 \left(v,\operatorname{Ad}_{g^{-1}}x+
                   \widetilde g^{-1}v(\widetilde g)\right).
\end{equation}
This is the change in \eqref{eq:L1}, including the determinant
correction.  Hence \eqref{eq:T2} identifies the relative kernel
formulas at fixed plot parameters.  To obtain a comparison of
D-module kernels, we must also verify formal transport when the plot
parameter moves.

\emph{2. Crystalline transfer and independence of choices.}
Consider two parameter lifts across a fixed nilpotent ideal.  Use
common full frames, passing to a common deeper level if necessary.
After smooth chart refinement, the two modified bundles lift to maps
$q_1,q_2:W\to Y$ to the same smooth bundle chart, agreeing on reduction.
Choose their disk and determinant frames with matching reductions;
other choices are handled by the arc descent already proved.  The
change between the two lifts is a formal arrow of the replacement
groupoid of $A_Y=\mathcal T_Y$.  It consists of the nearby chart maps,
a trivialization of the deformation off the markings, and its Laurent
replacement in the chosen disk frames, including the determinant lift.

We describe its action to all nilpotent orders.  Locally split the
transitive anchor of $A_Y$.  Lifts of coordinate derivatives together
with outside-current directions give coordinates for formal arrows:
first move the target along the lifted coordinate directions, then
apply a formal outside automorphism.  The first movement may be
constructed recursively by its formal flow, integrating the Laurent
vector and its central component through the logarithmic derivative.
It remains a replacement arrow because the framed gluing relation is
preserved at each nilpotent order.  Equivalently, construct the outside
trivializations order by order; their obstruction vanishes by the
affineness of $X-E$.  Formal isotropy is the exponential of outside
currents, with the indicated central splitting.

The algebra of finite tangent distributions on this formal groupoid
is therefore $U(A_Y)$.  Indeed its order filtration has associated
graded the symmetric powers of the coordinate and outside-current
tangents, giving the usual Taylor proof of PBW.  Only finite Taylor
expressions are needed: the coordinate displacements are nilpotent,
and at each fixed nilpotent order a Laurent section has a finite pole
bound and uses finitely many tangent expressions.  No dual of all
functions on an infinite-dimensional formal germ is used.  Formal
composition gives the composition of these distributions.

For clarity, this identifies the derived transfer maps as well.
The two module pulls use
\[
 \mathcal O_W\otimes_{\mathcal O_Y,q_i}U(A_Y),\qquad i=1,2,
\]
as right $U(A_Y)$-modules with their inverse-image algebroid actions.
Multiply by the Taylor distribution of the formal arrow to send the
second generating section to the first transfer.  This map is right
linear.  Differentiating composition shows that it intertwines the
inverse-image algebroids by \eqref{loc:gauge}.  After tensoring with
the chart TDO, the same distribution becomes the ordinary TDO Taylor
transfer, with the determinant-frame factor.  A nil-unit change of
that lift acts by its inverse-section power in the arrow from the
second pull to the first, on both sides, because the central scalar is
$-s$.

One can verify the transfer quotient directly by PBW.  The anchor
kernel pulls back from $Y$; order its factors first.  The enveloping
transfer is then PBW-free in these kernel directions, so tensoring to
the kernel quotient computes its derived tensor as well.  Equivalently,
the associated-graded Koszul complex has no additional homology.
The resulting quotient is precisely the ordinary TDO transfer.

The result is independent of the outside trivialization used to
construct the formal arrow.  Two choices differ by a formal outside
automorphism.  On a fixed nilpotent test its Taylor distribution is
a finite unit $u$ in the enveloping algebra of the target
inverse-image algebroid, lying in the anchor-kernel subalgebra.
Its determinant-trivial lift maps to $1$ in the TDO quotient.
The inner-conjugation bimodule is canonically identified by
multiplication by $u$, with $u^{-1}$ for the reverse direction;
this intertwines the corresponding coefficient-module actions.
On the TDO quotient the identification is the identity.  These
bimodule identifications compose by multiplication of units.
Applying the functorial two-sided bar construction therefore gives
a coherent trivialization of the ambiguity after derived tensor,
including all higher homotopies.

Thus tensoring these transfer diagrams with a complex of coefficient
modules proves the required comparison, including derived relations
in compact presentations.  On the coefficient module the formal
arrow acts by the Taylor form of \eqref{eq:L1}, since its derivatives
are the stated Lie action.  Formal-arrow composition gives every
cocycle identity.  Refining charts or changing arc frames uses the
same transfer and Harish--Chandra formulas.

\emph{3. Integration and insertion.}
We have proved equality after every $!$-plot pull, with all the
infinitesimal lift comparisons.  The plot criterion proved in
Proposition~\ref{loop:strong-action} therefore identifies the parameter D-module kernels.  Localization
with parameter coefficients is continuous and $D(S)$-linear.
Star-pairing these kernels with $K$ commutes with localization, by
relative tensor and $!$ base change, and gives \eqref{eq:T6}.  The
calculation is schematic on finite bounds, so it is valid on the
smooth bundle charts used above.

Finally insert into $E$.  The modification represented by $g$ has
poles only at $T$.  In an $E$-frame its Laurent datum consequently
belongs to the intermediate meromorphic group: clear powers of the
old divisor before completing, as in
Proposition~\ref{loc:propagation}.  Conjugation identifies the
intermediate-algebra induction bars with the modified old-module
bars.  Termwise their Lie actions are intertwined by
\eqref{loc:gauge}, with the same relative determinant.  The derivative
on the added enveloping factor is the conjugation derivative; together
with the derivative on the coefficient module this is exactly
\eqref{eq:L1} on the induced module.  Hence the preceding formal Taylor
comparison also intertwines the bars.  The old level depth suffices:
only poles at the old divisor are cleared, and the new positions
carry positive arcs.  Arc changes use Harish--Chandra equivariance.
The plot comparison and star integration now prove the insertion
assertion.  At any enlarged finite bound, imposing the old pole
conditions is a closed embedding, so closed-image/$!$-pull formulas
supply the stated image of $K$.  All these comparisons are unchanged
at deeper levels.
\end{proof}

\subsection{Whittaker averaging as a unipotent period}

Localization and its kernel compatibility are now available in
families.  We use them to express the left side of \eqref{eq:T1},
before the scalar normalization in $\Lambda_T$, as one exponential
period.

Let $q_N^{\mathrm{lv}}:B_N^{\mathrm{lv}}\to S$ be projection and let
$\psi_0$ be the zero-defect function in \eqref{eq:W1}, pulled back to
this stack.  Define
\[
 \begin{split}
 \operatorname{Per}_{G,E}^{\mathrm{lv}}(M)
   &=(q_N^{\mathrm{lv}})_*
     \left(\operatorname{Exp}(-\psi_0)\otimes^!
                  p^!\operatorname{Loc}_{G,E}^{\mathrm{lv}}(M)\right),\\
 \operatorname{Per}_{N,E}^{\mathrm{lv}}(M)
   &=(q_N^{\mathrm{lv}})_*
     \left(\operatorname{Exp}(-\psi_0)\otimes^!
                    \operatorname{Loc}_{N,E}^{\mathrm{lv}}(M)\right).
 \end{split}
\]
Their projections are safe, since stabilizers are unipotent, and these
functors have the preceding parameter compatibilities.  Put
\[
 \delta=\dim(B_N^{\mathrm{lv}}/S)
             =-\chi(X,\mathfrak n^{\mathrm{lv}}).
\]
The periods with shift $[-2\delta]$ agree under deeper nonzero-level
pull comparisons.  This is the smooth normalization for the
unipotent torsors that forget the extra level.

For $W'=\mathbf d(K)$, where $K$ is supported on $Z$, the pairing
\eqref{eq:loop-whittaker-functional} expresses the unnormalized left side
as
\begin{equation}\label{eq:T7}
 (q_Z)_*\left(K\otimes^!\mu^!R_T
                              \operatorname{Loc}_T(M)\right).
\end{equation}
The maps are $q_Z:Z\to S$ and the uniformization map $\mu$ used in
the Whittaker model.

\begin{proposition}\label{loc:averaging}
For the deep level chosen in Proposition~\ref{loc:kernel},
Expression~\eqref{eq:T7} is naturally isomorphic to
\begin{equation}\label{eq:T8}
 \operatorname{Per}_{G,T}^{\mathrm{lv}}(K\star M)[-2\delta].
\end{equation}
The isomorphism respects parameter operations, increases of finite
bounds, and deeper-level comparisons.
\end{proposition}

\begin{proof}
We first recover $\mu^!R_T(F)$ on $Z$, for any family $F$ of bundle
D-modules, by finite-loop averaging.  We then pair this formula with
$K$ and exchange the two integrations; the dimensions of the unipotent
presentation give the period shift.

Let $L_l\subset LN_T$ be the compact subgroup
allowing pole order $l\operatorname{ht}(\alpha)$ times the marking
divisor in each root direction $\alpha$.  The divisor counts repeated
marks.  For fixed $l$ and $Z$, a sufficiently deep normal congruence
$J\subset L_l$ makes the action, determinant, and character formulas
factor through a finite-dimensional quotient.  Normalized star
averaging evaluated at $g$ integrates the bundle pull at $ng$
against $\operatorname{Exp}(-\chi(n))$ over that quotient, with shift
minus twice its relative dimension.  Passing to a deeper quotient
leaves the result unchanged by unipotent de Rham acyclicity.  The
$N$-equivariance of $R_T(F)$ and its counit to the bundle pull give a
natural comparison from $\mu^!R_T(F)$ to this averaging expression.

We show that the comparison is an isomorphism for sufficiently large
$l$.  Stratify $Z$ first by coincidences and then by semi-infinite orbit
type.  Only finitely many types occur: the Pl\"ucker orders are bounded
above and below by the lattice bounds in a finite collection of
representations.  On each stratum, divide the Pl\"ucker maps by their
prescribed zero orders.  This produces the genuine Borel extension
with its Cartan lattices.  Choose unipotent frames on the disks and
compare to the punctured trivialization.  Locally on the parameter
stratum the result has the form $n_0t^\lambda$, with $\lambda$ not
assumed dominant.  These representatives have bounded Laurent poles;
a finite cover of the bounded strata gives a common allowance $l$.
The calculation ignores nilpotent structure only when forming the
stratification; the comparison of D-modules is pulled by $!$ base
change, so it applies with arbitrary parameter coefficients.

The stabilizer of $t^\lambda$ is
\[
 J_\lambda=t^\lambda H_{N,T}^0t^{-\lambda}\subseteq L_l
\]
for sufficiently large $l$.  Here the superscript $0$ means zero level
divisor, hence full arcs.  The residual integration along its finite
unipotent quotient is zero unless $\chi|_{J_\lambda}=0$: the de Rham
cohomology of a nontrivial exponential on an additive root quotient
vanishes.  Testing the simple roots shows that this condition is
exactly dominance of $\lambda$.  The determinant on the stabilizer
has its canonical splitting.

For dominant $\lambda$, the quotient $L_l/J_\lambda$ presents the
genuine bundle data with Cartan bundle
$T_\lambda=T_0(-\sum_y\lambda_y y)$, modulo a smooth finite-dimensional
unipotent group.  To see this in families, enlarge the root lattices
at the markings to those defining $L_l$, leaving the group unchanged
on their complement.  On each locally closed parameter stratum $S_D$,
choose $l$ so that
$R^1\pi_*$ vanishes for every enlarged root line, where
$\pi:X\times S_D\to S_D$.  Then their $\pi_*$ are locally free and
commute with base change.  Successive vector extensions show that
the enlarged group bundles are locally trivial in families; their trivializations form a torsor for the
unipotent group of global sections.  Recovering the original lattice
is exactly the quotient $L_l/J_\lambda$.  The residue convention
makes the exponential descend to this presentation.

Enlarge $l$ once more so that the chosen $n_0$ lies in $L_l$.
Pulling this presentation to the stratum identifies normalized
averaging with the conjugated pull after star image along the
stratum map $q_D$ in \eqref{eq:W1}.  Indeed, $!$ base change gives
that correspondence; integration along its smooth unipotent
presentation fibers, together with the averaging normalization,
gives precisely $[-2\dim_{\mathrm{rel}}q_D]$.  The averaging correspondence
and the Whittaker stratum projector
in \eqref{eq:W1} are thus $!$-base changes of the same relative quotient
presentation.  Their counits induce the identity of that projector,
with the displayed dimension shift.  This proves the comparison for
arbitrary coefficient D-modules on $S_D$.  The strata are finite, so
their $!$ restrictions
prove the assertion on $Z$.  Increasing $l$ uses the same averaging
projection and trace, hence gives the same map.

Apply this calculation inside \eqref{eq:T7}.  The level choice allows
us to project the averaging variable to
$L_l/H_{N,T}^{\mathrm{lv}}$.  Its element represents a point of
$B_N^{\mathrm{lv}}$, and the product $ng$ is precisely the level
modification in \eqref{eq:T6}.  The line and phase descend with this
map.  Integration over the unipotent difference changes the
normalization to $[-2e_l]$, where
\[
 e_l=\dim(L_l/H_{N,T}^{\mathrm{lv}}).
\]
By the preceding enlarged-root argument, this quotient presents
$B_N^{\mathrm{lv}}$ modulo the unipotent group of global sections of
the enlarged bundle.  If that group has dimension $f_l$, then
$\delta=e_l-f_l$.  Integration over its presentation fibers changes
$[-2e_l]$ to $[-2e_l+2f_l]=[-2\delta]$.
The descended character is $\operatorname{Exp}(-\psi_0)$, and the
determinant splittings agree by multiplicativity along the root
filtration.  The remaining integration against $K$ is exactly
\eqref{eq:T6}, proving \eqref{eq:T8}.

All shifts here follow the fixed smooth/left convention: integrating
a dualizing pull over an affine-space fiber of dimension $e$
contributes $[2e]$.  Every calculation takes place at finite bounds
and finite congruence quotients, and the comparison maps are the
normalized pull, trace, and base-change maps.  Their compatibility
under refinement proves the asserted naturality.
\end{proof}
==== END Global-quantum-geometric-Langlands-at-irrational-level//10-localization.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//11-unipotent-period.tex ====
\section{The unipotent period map}\label{sec:unipotent}

We construct a map from semi-infinite cohomology to the period of
localization.  Its underlying map is the inclusion of constant functions
on a framed bundle space.  The main point is to identify the complexes
on both sides with their correct determinant lines, and to make this map
compatible with changing the marked divisor and its formal lift.
The character-twisted period construction of
\cite[Section~12.5.17 and Theorem~12.5.18]{GLCII}, and its extension to
reductive-group inputs in \cite[Theorem~12.8.8]{GLCII}, provide the
corresponding Ran comparison.  Here we construct the map before Ran
integration, retaining the relative and level compatibilities needed
for the family comparison.

\subsection{The normalization and the statement}

Keep the marked divisor $E$, the chosen level divisor, and the
Harish--Chandra categories of Section~\ref{sec:localization}.  Write
$h_E^0$ for the full $N_0$ arc Lie lattice and
$h=h_E^{\mathrm{lv}}$ for its level sublattice.  On either the $N$ or
the $G$ input category, define
\[
 B_E^0(M)=C^{\infty/2,h_E^0}
       \bigl(\operatorname{Lie}LN_E,M\otimes k_{-\chi}\bigr),
 \qquad
 B_E^{\mathrm{lv}}(M)=C^{\infty/2,h}
       \bigl(\operatorname{Lie}LN_E,M\otimes k_{-\chi}\bigr).
\]
These are the continuous extensions of the compact-object calculations
in \eqref{eq:L6}; additional structures are forgotten.  As in
Section~\ref{sec:loop}, after applying the forgetful functor $b$, we write the underlying
relative formulas over the smooth affine parameter scheme $S$ without
angle brackets.  Their crystalline versions on parameter
schemes use the angle-bracket convention.  The full-arc formula
$B_E^0$ has the latter interpretation directly; the level formula
requires the following relative determinant normalization.

Put
\[
 m=\operatorname{rank}(h_E^0/h),\qquad
 l^{\mathrm{lv}}=\det R\Gamma(X,\mathfrak n^{\mathrm{lv}}),\qquad
 \delta=-\chi(X,\mathfrak n^{\mathrm{lv}}).
\]
The exact sequence for the level divisor, together with change of
semi-infinite lattice, gives
\begin{equation}\label{eq:T9}
 \begin{aligned}
 \delta&=\delta_N+m,\\
 l^{\mathrm{lv}}
   &=l_N\otimes\det(h_E^0/h)^{-1},\\
 B_E^{\mathrm{lv}}
   &=B_E^0\otimes\det(h_E^0/h)[m].
 \end{aligned}
\end{equation}
These identities also hold for intermediate tuples: the quotient
$h_E^0/h$ depends only on the level divisor.

\begin{proposition}[Period map]\label{prop:unipotent-period}
For the $N$ Harish--Chandra category there is a natural continuous map
\begin{equation}\label{eq:T10}
 l^{\mathrm{lv}}[\delta]\otimes B_E^{\mathrm{lv}}
       \longrightarrow \operatorname{Per}_{N,E}^{\mathrm{lv}}.
\end{equation}
After shifting by $[-2\delta]$, its source identifies canonically with
$l_N[-\delta_N]\otimes B_E^0$.  In this normalization the map is a
map of crystalline operations, compatible with the parameter base
changes, lift comparisons, and deeper-level comparisons of
Section~\ref{sec:localization}.
\end{proposition}

We first construct the map on bounded ordinary modules over an affine
base.  Finite-dimensional localization will identify its target with
semi-infinite cohomology having an extra function-space coefficient.
We then construct the same comparison on the compact presentations of
the renormalized categories.  This second construction proves the
derived base-change assertion without an exactness assertion for
invariants under a fixed congruence subgroup.

\subsection{A finite presentation of the bundle space}

Abbreviate the bundle stack $B_N^{\mathrm{lv}}$ to $B_N$, and let
$\widetilde B_N$ denote its space of full frames along $E$.  Framed
replacement gives it a right $N$-loop action.  Specifying the full
frame already specifies the level structure, so $\widetilde B_N$ is
independent of level: it is also the space of ordinary $N_0$-bundles
with frames on these disks.

Use the height lattices $L_l$ from the proof of \eqref{eq:T8}, now
along $E$.  Enlarge $l$ until the first cohomology of every enlarged
root line vanishes, relatively over the affine base.  Set
\[
 H=H_{N,E}^{\mathrm{lv}},\qquad
 A_l=\operatorname{Lie}L_l.
\]
Let $K_l$ be the group of global sections of the enlarged unipotent
group scheme and put $k_l=\operatorname{Lie}K_l$.  It is a smooth
finite-dimensional unipotent group over the base.

\begin{lemma}\label{lem:unipotent-presentation}
Disk gluing identifies
\begin{equation}\label{eq:T11}
 B_N=[K_l\backslash L_l/H],\qquad
 \widetilde B_N=K_l\backslash L_l.
\end{equation}
For $e_l=\operatorname{rank}(A_l/h)$ and
$f_l=\operatorname{rank}k_l$, one has
\begin{equation}\label{eq:T12}
 \delta=e_l-f_l,\qquad
 l^{\mathrm{lv}}=\det(k_l)\otimes\det(A_l/h)^{-1}.
\end{equation}
The character $\chi$ on the presentation induces the function
$\psi_0$ on $B_N$.
\end{lemma}

\begin{proof}
Filter the enlarged group by root height.  Each successive quotient
is a vector group whose underlying vector bundle has vanishing
$H^1$.  Its torsors on $X$ therefore become trivial over an affine
chart of the parameter base.  Induction along this filtration
trivializes every enlarged-group torsor; two trivializations differ
by $K_l$.  Recovering the original bundle amounts to choosing its
lattice at $E$, which gives the double quotient in \eqref{eq:T11}.
With full frames the right quotient by $H$ is omitted.

The inclusion of global sections into sufficiently long jets is
fiberwise injective.  After increasing the jet order on the chosen
base chart, it realizes $K_l$ as a closed smooth subgroup of a finite
quotient of $L_l$.  Exponential coordinates compatible with root
height identify the successive quotient maps with affine-space
torsors.  Thus the quotients occurring here have smooth relative
scheme presentations.  Enlarging $l$ recovers the same framed space
with its right action.

For each root bundle the meromorphic-lattice sequence represents
$R\Gamma(X,\mathfrak n^{\mathrm{lv}})$ by global sections of the
enlarged bundle mapping to its quotient lattice at $E$.  Taking the
successive root-height pieces gives the two-term determinant
calculation
\[
 \det R\Gamma(X,\mathfrak n^{\mathrm{lv}})
       =\det(k_l)\otimes\det(A_l/h)^{-1}.
\]
Its Euler characteristic is $f_l-e_l$, proving both assertions in
\eqref{eq:T12}.  Finally, the residue calculation of \eqref{eq:W1}
identifies the character on this presentation with $\psi_0$.
\end{proof}

The construction of Lemma~\ref{lem:unipotent-presentation} uses the
root-height filtration and the underlying root bundles.  Consequently
it also applies to the family with bracket $t[-,-]$: the same
$H^1$-vanishing bounds and jet orders work over the auxiliary affine
line.  These geometric presentations and their untwisted structure
maps are equivariant for root height and for the simultaneous scaling
$t\mapsto ct$, $x\mapsto c^{-1}x$ in exponential coordinates.
Under the latter scaling the character has weight $-1$.  The
equivariance just stated will be used for the untwisted degeneration
in Section~\ref{sec:ran}.

\subsection{Localization at a finite quotient}

Choose a sufficiently deep normal congruence subgroup $J\subset L_l$
contained in $H$, with $K_l$ still embedded in $L_l/J$.  Such
subgroups are obtained by truncating the enlarged height coordinates
along the divisor.  For finitely many lattices we use a common
subgroup normal in all of them.  Bars denote the finite quotients;
in particular,
\[
 \bar Z=K_l\backslash\bar L_l,
 \qquad B_N=[\bar Z/\bar H].
\]
For an ordinary smooth module $M$, let $M[J]$ be the submodule killed
by $\operatorname{Lie}J$.  Normality makes it an
$(A_l,H)$-module, and the $M[J]$ exhaust $M$.  Initially, for a bounded
complex, apply this construction termwise.

The finite-action localization of $M[J]$ is the object
$\mathcal V_l(M[J])$ descending from the left-module formula
\begin{equation}\label{eq:T13}
 \mathscr D_{\bar Z}\otimes^L_{U(\bar A_l)}M[J].
\end{equation}
The operator map uses infinitesimal right translations on $\bar Z$;
Harish--Chandra equivariance supplies descent along $\bar H$.
This is the transitive action-algebroid localization of
\eqref{eq:T2}.

If $J'\subset J$, the map $\bar Z'\to\bar Z$ is a torsor under
$J/J'$.  The new action algebroid is the inverse image of the old
one: the added Lie quotient is exactly the relative tangent of this
torsor.  Inflation of the module is therefore pullback as an
action-algebroid module, and \eqref{eq:T13} pulls back to the formula
for $J'$.  For $l\leq l'$, choose a common $J$ and use the inclusion
of the action algebroids on the common presentation of the framed
space.  These maps supply the transition maps in both indices.

Taking the colimit first over $J$ and then over $l$ gives
$\operatorname{Loc}_{N,E}^{\mathrm{lv}}(M)$.  To check the claim,
pull to a chart with a full frame.  Formula~\eqref{eq:T13} there is
the vector/Lie-pair formula for framed changes with Laurent component
in $A_l$.  Replacing a frame modulo $J$ by a full frame only supplies
the components in $\operatorname{Lie}J$.  These components are
uniquely determined because $J$ acts freely, and they act trivially
on $M[J]$.  The stabilizer Lie kernel thus projects isomorphically to
the kernel in the finite action-algebroid calculation.  Increasing
$l$ exhausts the out-current kernel in \eqref{eq:T2}.

This also explains why the construction is valid on bounded derived
inputs.  For a fixed $l$ the kernel on a framed chart has finite rank
$f_l$.  Its Chevalley complex commutes with the exhaustive colimit
of the termwise modules $M[J]$.  The resulting bounded-input
calculation is therefore the ordinary derived Chevalley calculation
with coefficients in $M$.  Exactness of the operation $M\mapsto
M[J]$ at an individual $J$ is not needed.  The comparison to
\eqref{eq:T2} is the colimit of the corresponding maps on out-current
chains.

\subsection{The determinant and cohomological shift}

We now calculate the period of \eqref{eq:T13}.  This is the step
which determines both the normalization of the map and its direction.

\begin{lemma}\label{lem:unipotent-finite-period}
After applying $b$, the finite-action localization satisfies
\begin{equation}\label{eq:T14}
 \begin{aligned}
 &(q_N^{\mathrm{lv}})_*
   \bigl(\operatorname{Exp}(-\psi_0)
                 \otimes^!\mathcal V_l(M[J])\bigr)\\
 &\quad\simeq l^{\mathrm{lv}}[\delta]\otimes
 C^\bullet\!\left(\bar A_l,
    M[J]\otimes\mathcal O(\bar Z)_{-\chi}\right)
       \otimes\det(A_l/h)[e_l].
 \end{aligned}
\end{equation}
Functions carry the derivative action of the right translations;
the subscript $-\chi$ adds the indicated Lie character.  The
isomorphism is natural for congruence refinement and for enlargement
of the height lattice.
\end{lemma}

\begin{proof}
Let $z=\dim(\bar Z/S)$ and $a=\operatorname{rank}\bar A_l$.
Compute ordinary relative left de Rham cohomology on $\bar Z$ and
then shift by $[2\delta]$.  Smooth pullback along the
$\bar H$-presentation and unipotent de Rham acyclicity identify this
with the stated stack period.  Ordinary left de Rham cohomology uses
right tensor with $\Omega^{\mathrm{top}}_{\bar Z}$, with the
exponential twist, and shift $[-z]$.  Since $\bar Z$ is affine,
\eqref{eq:T13} thus gives Lie chains over $U(\bar A_l)$ with this
right coefficient module.

Use the invariant volume line
$\det(\bar A_l/k_l)^{-1}$.  Converting right Lie derivatives on
functions times volume to left derivatives produces the character
$-\chi$.  The groups are unipotent, so their determinant actions
have no modular character.  Finite-dimensional Lie Poincar\'e
duality converts chains to cochains, contributing
$\det(\bar A_l)[a]$.  The resulting line is
\[
 \det(\bar A_l/k_l)^{-1}\otimes\det(\bar A_l)
       =\det(k_l)
       =l^{\mathrm{lv}}\otimes\det(A_l/h).
\]
Because $z=a-f_l$ and $\delta=e_l-f_l$, the resulting shift is
\[
 2\delta-z+a=2\delta+f_l=\delta+e_l.
\]
This proves the line and shift in \eqref{eq:T14}.

It remains to identify the transition maps under this conversion.
For height enlargement the map before Poincar\'e duality is the
inclusion of Lie chains.  Afterwards it is the cochain inclusion
followed by wedging to top degree in the added lattice, with the
relative determinant and shift displayed in \eqref{eq:T14}.  On a
common finite presentation, the determinant comparisons follow from
the exact sequences for the tangent spaces.  Equivalently, the
meromorphic-lattice sequences and the vanishing of the enlarged
root-bundle $H^1$ identify $A_{l'}/A_l$ with $k_{l'}/k_l$ in these
determinant comparisons.  We always use graded determinant lines,
so the wedge maps include the signs of the graded Poincar\'e
conversion.

For congruence refinement the map is inflation of cochains together
with pullback of functions.  One can see this directly from the
Spencer calculation.  If $A$ is a transitive action algebroid on a
smooth scheme and $I=\ker(A\to T)$, its sheaf form is
\begin{equation}\label{eq:unipotent-spencer}
 \operatorname{DR}^{\mathrm{ord}}
   \bigl(\mathscr D\otimes^L_{U(A)}R\bigr)
 \simeq C^\bullet(A,R\otimes\det I)
                           [\operatorname{rank}I].
\end{equation}
Here $\det I$ carries adjoint transport.  Right tensor with top
forms gives Spencer chains; the identity
$\det A\otimes\Omega^{\mathrm{top}}=\det I$ then gives
\eqref{eq:unipotent-spencer}.  The Lie derivative on the two
determinant factors is exactly adjoint transport on $\det I$.
For smooth inverse image of the algebroid, the relative tangent
contributes ordinary smooth de Rham pullback.  Locally in smooth
coordinates this is the product Spencer formula, with ordinary
pullback of forms in the new coordinates.  The Spencer transfer
map is canonical, so these local descriptions glue.  Applying it
to $\bar Z'\to\bar Z$ proves the asserted inflation formula.
The same argument applies to derived coefficient modules: the
bimodule and Spencer resolutions use locally free tangent and Lie
sheaves, and their maps are defined before derived tensor.
\end{proof}

\subsection{The period on bounded inputs and the constant-function map}

After applying $b$, let $M$ be a bounded ordinary Harish--Chandra
complex over the fixed affine parameter scheme $S$.  Take the
double colimit in \eqref{eq:T14}.  For fixed $l$, refinement
of $J$ gives continuous cochains for the compact Lie algebra $A_l$,
using its discrete dual.  Height enlargement gives the semi-infinite
wedge relative to $h$: compact modes contribute cochains, while
modes added outside $h$ contribute the determinant conversion of
chains.  This identifies the bounded ordinary period calculation as
\begin{equation}\label{eq:unipotent-coefficient-period}
 \begin{aligned}
 \operatorname{Per}_{N,E}^{\mathrm{lv}}(M)
 &\simeq l^{\mathrm{lv}}[\delta]\otimes{}\\
 &\quad C^{\infty/2,h}\!\left(
   \operatorname{Lie}LN_E,
   M\otimes\mathcal O(\widetilde B_N)\otimes k_{-\chi}\right).
 \end{aligned}
\end{equation}
Here the functions on the full framed space are the colimit of the
functions on its finite congruence presentations.

This identification includes the differentials.  Choose a
height-compatible topological basis of root modes.  The finite
Chevalley differentials have their usual action and bracket terms.
Congruence inflation is a map of complexes because the congruence
Lie algebras are ideals in $A_l$.  For height enlargement, the terms
outside the old lattice vanish after wedging to top degree: the
old lattice is a subalgebra, and its trace on the intervening quotient
is zero.  Indeed brackets strictly raise root height, so all the
normal-ordering trace terms vanish, including those on lattice
quotients.  The differential in the colimit is therefore the
normal-ordered semi-infinite differential.  For bounded complexes
the ordinary totalization is quasi-isomorphism invariant, since the
coefficient-complex direction is bounded.

The inclusion
\[
 \mathcal O_S\longrightarrow\mathcal O(\widetilde B_N)
\]
of constant functions is equivariant for the loop action.  Applying
semi-infinite cohomology gives \eqref{eq:T10} on bounded ordinary
inputs.  We have thus constructed the map and checked its line,
shift, and congruence transitions.  To obtain the assertion for
families, we must still verify that the construction descends through
the derived presentations and is independent of the formal lift.

\subsection{Derived base change and lift comparisons}

We extend this bounded-input comparison and its constant-function
map to the renormalized category, and then verify its crystalline
compatibilities.
For fixed $l$, use the renormalized Harish--Chandra category of
$(A_l,H)$ as the colimit of its finite-quotient presentations, as in
the compact-Lie calculation in the proof of
Proposition~\ref{loop:strong-action}.  Operations on arbitrary
unbounded inputs will be defined through this categorical colimit
and continuous extension; the termwise $M[J]$ exhaustion above is
used only to identify the bounded ordinary calculation.
On a fixed finite
quotient, construct \eqref{eq:T13} and \eqref{eq:T14} with derived
tensor and the Spencer resolution.  On the cochain side allow all
further congruence refinements, with the pulled function spaces.
Lemma~\ref{lem:unipotent-finite-period} shows that this computes the
same period.  Also form the continuous cochain calculation without
the function space, again as a colimit over further refinements.
The latter transition maps need not be isomorphisms.  Constants give
a natural transformation from this second calculation to the first.

Every operation just described commutes with the relevant derived
base extensions on compact presentations.  At a finite quotient,
the Chevalley and Spencer resolutions have relatively flat terms;
the unipotent-equivariant descent calculations have the same property.
Thus the assertions follow from derived tensor and then colimit.
In the parameter diagrams one uses finite-Tor-dimension changes
between smooth bases, and flat lift changes on infinitesimal
neighborhoods.  This construction does not assert that a fixed
ordinary invariant submodule $M[J]$ commutes with a nonflat base
change.

Compatibility with changing $l$ holds at the same derived level.
For two lattices, compute on a finite quotient of a common larger
lattice and choose a common normal $J$.  Restriction to the smaller
Lie algebra is ordinary derived restriction followed by the
finite-quotient operation.  It commutes with base extension even
when its value is not compact.  For example, restriction of a free
inducing generator has a PBW filtration for the finite-rank difference
of the lattices.  Each term is a module over the smaller Lie algebra,
and their exhaustion computes the restriction and commutes with
base extension.  Equivalently, this is restriction for the induction
monads.  The algebroid inclusion and the top-wedge maps of
Lemma~\ref{lem:unipotent-finite-period} are defined on this common
finite quotient and therefore remain compatible after colimit.

Restriction from the original loop Harish--Chandra category to
$(A_l,H)$ is likewise restriction of induction-monad modules.  The
equivariance category is the same on both sides, so the monadic
calculation of Proposition~\ref{loop:hc-renormalization} makes this
restriction continuous and $t$-exact after applying $b$.  Its agreement with the bounded construction can be checked
on an inducing generator.  If $W$ is its locally free inducing
representation, then
\[
 \operatorname{Ind}_{h}^{\operatorname{Lie}LN_E}W
   =\mathop{\operatorname{colim}}_{l'\geq l}
                         \operatorname{Ind}_{h}^{A_{l'}}W.
\]
For a fixed $l'$, take $J$ normal also in $L_{l'}$ and sufficiently
deep for $W$.  The resulting PBW-flat module factors through the
finite Lie quotient.  It yields precisely the colimit and
constant-function map constructed above.

We now identify the natural transformations in these comparisons.
Each finite presentation maps naturally to the bounded
calculation by inflation and ordinary comparison; for localization
this is the action-algebroid map after pullback to full-frame charts.
On bounded inputs in the loop category, comparison with the
continuous calculation is obtained by evaluating these maps on
maps from compact objects, using the fully faithful bounded
comparison of
Proposition~\ref{loop:hc-renormalization}.  The displayed exhaustion proves that
these comparison maps are isomorphisms on inducing generators,
hence on their thick closure.  Finally, the comparisons with
\eqref{eq:T2} and \eqref{eq:L6} are Lie-algebroid inclusions and
the same wedge identifications.  Their terms on the inducing
generators are locally free or filtered unions of flat modules.
They therefore commute with the derived base changes in question.

It remains to compare lifts.  The full-frame presentation
\eqref{eq:T11} uses the absolute formal curves and is independent of
the lift.  For two lifts, compare cofinal height lattices: a pole
bound coming from either lift is contained in a sufficiently large
bound coming from the other.  Relative divisor thickenings give
smooth lattice differences, including in the nilpotent comparison
diagrams.  With these common enlargements, the presentation by
$K_l$ and the congruence refinements apply on both sides.  All
comparison maps are the inflation and top-wedge maps already proved
compatible in Lemma~\ref{lem:unipotent-finite-period}.

For a change between nonzero levels, choose $J$ sufficiently deep
for both.  The same $\bar Z$ and $\bar A_l$ then calculate the two
periods; only the quotient by the level group changes.  Smooth
pullback, with normalization $[-2\delta]$, identifies their ordinary
de Rham calculations on $\bar Z$.  On the semi-infinite side,
\eqref{eq:T9} is exactly the change of lattice.  The determinant and
dimension formulas \eqref{eq:T12} identify the two normalizations.
These comparisons are transitive, including when a common deeper
level is needed.  We obtain a map of crystalline operations.
Continuous extension from compacts completes the proof of
Proposition~\ref{prop:unipotent-period}.

\subsection{Restriction from the reductive group}

To apply the period map to a $G$ input, one must compare the
continuous $N$ calculation after restriction with the bounded BRST
calculation used to define the $G$ operation.  This comparison is
needed because restriction need not preserve compact objects.

\begin{lemma}\label{lem:unipotent-restriction}
With the restriction functor of Section~\ref{sec:localization}, there
is a natural isomorphism of continuous operations
\begin{equation}\label{eq:T15}
 B_E^0\circ\operatorname{Res}\simeq B_E^0,
\end{equation}
where the right side denotes the operation on the $G$ input category.
Consequently \eqref{eq:T10} also gives the period map for $G$ inputs,
with the same parameter and level compatibilities.
\end{lemma}

\begin{proof}
On compact $G$ inputs, restriction is computed by bounded ordinary
modules.  On a bounded $N$ module there is a natural comparison from
the continuously extended $B_E^0$ to the bounded BRST formula:
evaluate the latter on maps from compacts and use the fully faithful
bounded comparison of
Proposition~\ref{loop:hc-renormalization}.  This map is the identity on
compact $N$ inputs.

For restriction of an inducing $G$ generator, use the outside-mode
PBW filtration \eqref{eq:T4}.  Each graded term is induced from a
representation exhausted in heart degrees by finite representations.
Bounded BRST complexes commute termwise with these filtered colimits
in a bounded range of coefficient degrees.  For derived filtered
diagrams in the same range, finite truncation triangles reduce this
assertion to the assertion in the heart.  The comparison therefore
computes the exhausted graded terms and then the filtration itself.
It is an isomorphism on restricted inducing generators and hence on
the thick category of compact $G$ inputs.  This proves
\eqref{eq:T15}, including naturality, by continuous extension.

Composing \eqref{eq:T10} with the localization comparison
\eqref{eq:T5} now gives the asserted $G$ map.  The preceding proof
also justifies the bounded BRST calculations on inserted exhausted
modules used below.  It makes no comparison of ordinary and
renormalized formulas on arbitrary unbounded complexes.
\end{proof}
==== END Global-quantum-geometric-Langlands-at-irrational-level//11-unipotent-period.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//12-ran-period.tex ====
\section{Integrating the period map over the Ran space}
\label{sec:ran}

The period map of Section~\ref{sec:unipotent} becomes an isomorphism after
allowing additional marked points and integrating over them.  This is the
remaining ingredient in the pairing identity \eqref{eq:T1}.  We first prove
an incidence calculation for principal parts, then apply it to unipotent
cochains and to the PBW filtration for restriction from $G$ to $N$.
Throughout, the level divisor is supported on the original tuple $T$.
Additional markings carry zero level.  When making underlying-complex
calculations, we apply the forgetful functor $b$ of
Section~\ref{sec:loop} in the original tuple variables.  The added
markings retain their relative D-module operations.

For an intermediate tuple $U\supseteq T$, insertion and the map
\eqref{eq:T10}, or its restriction version \eqref{eq:T15}, give
\begin{equation}\label{eq:T16}
 l_N[-\delta_N]\otimes
 \int_{V\supseteq U} B^0_V\bigl(\operatorname{ins}_{U,V}M\bigr)
 \longrightarrow \operatorname{Per}^{\mathrm{lv}}_U(M)[-2\delta].
\end{equation}
Here the target is identified using propagation \eqref{eq:T3} and the Ran
trace.  The symbol $\operatorname{Per}$ denotes the $N$-period for an
$N$-input and the $G$-period for a $G$-input.  After applying $b$ in the old variables, we
may equivalently use the source
$l^{\mathrm{lv}}[\delta]\otimes B^{\mathrm{lv}}$ and unshifted periods,
by \eqref{eq:T9}.

\begin{proposition}\label{prop:ran-integrated-period}
The map \eqref{eq:T16} is an isomorphism for unipotent Harish--Chandra
inputs at every intermediate tuple $U\supseteq T$.  With $U=T$, it is
also an isomorphism for every $G$ Harish--Chandra input whenever the
level divisor is a nonzero admissible thickening of the entire tuple $T$.
These isomorphisms are
continuous, linear over the D-module category of the original parameter
scheme, and compatible with insertion, deeper levels, and the parameter
comparisons of Sections~\ref{sec:localization} and~\ref{sec:unipotent}.
\end{proposition}

The base-change convention in this statement is essential.  Integration is
relative to the old base.  After applying $b$, crystal base change computes it
by the same Ran colimits and relative de Rham operations in the added
marking schemes.  The proofs below consequently retain the crystal
structures in these added variables.

\subsection{Principal parts and incidence}

Let $J_0$ be a vector bundle on the curve over an affine base.  For an
additional tuple $V$, let $P_V(J_0)=\operatorname{PP}(J_0,V)$ be the full
relative principal parts: the filtered union over all pole orders, modulo
regular sections, pushed from the divisor support.  We equip this object
with its principal-part crystal in the varying tuple, in the
angle-bracket dualizing normalization.  On the curve set
\[
 \mathcal I_X(J_0)=
 \bigl(\mathscr D_X\otimes_{\mathcal O_X}
               (J_0\otimes\Omega_X^{-1})\bigr)^{\langle X\rangle}.
\]
All these objects and operations are relative to the affine base.  Write
$i$ for the closed incidence $z\in V$ in the product of the curve with the
tuple scheme, and $p_V$ for projection from that product to the tuple
scheme.

\begin{lemma}\label{lem:ran-principal-parts}
There is a natural identification
\begin{equation}\label{eq:T17}
 P_V(J_0)\simeq
 p_{V,*}i_*i^!\bigl(\mathcal I_X(J_0)
                         \boxtimes\omega_{\mathrm{tuple}}\bigr).
\end{equation}
Under this identification, forgetting supports is the canonical map
$P_V(J_0)\to R\Gamma(X,J_0)[1]$.  For every finite list of vector bundles
$J_1,\ldots,J_r$ and every old tuple $U$, these maps induce an isomorphism
\begin{equation}\label{eq:T18}
 \int_{V\supseteq U}\mathop{\bigotimes}\limits_{j=1}^r{}^{!} P_V(J_j)
 \xrightarrow{\ \sim\ }
 \bigotimes_{j=1}^r R\Gamma(X,J_j)[1].
\end{equation}
The assertion includes $r=0$, with the tensor units understood, and is
compatible with permutations, shifts, and symmetric powers.
\end{lemma}

\begin{proof}
Apply the localization triangle to the complement of the incidence.  This
complement is the complement of the Cartier marking divisor, with
multiplicities allowed.  In our pushforward convention the relative-curve
de Rham complex of $\mathcal I_X(J_0)$ computes $J_0[1]$.  Indeed, passing
from a left induced module to a right module tensors by $\Omega_X$;
the factor $\Omega_X^{-1}$ in the definition cancels it, and the
angle-bracket normalization supplies the smooth-curve integration shift,
exactly as in \eqref{eq:T14}.  Localizing differential operators gives the
same calculation on the open complement.  The complex with supports is
therefore the principal-part quotient.  Its map to the full de Rham
complex is the boundary map
$P_V(J_0)\to R\Gamma(X,J_0)[1]$ of the meromorphic resolution.

This is an identification of crystals, including across collisions.
Under two lifts of a tuple agreeing on reduction, localize the same
bundle by the two divisor equations.  Their supports agree on reduction,
so their localizations agree after passing to all powers of those
equations.  These cofinal pole systems give the principal-part comparison;
in local coordinates it is differentiation in the marking variable with
the curve variable fixed.  Thus the support calculation gives the
specified comparison, rather than an identification only on geometric
fibers.

For $r$ factors, use $r$ curve variables $z_1,\ldots,z_r$ and the incidence
conditions $z_j\in V$.  ! base change and K\"unneth identify the tensor
product in \eqref{eq:T18} with the corresponding product support
calculation.  No smoothness of the intersections of these incidences is
needed: each support operation is defined by its localization triangle.
Project this incidence space to $X^r$.  Its relative Ran direction consists
of finite subsets containing both $U$ and all the specified $z_j$.
Universal homological contractibility identifies integration in this
direction with the identity, by the Ran trace
\cite[Theorem~1.6.5, Section~2.5.12, and Corollary~2.5.13]
{GaitsgoryContractibility}.  The resulting morphism is precisely the
one obtained by forgetting the supports in \eqref{eq:T17}, so it is the
map displayed in \eqref{eq:T18}.

All operations commute with the finite permutation actions and with
shifts.  In characteristic zero, symmetric powers are direct summands of
tensor powers.  The same calculation therefore applies to every symmetric
power, and continuity permits their direct sum.
\end{proof}

\subsection{The unipotent calculation}

We begin by identifying the period map on induced generators.  At a tuple
$V$, put $H=H_{N,V}^{\mathrm{lv}}$ and $h=\operatorname{Lie}H$.  Let $W$ be
an inducing representation, and let
$\operatorname{ev}_V:B_N\to BH$ classify the frame torsor
$\widetilde B_N\to B_N$.  The symbols $B_N$ and $\widetilde B_N$ here
include the fixed level data, as in Section~\ref{sec:unipotent}.

\begin{lemma}\label{lem:ran-shapiro}
With continuous cochains and the Harish--Chandra convention of
\eqref{eq:T13}, there are natural identifications
\begin{equation}\label{eq:T19}
\begin{aligned}
 B_V^{\mathrm{lv}}(\operatorname{Ind}W)
   &\simeq C^\bullet(h,W),\\
 (l^{\mathrm{lv}})^{-1}[-\delta]\otimes
       \operatorname{Per}_{N,V}^{\mathrm{lv}}(\operatorname{Ind}W)
   &\simeq C^\bullet\bigl(h,W\otimes
                                  \mathcal O(\widetilde B_N)\bigr)\\
   &\simeq R\Gamma\bigl(B_N,\operatorname{ev}_V^*W\bigr).
\end{aligned}
\end{equation}
The map \eqref{eq:T10} becomes the cochain pullback induced by
$W\to W\otimes\mathcal O(\widetilde B_N)$.
\end{lemma}

\begin{proof}
Use the increasing compact Lie lattices $A_l$ from \eqref{eq:T14}.
For a fixed $l$, a finite stage of $W$, and, when present, a finite stage
of functions, choose a sufficiently deep normal quotient $J$ through
which the requisite actions factor.  Finite-dimensional Lie Shapiro then
computes induction to $A_l/J$.  In cochain form its map inserts the
inducing vectors and wedges with the top exterior power of the
complementary modes.  The latter is exactly the determinant conversion
in \eqref{eq:T14}.  Thus the comparison is the map of Lie chains defining
\eqref{eq:T10}; with functions included, the projection formula for Lie
induction gives the same map.  The character $-\chi$ restricts to zero on
$h$, so it does not change these finite Shapiro computations.

There are two compatible exhaustions.  Deepening $J$ gives cochain
inflation, and increasing $A_l$ gives the same complementary top wedge as
in the semi-infinite transition maps.  Every inducing-module element
occurs for some $l$.  For any chosen finite representation and function
stages, the quotient can then be taken sufficiently deep and normal for
all of those data.  Taking their filtered colimit proves the first two
identifications of \eqref{eq:T19}, together with the asserted map.  In
the function calculation the stages are
$\overline Z=\widetilde B_N/J$ from \eqref{eq:T11}.

In characteristic zero, algebraic cohomology of a split unipotent group
is computed by its Lie cochains, also relatively over the present base.
For completeness, resolve a representation by cofree representations of
the form functions tensored with a representation.  Their Lie cochain
complex is the algebraic de Rham complex of the unipotent group, hence is
acyclic in positive degrees; successive vector-group extensions give
the relative assertion.  Apply this at the finite quotients and then
pass to the displayed colimit.  Descent for the frame torsor gives the
last identification in \eqref{eq:T19}.

These identifications also compute the crystals in the added markings.
Across a nilpotent lift comparison, the arcs and the frame torsor use the
same formal curves.  The finite lattice bounds have common enlargements,
and the quotient levels have common refinements, so the preceding
cochain maps compare on a cofinal system.  Evaluation is the map of those
same frame torsors.  The resulting comparison is therefore the
parameter comparison, not merely the underlying pointwise Shapiro map.
\end{proof}

Since $H$ is pro-unipotent, the induced trivial representation suffices
for the generation test of \eqref{eq:L2}.  Proposition
\ref{prop:ran-integrated-period} for $N$ consequently reduces to proving
that evaluation induces an isomorphism
\begin{equation}\label{eq:T20}
 \int_{V\supseteq U}
      C^\bullet\bigl(\operatorname{Lie}H_{N,V}^{\mathrm{lv}},
                    \mathcal O\bigr)
 \longrightarrow R\Gamma(B_N,\mathcal O).
\end{equation}
Indeed, the right side in \eqref{eq:T19} is the constant family in the
added markings.  Propagation \eqref{eq:T3} identifies it with the family
from the old tuple, and integration of that ! pull is the identity.

\begin{lemma}\label{lem:ran-unipotent-cohomology}
The evaluation map \eqref{eq:T20} is an isomorphism, with its relative
parameter comparisons.
\end{lemma}

\begin{proof}
First suppose that the group is the additive group of a vector bundle
$J$ on the curve.  Residue duality identifies continuous arc cochains
with
\[
 \operatorname{Sym}\bigl(P_V(J^*\otimes\Omega_X)[-1]\bigr),
\]
where tensor operations use the ! family convention.  A two-term perfect
complex for $R\Gamma(X,J)$ presents the stack of $J$-torsors as a vector
stack.  Computing its functions by additive-group cochains gives
\[
 R\Gamma(B_J,\mathcal O)
   \simeq\operatorname{Sym}\bigl(R\Gamma(X,J)^*[-1]\bigr).
\]
Serre duality identifies $R\Gamma(X,J)^*[-1]$ with
$R\Gamma(X,J^*\otimes\Omega_X)$.  Thus Lemma
\ref{lem:ran-principal-parts}, applied degree by degree and with the
shift $[-1]$, identifies these two symmetric algebras after integration.

It also identifies their actual comparison map.  Evaluation at a finite
jet is restriction of linear complexes,
\[
 R\Gamma(X,J)\longrightarrow\Gamma(X,J|_{V_{\mathrm{thick}}}).
\]
The shifted dual is its map on cochain generators.  Serre duality and
the lattice/support exact sequence identify this dual map with forgetting
supports in \eqref{eq:T17}.  Passing to all thickened divisors gives the
full principal-part map.  Over an infinitesimal test on which two tuples
agree on reduction, their thickenings are cofinal; the restriction maps
and the dual support maps consequently give the same crystal comparison.
This checks the derived map before truncating the linear complexes, and
proves \eqref{eq:T20} for the vector group.

We pass from the vector group to $N$ by a deformation whose gradings
control specialization.  In exponential coordinates replace the bracket
on $\mathfrak n^{\mathrm{lv}}$ by $t[-,-]$, over $\mathbb A^1_t$.
The positive root-height decomposition makes this construction global.
At $t=0$ the group is the additive group of
$J=\mathfrak n^{\mathrm{lv}}$, and at $t=1$ it is the original group.
The deformation concerns only the untwisted cochains and functions in
\eqref{eq:T20}.  Both sides and its evaluation map are computed
quasi-coherently in $t$, while retaining the relative D-module operations
in the added marking schemes.

These computations commute with specialization.  On the source, the
continuous Chevalley--Eilenberg complexes use locally free discrete duals
and their filtered unions.  On the target, use the presentations
\eqref{eq:T11}, taking the root-lattice enlargement sufficiently large
that $H^1$ vanishes in each of the finitely many root pieces.  The same
choice works over $\mathbb A^1_t$, since these underlying vector bundles
are unchanged.  The height lattices, their global-section subgroups
$K_l$, and the deep congruence quotients are invariant under both scalings
used below, by their exponential-coordinate definitions.  Finite quotient
presentations and their cochain computations are flat over the $t$-line.  Common deeper quotients compare
them with the exhausted presentation.  Evaluation uses these very
presentations, so its map commutes with specialization as well.

Here is the bound that makes the special fiber decisive.  Give a
coordinate, or a linear cochain, belonging to a height-$m$ root piece
first weight $m>0$.  Give $t$ first weight zero.  A second action is
\[
             t\longmapsto c t,\qquad x\longmapsto c^{-1}x
             \quad\text{in exponential coordinates}.
\]
It acts by group isomorphisms in the family: scaling the variables by
$c^{-1}$ compensates for scaling the bracket parameter by $c$.
Thus a linear coordinate or cochain has second weight $-1$, and $t$ has
second weight $1$.  In first weight $n\geq0$, every monomial has at most
$n$ such linear factors, and hence has second weight at least $-n$.
This bound applies to arc cochains and to the function modules in the
finite unipotent presentations.  In particular, functions on
$K_l\backslash(L_l/J)$ form a graded submodule of
$\mathcal O(L_l/J)$.  The bound therefore passes to these quotient
function modules, to the cochain complexes computing the quotient
stacks, and to their cohomology.  At finite stages relative unipotent cohomology may be
computed by finite-range Lie cochains, so these are actual graded
complexes.  The first grading is a direct sum, not a completion.

All maps in the added marking variables preserve both gradings.  Relative
de Rham pushforward, followed by the Ran colimits, preserves their direct
sums and the lower bound $-n$ in every first weight.  Let $Q_n$ be the
first-weight-$n$ part of the cone of \eqref{eq:T20} for the family.  The
vector-group calculation says
$Q_n\otimes^{\mathbf L}_{\mathcal O[t]}\mathcal O=0$ at $t=0$.
The associated exact triangle says that multiplication by $t$ is an
isomorphism on $Q_n$.  On cohomology it would therefore give isomorphisms
between successive second weights.  A nonzero class could be pulled back
repeatedly to weights below $-n$, which is impossible.  Hence every $Q_n$
vanishes.  Their direct sum is the full cone, so the family map, and in
particular its specialization at $t=1$, is an isomorphism.
\end{proof}

Together with Lemma~\ref{lem:ran-shapiro}, this proves Proposition
\ref{prop:ran-integrated-period} for $N$.  The factor
$l^{\mathrm{lv}}[\delta]$ is constant in the added markings and can be
tensored through the calculation.  We next compare restriction from $G$
with localization; this supplies the passage to the reductive group.

\subsection{Restriction and the two insertion comparisons}

For $G$, start at $U=T$ with a nonzero level covering the original tuple,
as in the congruence calculation of Section~\ref{sec:localization}.
Fix any such level.  The additional depth required by a finite
Grassmannian kernel will enter only when we apply \eqref{eq:T6} below.
Combining \eqref{eq:T5}, propagation, and trace gives a map
\begin{equation}\label{eq:T21}
 \int_{U\supseteq T}
  \operatorname{Loc}_{N,U}^{\mathrm{lv}}
       \bigl(\operatorname{Res}\operatorname{ins}_{G,T,U}M\bigr)
 \longrightarrow p^!\operatorname{Loc}_{G,T}^{\mathrm{lv}}(M).
\end{equation}

The reductive comparison fits into the following square.  To display its
four vertices without repeating the insertion expressions, set
\[
 I_U=\operatorname{Res}\operatorname{ins}_{G,T,U}M,
 \qquad c_N=l_N[-\delta_N],
\]
and define
\[
\begin{aligned}
 \mathsf B_2(M)
   &=c_N\otimes\int_{T\subseteq U\subseteq V}
       B_V^0\bigl(\operatorname{ins}_{N,U,V}I_U\bigr),\\
 \mathsf B_1(M)
   &=c_N\otimes\int_{T\subseteq V}B_V^0(I_V),\\
 \mathsf P_N(M)
   &=\int_{T\subseteq U}\operatorname{Per}_{N,U}^{\mathrm{lv}}(I_U)
                                            [-2\delta],\\
 \mathsf P_G(M)
   &=\operatorname{Per}_{G,T}^{\mathrm{lv}}(M)[-2\delta].
\end{aligned}
\]
The level remains on $T$, so $\delta$ and $c_N$ are constant in the added
markings.  The maps form the commutative diagram
\begin{equation}\label{eq:ran-reductive-square}
\begin{tikzcd}[column sep=8em,row sep=4em]
 \mathsf B_2(M)
   \arrow[r,"\text{integrated $N$ case of \eqref{eq:T16}}","\sim"']
   \arrow[d,"\text{insertion comparison}"']
 &\mathsf P_N(M)
   \arrow[d,"\text{period of \eqref{eq:T21}}"]\\
 \mathsf B_1(M)
   \arrow[r,"\text{$G$ case of \eqref{eq:T16}}"']
 &\mathsf P_G(M).
\end{tikzcd}
\end{equation}
The top map is already an isomorphism by the unipotent calculation.  The
right map applies the period operation to \eqref{eq:T21}.  For the left
map, use the natural insertion comparison
\[
 \operatorname{ins}_{N,U,V}I_U\longrightarrow I_V
\]
and the integration counit for forgetting $U$.  By \eqref{eq:T9}, this is
the map \eqref{eq:T23} in the full-arc normalization displayed here.
The square commutes because \eqref{eq:T10} is natural, the restriction
map \eqref{eq:T5} commutes with propagation \eqref{eq:T3}, and the
integration traces compose; the $G$-period map uses exactly the resulting
restriction map \eqref{eq:T15}.  We will prove that the right vertical
map is an isomorphism using \eqref{eq:T21}, and then that the left
vertical map is an isomorphism using \eqref{eq:T23}.  This will prove the
bottom map is an isomorphism.

\begin{lemma}\label{lem:ran-restriction-localization}
The map \eqref{eq:T21} is an isomorphism.
\end{lemma}

\begin{proof}
Pull to bundle charts on $B_N$ and apply $b$ in the old tuple
variables.  ! base change commutes with the indicated integrations.  For this
isomorphism test we may forget differential operators in the bundle-chart
variables, while retaining the D-module operations in the added markings.
By pro-unipotence at the initial congruence level and \eqref{eq:L2}, it
suffices to use the arc-induced generator with trivial inducing
representation.  At a chart bundle $P$ with unipotent data, filter the
calculation \eqref{eq:T2} by total PBW degree, including the Lie-chain
factors.  The associated graded on the left, before integration, is
\begin{equation}\label{eq:T22}
 \operatorname{Sym}\!\left(
   \operatorname{Cone}\!\left(
       \mathfrak n^{\mathrm{lv}}_{P,\mathrm{out},U}
          \longrightarrow P_U(\mathfrak g^{\mathrm{lv}}_P)
                         \right)\right).
\end{equation}
On the right the out-current term is
$\mathfrak g^{\mathrm{lv}}_{P,\mathrm{out},U}$ instead.  Its cone is
$R\Gamma(X,\mathfrak g^{\mathrm{lv}}_P)[1]$ by the meromorphic resolution.
The comparison is induced by inclusion of out currents.

To apply integration to this filtration, we check it in the added
markings.  The bundle $P$ and its level are fixed over these variables.
Across nilpotent lifts of the added markings, its disk frames are frames
on the same formal curves.  Changes of frame belong to the regular level
arc group and preserve the PBW filtration by Harish--Chandra equivariance.
The out-current and principal-part complexes use these same comparisons.
Hence \eqref{eq:T22} is a filtered crystal calculation in the added
markings.  There is no assertion here that this is a filtration by
D-submodules in the bundle-chart variables; those variables have already
been forgotten for the isomorphism test.

Set $J_P=\mathfrak g^{\mathrm{lv}}_P/\mathfrak n^{\mathrm{lv}}_P$.
The smooth root lattices make
$0\to\mathfrak n^{\mathrm{lv}}_P\to
\mathfrak g^{\mathrm{lv}}_P\to J_P\to0$ an exact sequence of vector
bundles.  Affineness of the complement of $U$ and the principal-part
sequence give a diagram of exact triangles.  Its common first term is
$K_P=R\Gamma(X,\mathfrak n^{\mathrm{lv}}_P)[1]$; its middle terms are the
two cone-generating complexes just described; and its quotient map is
\[
        P_U(J_P)\longrightarrow R\Gamma(X,J_P)[1].
\]
More explicitly, the two rows of the diagram are
\[
\begin{array}{cccccc}
 K_P&\longrightarrow&
 \operatorname{Cone}(\mathfrak n^{\mathrm{lv}}_{P,\mathrm{out},U}
                  \to P_U(\mathfrak g^{\mathrm{lv}}_P))
 &\longrightarrow&P_U(J_P)&\longrightarrow K_P[1],\\[2pt]
 K_P&\longrightarrow&R\Gamma(X,\mathfrak g^{\mathrm{lv}}_P)[1]
 &\longrightarrow&R\Gamma(X,J_P)[1]&\longrightarrow K_P[1],
\end{array}
\]
with identity on $K_P$ and the forget-support map on the quotients.
For each symmetric degree $r$, the extension filtration has $r+1$ steps,
with pieces given by tensor products of a symmetric power of $K_P$ and a
symmetric power of the corresponding quotient.  The former is constant
in $U$; Lemma~\ref{lem:ran-principal-parts} identifies the integrated maps
on the latter.  Thus the comparison is an isomorphism in each symmetric
degree.  Finite induction through the extension filtration, and then the
exhaustive PBW colimit, prove \eqref{eq:T21}.
\end{proof}

Lemma~\ref{lem:ran-restriction-localization}, followed by the period
operation, proves that the right vertical map in
\eqref{eq:ran-reductive-square} is an isomorphism.  It remains to prove
the same assertion for the left vertical map.  By \eqref{eq:T9}, we may
remove the common invertible normalization and use the level-lattice
form of this comparison.  Thus we must show that
\begin{equation}\label{eq:T23}
\begin{aligned}
 &\int_{T\subseteq U\subseteq V}
 B_V^{\mathrm{lv}}\bigl(
    \operatorname{ins}_{N,U,V}\operatorname{Res}
                    \operatorname{ins}_{G,T,U}M\bigr)\\
 &\hspace{30mm}\longrightarrow
 \int_{T\subseteq V}B_V^{\mathrm{lv}}\bigl(
                 \operatorname{Res}\operatorname{ins}_{G,T,V}M\bigr)
\end{aligned}
\end{equation}
is an isomorphism.  Both nested integrals use the ordinary relative
parameter operations, with ! pulls and their integration counits.

\begin{lemma}\label{lem:ran-two-insertions}
The map \eqref{eq:T23} is an isomorphism.
\end{lemma}

\begin{proof}
Again use the trivially induced generator at the old congruence level.
Filter restriction by the number of outside modes as in \eqref{eq:T4}.
Insertion \eqref{eq:L9}, followed by Lemma~\ref{lem:ran-shapiro}, identifies
the map in outside degree $i$ with the integrated map of complexes
\begin{equation}\label{eq:T24}
\begin{aligned}
 C^\bullet\!\left(h_V,
       \operatorname{Sym}^{i}P_U
          (\mathfrak g^{\mathrm{lv}}/\mathfrak n^{\mathrm{lv}})\right)
 \longrightarrow
 C^\bullet\!\left(h_V,
       \operatorname{Sym}^{i}P_V
          (\mathfrak g^{\mathrm{lv}}/\mathfrak n^{\mathrm{lv}})\right).
\end{aligned}
\end{equation}
Here $h_V=\operatorname{Lie}H_{N,V}^{\mathrm{lv}}$.  The map on the
coefficients is the actual principal-part inclusion: in the inserting
induction, the intermediate Laurent vectors have poles only on $U$.
For a fixed outside degree and finite pole bound the inducing
representations in \eqref{eq:T4} are finite stages in heart degree; choose
a common sufficiently deep quotient, apply Shapiro there, and pass to the
filtered union of these stages.  This gives \eqref{eq:T24}, including its
map.  The old level remains fixed, so all these filtrations use the
formal-curve comparisons in the added variables already checked for
\eqref{eq:T22}.

We reduce \eqref{eq:T24} to tensor products of principal parts by two
finite filtrations.  First fix $i$.  The heights of the root and Cartan
pieces of $\mathfrak g/\mathfrak n$ lie in a finite interval, say
$[a,b]$; a degree-$i$ symmetric monomial has height in $[ia,ib]$.
The arc-unipotent action strictly raises height.  The resulting finite
filtration, of length at most $i(b-a)+1$, has trivial action on its
associated graded.  Thus each piece is the tensor product of
$C^\bullet(h_V,\mathcal O)$ with a fixed tensor/symmetry expression in
coefficient principal parts.

Next decompose the plain cochain complex by the positive height grading
used in Lemma~\ref{lem:ran-unipotent-cohomology}.  In first weight $n$,
cochain degree is at most $n$, since every linear cochain has positive
height at least one.  Its cochain-degree filtration is therefore finite.
By residue duality each of its terms is a tensor, shift, and alternating
summand of the principal parts of the dual root bundles with
$\Omega_X$.  Consequently the comparison on every such term is a
comparison between a finite list of principal parts taken at $U$ and at
$V$, on the left, and the same list all taken at $V$, on the right.

For these terms use the incidence proof of \eqref{eq:T18}.  Both sides
project to the same power of the curve.  On the left, $U$ must contain
$T$ and the curve variables assigned to its list; $V$ must contain $U$
and the remaining variables.  Integrating first in the larger tuple and
then in the smaller tuple contracts these two Ran directions.  On the
right, a single tuple $V$ contains $T$ and all the curve variables, and
contracts by the same trace.  The comparison is the isomorphism between
these two contractions: the inclusion of incidences $z\in U\Rightarrow
z\in V$ uses exactly the forget-support maps of \eqref{eq:T17}, whose
traces compose.

Finite induction through the cochain-degree and coefficient-height
filtrations proves the assertion in each first weight and outside
symmetric degree.  Take the direct sum of first weights and then the
exhaustive outside-mode filtration.  Continuity of the cochain,
insertion, and integration constructions permits these colimits.
This proves \eqref{eq:T23}.
\end{proof}

The two insertion comparisons finish the reductive case of Proposition
\ref{prop:ran-integrated-period}.  Every comparison was made in the
parameter operations, with common deeper levels on any finite collection
of bounds.  It follows by continuous extension that the resulting
isomorphism has all the compatibilities stated there.

\subsection{Normalization and descent from Grassmannian tests}

We now complete the proof of Theorem~\ref{loc:period-comparison}.
Start with a test
$W'=\mathbf d(K)$, where $K$ lies in a finite closed Grassmannian bound.
Choose the sufficiently deep level required by \eqref{eq:T6}.  Formula
\eqref{eq:T8} expresses the unnormalized pairing as a shifted period.
Proposition~\ref{prop:ran-integrated-period}, together with the
normalization of localization, cancels $l_N[-\delta_N]$ and gives
\begin{equation}\label{eq:T25}
       \int_{V\supseteq T}
       B^0_V\bigl(\operatorname{ins}_{T,V}(K\star M)\bigr).
\end{equation}
In particular all auxiliary level determinants and dimension shifts have
canceled; the semi-infinite lattice remaining in this formula is the full
arc lattice $h_V^0$.

The insertion-with-action assertion of \eqref{eq:T6} and the descriptions
\eqref{eq:L3} and \eqref{eq:L6} identify \eqref{eq:T25} with the right
side of \eqref{eq:T1}.  More precisely, compare $B^0$ on the
Harish--Chandra categories with the Whittaker coinvariant projection
after their forgetful maps to $\mathcal M$, followed by
$\operatorname{oblv}_W$.  After applying
$b$, the forgetful maps to $\mathcal M^0$ preserve compact objects by
the finite relative Chevalley calculation of
Proposition~\ref{loop:hc-renormalization}.  Both operations on these
compact presentations are the BRST complex \eqref{eq:L6} with lattice
$h_V^0$; continuity then gives their agreement on all inputs.  Insertion
of $\mathbf d(K)$ is the closed Grassmannian image from the tests version
of \eqref{eq:L9}.  Thus the comparison is a comparison in parameters,
including moving kernels.  Every operation used here is the finite-bound
pairing followed by continuous extension.

We must still descend the comparison from these tests to all of $d_T$.
The construction is compatible as the bounds grow: for any finite set of
bounds, take one common sufficiently deep level, and use the pull and
period comparisons when deepening further.  We have therefore constructed
the comparison after precomposition by
\[
 \mathbf d:D_{\mathcal L^{-s}}(\operatorname{Gr}_T)\longrightarrow d_T.
\]
Precomposition along this functor is fully faithful on continuous
functionals.  Indeed, its transpose is the full embedding of the
positive-character Whittaker category on the Grassmannian, by
\eqref{loop:whittaker-coinvariants}--\eqref{eq:loop-whittaker-functional}
and the normalization
\eqref{eq:W2}.  After currying, maps between the two pairings are therefore
computed fully faithfully after precomposition by $\mathbf d$.
The isomorphism just constructed, and its inverse, descend uniquely.

For a pairing valued in D-modules on a base, apply the same argument after
pairing with the dual of the parameter D-module category.  It reduces the
claim to continuous Vect-valued pairings, to which the preceding full
faithfulness applies.  The argument also applies with any list of
parameter actions, so the descended isomorphism is parameter-linear and
respects the action maps.  This proves \eqref{eq:T1} with all of its
stated compatibilities.

\subsection{The Ran pairing and reversal of levels}

To pass to independent marking tuples, write $C=C_{\mathrm{Ran}}$ and
$\check C=\check C_{\mathrm{Ran}}$.  In this subsection only, $d=d_{\mathrm{Ran}}$
denotes the tests category, rather than the numerical shifted level.
The equivalences \eqref{eq:L11}, in particular
$C\simeq\check d$ and $\check C\simeq d$, extend to Ran by their diagonal
comparisons; the categories presented by ! systems also use the colimit
by closed-image operations.  Integrating normalized localization from
tuple schemes gives continuous functors
\[
                \Lambda:C\longrightarrow A,
       \qquad \check\Lambda:\check C\longrightarrow B'.
\]
Integration here has the stack and sheaf conventions of the earlier
parameter operations; on $X^I$ it is ordinary proper de Rham pushforward.

\begin{proposition}\label{prop:ran-pairing}
For $M\in C$ and $\check M\in\check C$, there is a canonical isomorphism
of bilinear forms
\begin{equation}\label{eq:T26}
 \bigl\langle R_A\Lambda(M),\check\alpha(\check M)\bigr\rangle
 \simeq
 \bigl\langle R_{B'}\check\Lambda(\check M),\alpha(M)\bigr\rangle.
\end{equation}
It is compatible with parameter actions, permutations, and diagonals.
\end{proposition}

\begin{proof}
Represent the inputs over independent marking schemes.  On their product,
insert each list into the union and apply \eqref{eq:T1}; then integrate
over the two original marking schemes.  We check that the resulting
pairings are precisely the displayed ones.

For Whittaker coefficients, use \eqref{eq:W2} with the tests extended by
closed push from the old pole data, where no extra poles have yet been
allowed.  ! restriction of the coefficient to these old pole bounds is
its old value.  This is the closed parameter/pole comparison established
in the tests-insertion calculation after \eqref{eq:L9}; equivalently,
compute \eqref{eq:W1} away from the union of marking points, where the
auxiliary correspondences are unchanged.  On the localization side use
propagation \eqref{eq:T3} with zero level divisor at the additional
markings.  ! projection formula and base change, followed by de Rham
integration, consequently compute the first pairing in \eqref{eq:T26}.
On the other side of \eqref{eq:T1}, the identification \eqref{eq:L12}
and the common underlying-complex functor \eqref{eq:walg-forgetful}
give the second one.

These are maps on the actual marking schemes.  Their permutation and
diagonal compatibilities were retained in the parameter constructions;
the level choices can be deepened to common choices in each comparison
diagram.  The same argument works for closed pushes in tuple coordinates,
using the parameter-linear calculation and the same closed-embedding
adjunction maps.  These diagrams present the Ran categories and their
continuous functors, so the isomorphism descends to Ran as asserted.
\end{proof}

Finally, the period calculation applies with simultaneous reversal of the
characters: take negative character in \eqref{eq:L1}--\eqref{eq:L3} and
\eqref{eq:T1} on their original-twist categories, and positive character
on their opposite categories.  It also applies to any other pair of
irrational inverse shifted forms.  The proof used no order or positivity
condition on those forms.

For negated shifted forms, one must first recompute the affine levels:
the affine level is the new shifted form minus the dual Coxeter number.
The determinant exponents are then exactly those prescribed in
\eqref{eq:L1}.  One may subsequently transport the comparison by integral
powers of the determinant.  These tensor identifications respect the
formulas: the pole, Grassmannian, and Hecke maps carry their relative
determinant; the unipotent bundle and action maps use the filtration
trivialization and orbit maps of \eqref{eq:W1}; and the two powers cancel
in opposite pairings.  This is the sign convention used in
Section~\ref{sec:equivalence}.
==== END Global-quantum-geometric-Langlands-at-irrational-level//12-ran-period.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//13-equivalence.tex ====
\section{The global equivalence and its central data}
\label{sec:equivalence}

We now combine the period comparison with the Plancherel identity.  The
period comparison first produces a functor out of a localization quotient.
The two Plancherel identities, one for each group, then provide its two
inverses.  We finish by describing the compatibility with localization,
components, central characters, and central torsors.

\subsection{The four localizations}

Recall the shifted levels and determinant exponents
\[
 a=-c,\qquad d=\frac1{ra},\qquad
 k_a=a-h^\vee,\qquad k_d=d-\check h^\vee,\qquad
 s=-\frac{k_a}{2h^\vee},\qquad
 s'=-\frac{k_d}{2\check h^\vee}.
\]
Thus the four bundle categories used below are
\[
 \begin{array}{ll}
 A=D(L_G^s),& A'=D(L_G^{-s}),\\
 B=D(L_{G^\vee}^{-s'}),&B'=D(L_{G^\vee}^{s'}).
 \end{array}
\]
All contain every component of the corresponding bundle stack.  Use the
four coefficient functors
\[
 R:A\longrightarrow\mathcal D,\qquad
 R':A'\longrightarrow\mathcal D',\qquad
 S:B\longrightarrow\mathcal C,\qquad
 S':B'\longrightarrow\mathcal C'.
\]
The Whittaker twists and characters specifying their targets are
\[
 \begin{array}{c|c|c|c}
 \text{category}&\text{group}&\text{determinant exponent}&
       \text{Whittaker character}\\ \hline
 \mathcal D&G&s&+\\
 \mathcal D'&G&-s&-\\
 \mathcal C&G^\vee&-s'&-\\
 \mathcal C'&G^\vee&s'&+
 \end{array}
\]
in the notation of \eqref{eq:W1}.  The opposite-character pairings
\eqref{eq:W2} give
\[
 \mathcal D'=\mathcal D^\vee,\qquad
 \mathcal C'=\mathcal C^\vee.
\]
Here and below these equalities denote the specified equivalences, and
\(\vee\) denotes categorical duality.

The Ran form of \eqref{eq:L11} identifies
\[
 \mathcal C\simeq\operatorname{KL}_G(a)_{\operatorname{Ran}},\qquad
 \mathcal D'\simeq\operatorname{KL}_{G^\vee}(d)_{\operatorname{Ran}}.
\]
Apply the same comparison to \(-a,-d\), with negative coefficient
characters and positive test characters as at the end of
Section~\ref{sec:ran}.  Its localization exponents differ from the
required opposite exponents by one.  For example,
\[
 -\frac{-a-h^\vee}{2h^\vee}=-s+1.
\]
Tensor by \(L_G^{-1}\) on this localization side and by the opposite
power on its test side; use the corresponding transports for the dual
group.  This gives
\[
 \mathcal C'\simeq\operatorname{KL}_G(-a)_{\operatorname{Ran}},\qquad
 \mathcal D\simeq\operatorname{KL}_{G^\vee}(-d)_{\operatorname{Ran}}.
\]
We use these constructed comparisons throughout.  In particular, no
additional identification between a Kac-module duality and
\eqref{eq:W2} is required.

Let
\[
 L:\mathcal C\longrightarrow A,\qquad
 M':\mathcal D'\longrightarrow B',\qquad
 L':\mathcal C'\longrightarrow A',\qquad
 M:\mathcal D\longrightarrow B
\]
be the four normalized localization functors.  They include the integral
transports just described.  Their normalization is the one used in the
period comparison: induced-stack localization in angle brackets, tensored
with \(l_N^{-1}[\delta_N]\).

\begin{proposition}\label{equivalence:localization}
Each of \(L,M',L',M\) is a continuous localization: its right adjoint is
fully faithful.  Its categorical transpose is therefore fully faithful.
\end{proposition}

\begin{proof}
We use the arbitrary-level localization theorem
\cite[Theorem~13.4.2]{GLCII}.  For a connected reductive group over a field
of characteristic zero, a smooth complete curve, a level \(\kappa\), and
a quasi-compact open \(j:U\hookrightarrow\operatorname{Bun}_H\), this
asserts that
\[
 j^*\operatorname{Loc}_{H,\kappa}:
 \operatorname{KL}(H)_{\kappa,\operatorname{Ran}}
 \longrightarrow D\text{-}\operatorname{mod}_{\kappa}(U)
\]
is a localization.  The target carries the stack twisting associated to
that same level.  Sections~10.1--10.5 of \cite{GLCII} construct this
localization for arbitrary levels; the restriction to the critical level
begins in Section~14 of that work.

Its localization is the Beilinson--Drinfeld localization calculated by
the frame coinvariants in \eqref{eq:T2}.  The arc-equivariant frame
identification \eqref{eq:L1} identifies the affine level with the form
\(-s\operatorname{Kil}\), or its counterpart for the other three
functors.  Proposition~10.1.11 of \cite{GLCII} associates the form
\(-\operatorname{Kil}=2\operatorname{crit}\) with
\(d\log(L_H)\).  Consequently \(-s\operatorname{Kil}\) gives
\(s\,d\log(L_H)\), exactly the determinant twisting displayed here.
Thus the source, the stack twisting, and the localization functor agree
with the ones to which the cited theorem applies.  Tensoring by a
constant invertible line, shifting, and transporting by an integral
power of a determinant line are equivalences and preserve the
localization property.

By the irrational-twist boundedness result of
Section~\ref{sec:foundations}, restriction to a sufficiently large
quasi-compact open is an equivalence for each of \(A,A',B,B'\).  Since
restriction to an open satisfies \(j^*=j^!\), the cited localization
assertion therefore gives the assertion on the full bundle categories.

Finally, for a localization \(Q:\mathcal E\to\mathcal F\),
precomposition identifies functors on \(\mathcal F\) with functors on
\(\mathcal E\) that invert the morphisms inverted by \(Q\).  It is
fully faithful.  Restricting to continuous functors with values in
\(\operatorname{Vect}\) proves that
\(Q^\vee:\mathcal F^\vee\to\mathcal E^\vee\) is fully faithful.
This argument uses the ordinary localization property and does not
require the right adjoint of \(Q\) to preserve colimits.
\end{proof}

\subsection{Construction and the two inverses}

All categories in the following calculation are dualizable.  The period
identity \eqref{eq:T26}, followed by its application to the negated
shifted forms, gives natural isomorphisms
\begin{equation}\label{eq:G1}
 RL\simeq(M')^\vee(S')^\vee,
 \qquad
 R'L'\simeq M^\vee S^\vee.
\end{equation}
These are identities of continuous functors, with the family and
insertion compatibilities proved in Section~\ref{sec:ran}.

\begin{lemma}\label{equivalence:factorization}
There are continuous functors
\(f:A\to(B')^\vee\) and \(f':A'\to B^\vee\), with natural
isomorphisms
\begin{equation}\label{eq:G2}
 \begin{aligned}
 R&\simeq(M')^\vee f,&\qquad (S')^\vee&\simeq fL,\\
 R'&\simeq M^\vee f',& S^\vee&\simeq f'L'.
 \end{aligned}
\end{equation}
They are uniquely determined, together with these isomorphisms, by
\eqref{eq:G1} and the fully faithful transposes of localization.
\end{lemma}

\begin{proof}
Every object of \(A\) is isomorphic to \(L(C)\) for some
\(C\in\mathcal C\).  The first identity of \eqref{eq:G1} therefore
places every value of \(R\) in the essential image of \((M')^\vee\).
Full faithfulness supplies the factorization \(R\simeq(M')^\vee f\),
including its action on morphisms and all coherence data.  The resulting
functor is continuous: for a diagram in \(A\), the comparison from the
colimit of its \(f\)-images to the \(f\)-image of its colimit becomes an
isomorphism after applying \((M')^\vee\), because both that embedding
and \(R\) preserve colimits.  Full faithfulness detects the
isomorphism.  Cancelling the fully faithful embedding in the first
identity of \eqref{eq:G1} gives \((S')^\vee\simeq fL\).
The second identity gives \(f'\) and the other two comparisons by the
same argument, with \(L',R',M^\vee,S^\vee\) in place of
\(L,R,(M')^\vee,(S')^\vee\).
\end{proof}

Let
\[
 e_A:(A')^\vee\xrightarrow{\sim}A,
 \qquad e_B:(B')^\vee\xrightarrow{\sim}B
\]
be the pseudo-identity equivalences of Section~\ref{sec:pseudoidentity},
with shifts \([2\dim Y_G+2\delta_N(G)]\) and
\([2\dim Y_{G^\vee}+2\delta_N(G^\vee)]\), respectively.  They evaluate
a functional on the first slot of the shriek diagonal kernel with the
indicated shift.  The Plancherel identity \eqref{eq:P1} reads
\begin{equation}\label{eq:G3}
 e_A(R')^\vee R\simeq\operatorname{Id}_A,
 \qquad
 e_B(S')^\vee S\simeq\operatorname{Id}_B.
\end{equation}

\begin{proof}[Proof of the equivalence in Theorem~\ref{thm:main}]
Define
\begin{equation}\label{eq:G4}
 \Phi=e_Bf:A\longrightarrow B.
\end{equation}
We display the two inverse candidates to keep their domains and the
role of the two groups explicit:
\[
 \begin{aligned}
 \Psi_{\mathrm{left}}
   &=e_A(f')^\vee M(M')^\vee e_B^{-1}:B\longrightarrow A,\\
 \Psi_{\mathrm{right}}
   &=L(L')^\vee(f')^\vee:B\longrightarrow A.
 \end{aligned}
\]
Transposing \(R'\simeq M^\vee f'\), and using biduality, gives
\((R')^\vee\simeq(f')^\vee M\).  The first identity of
\eqref{eq:G3} and \eqref{eq:G2} consequently imply
\[
 \Psi_{\mathrm{left}}\Phi
 \simeq e_A(f')^\vee M(M')^\vee f
 \simeq\operatorname{Id}_A.
\]
Transposing \(S^\vee\simeq f'L'\) gives
\(S\simeq(L')^\vee(f')^\vee\).  The second identity of
\eqref{eq:G3} therefore implies
\[
 \Phi\Psi_{\mathrm{right}}
 \simeq e_BfL(L')^\vee(f')^\vee
 \simeq\operatorname{Id}_B.
\]
These two inverses agree, since
\[
 \Psi_{\mathrm{left}}
 \simeq\Psi_{\mathrm{left}}\Phi\Psi_{\mathrm{right}}
 \simeq\Psi_{\mathrm{right}}.
\]
Thus \(\Phi\) is fully faithful and essentially surjective.  This uses
neither a composite \(L^RL\) nor a left adjoint on arbitrary
non-holonomic Whittaker tests.

Finally,
\[
 s=\frac{c+h^\vee}{2h^\vee}
   =\frac{c-h^\vee}{2h^\vee}+1,
 \qquad
 -s'=\frac{-1/(rc)-\check h^\vee}{2\check h^\vee}.
\]
Tensoring the source category in Theorem~\ref{thm:main} by the actual
line \(L_G\) identifies it with \(A\), while \(B\) is already the
required target.  Composing this tensor equivalence with \(\Phi\)
proves the theorem in its stated determinant convention.
\end{proof}

\subsection{Localization, coefficients, and families}

\begin{theorem}\label{equivalence:compatibility}
The equivalence \(\Phi\) has the natural localization and coefficient
compatibilities
\begin{equation}\label{eq:G5}
 \begin{gathered}
 \begin{tikzcd}[ampersand replacement=\&,column sep=large,row sep=large]
 \mathcal C \arrow[r,"L"] \arrow[dr,"e_B(S')^\vee"']
   \& A \arrow[d,"\Phi"] \\
   \& B
 \end{tikzcd}
 \\[1ex]
 R\simeq(M')^\vee e_B^{-1}\Phi.
 \end{gathered}
\end{equation}
They hold with the full Ran family structures: they commute with
collisions, insertion of vacua, disjoint products, and the base operations
in the local comparison and period construction.  The first
compatibility uniquely determines \(\Phi\), with its specified
comparison, on the localization quotient.
\end{theorem}

\begin{proof}
Both identities follow from \eqref{eq:G2} and the definition of
\(\Phi\).  Their maps are obtained by lifting the period pairings
through fully faithful embeddings.  Such a lift is unique with its
comparison, so the family compatibilities of \eqref{eq:T26} lift as
well: each coherence diagram becomes the already commuting period
diagram after applying the embedding.  Finally, precomposition with
the localization \(L\) is fully faithful on functor categories, which
proves the uniqueness assertion.
\end{proof}

This gives the unramified localization--Whittaker--Poincar\'e diagram
formulation, as in \cite[Section~4]{GaitsgoryQGL}, with all signs and
shifts fixed.  For any list of distinct insertion points, the
identification \eqref{eq:L11} matches the corresponding KL modules
with the dual-group Whittaker modules.  On that fixed-point category,
the fixed-mark comparison following \eqref{eq:P5} identifies
\(e_B(S')^\vee\) with holonomic shriek Poincar\'e and its continuous
extension.  It is shriek pushforward from the pole model, or from
actual reductions on its strata, with their exponential factors.
The description \eqref{eq:W3} determines the functor on the whole
fixed-point category.  Weyl modules match Whittaker standards with the
label lines of \eqref{eq:L11}; the zero label supplies the vacuum
comparison.

The localization in \eqref{eq:G5} includes
\(l_N^{-1}[\delta_N]\).  With that last normalization removed, its
right-hand side is correspondingly tensored by \(l_N[-\delta_N]\).
The KL comparison uses inverse shifted forms.  After the frame sign and
Cartan shifts in \eqref{eq:L1} and the integral transport above, it
becomes the negative-reciprocal relation between the determinant twists
in Theorem~\ref{thm:main}.  In particular, the compatibility identifies
a functor with specified natural maps, rather than merely identifying
the two categories abstractly.  Its canonicity is relative to the fixed
pinnings, Whittaker characters, orientation normalizations, and theta
characteristic.

\subsection{Components and central characters}

The actual root data give canonical identifications
\[
 X^*(Z_G)\simeq\pi_1(G^\vee),\qquad
 \pi_1(G)\simeq X^*(Z_{G^\vee}).
\]
We use them without passing to a simply connected or adjoint form.
Because the adjoint determinant line has trivial central inertia,
each bundle category decomposes by a central character and a component.

\begin{proposition}\label{equivalence:sectors}
If a source object has central character
\(\eta\in X^*(Z_G)\) and component \(p\in\pi_1(G)\), its image
under \(\Phi\) has central character \(p\) and component \(-\eta\).
Thus, writing character first and component second,
\begin{equation}\label{eq:G6}
 (\eta,p)\longmapsto(p,-\eta).
\end{equation}
\end{proposition}

\begin{proof}
The KL categories over tuples are graded by total central character.
The constant central arcs act using their canonical splitting.  The
localization frame formulas retain this action: changing a frame on
the right by \(h\) acts on its section generators by \(h\), with the
same sign as the infinitesimal formula \eqref{eq:L1}.  Localization
therefore carries the KL central character to bundle central inertia,
without inversion.

In the matched Whittaker cell a label \(\lambda\) gives Cartan bundle
\[
 T_D=T_0\Bigl(-\sum_i\lambda_i x_i\Bigr).
\]
Its component is \(-\sum_i\lambda_i\), since \(2\rho\) lies in the
coroot lattice.  Hence the local comparison sends the total KL central
character to the negative bundle component.  This verification on
cells gives the assertion on all objects: the two decompositions are
already defined on the full Grassmannian, and the open and closed
shriek restrictions to its strata jointly detect the assertion.
Parameter operations preserve both decompositions, so it also holds
on the Ran colimit.

Apply the first identity of \eqref{eq:G5}.  Poincar\'e with the shriek
diagonal preserves bundle components.  A KL input of central character
\(\eta\) therefore gives target component \(-\eta\).  Every source
object of character \(\eta\) is localized from that KL character
sector: localization is essentially surjective and preserves the
character decomposition, so projecting any preimage to the sector
produces a preimage there.  This proves the target component assertion.

For the target character, use \(R\simeq(M')^\vee f\).  The
coefficient functor \(R\) preserves the source bundle component
\(p\).  Under the dual-group local comparison, the part of
\(\mathcal D'\) with that component is the KL character sector
\(-p\).  Since \(M'\) realizes its character sectors, \(f\) sends
a source object of component \(p\) to a functional on \(B'\)
supported on central character \(-p\).  The two factors of the
shriek diagonal kernel have opposite central characters, by its
simultaneous central inertia and the diagonal splitting of the twist.
Evaluation by \(e_B\) consequently gives central character \(p\)
on \(B\), as asserted.
\end{proof}

\subsection{Central torsors and transgression}

We spell out the torsor compatibility because it records the global
form as a groupoid action, including its automorphisms.  Put \(Z=Z_G\).
For a \(Z\)-torsor \(Q\) on \(X\), let \(\tau_Q\) be central
translation on \(Y_G\), with contracted-product convention
\([qz,x]=[q,zx]\).  It canonically preserves the twisting: the
adjoint bundle of the translated bundle is canonically unchanged.
Use the same notation for the corresponding operation for
\(Z_{G^\vee}\)-torsors on \(Y_{G^\vee}\).

Let \(A_{G^\vee}\) be the finite kernel of
\(G^\vee_{\mathrm{sc}}\to G^\vee\).  The root datum gives the
perfect finite pairing
\(Z_G\times A_{G^\vee}\to\mathbb C^\times\).  Use complex
orientation and the exponential with \(2\pi\sqrt{-1}\) to identify
this kernel with the fundamental group.  We normalize the lifting
obstruction so that its integral over \(X\) is the component used in
Proposition~\ref{equivalence:sectors}.  Equivalently, a local
modification with gluing \(t^\lambda\) has relative obstruction
with integral \(-\lambda\).  This is the local lifting obstruction
around an oriented circle, with the same degree convention as the
Cartan bundle in the preceding proof.

The universal bundle on \(X\times Y_{G^\vee}\) has a gerbe of lifts
to \(G^\vee_{\mathrm{sc}}\), with obstruction class
\(\operatorname{obs}\) in degree two and coefficients in
\(A_{G^\vee}\).  Form \([Q]\cup\operatorname{obs}\), with
\(Q\) first, use the finite pairing, and integrate over \(X\).
The resulting class has degree \(1+2-2=1\) and defines a rank-one
local system \(u_Q\) with finite monodromy on \(Y_{G^\vee}\):
\[
 [u_Q]=\int_X\bigl\langle[Q]\cup\operatorname{obs}\bigr\rangle.
\]
This construction is performed on the
torsor and gerbe themselves, as maps to Eilenberg--MacLane spaces,
so it retains isomorphisms, automorphisms, and the coherent tensor law
in \(Q\).  Reversing the two groups gives \(u'_{Q'}\) on \(Y_G\)
for a \(Z_{G^\vee}\)-torsor \(Q'\).  Finite-coefficient comparison
and Riemann existence realize these systems algebraically on charts;
smooth descent gives the systems on the stacks.  Associated-line
monodromy is \(\gamma(z)\) for transport by \(z\) and character
\(\gamma\).  Tensor by these systems means dualizing-normalized
shriek tensor, or ordinary flat tensor inside angle brackets.

\begin{theorem}\label{equivalence:central}
The equivalence \(\Phi\) has coherent natural isomorphisms
\begin{equation}\label{eq:G7}
 \Phi\tau_Q^!\simeq u_Q^{-1}\otimes\Phi,
 \qquad
 \tau_{Q'}^!\Phi\simeq\Phi(u'_{Q'}\otimes-).
\end{equation}
They are natural on the full groupoids of central torsors and respect
their tensor products.  They and \eqref{eq:G6} remain valid after the
integral determinant transport to the convention of
Theorem~\ref{thm:main}.
\end{theorem}

\begin{proof}
We first describe two operations before passing to the global
localization quotient.

\emph{Twisting KL inputs.}
Over a tuple scheme, restrict \(Q\) to the formal neighborhoods of
the marked graphs.  Trivialize it locally and descend KL inputs by the
central arc action.  Denote the resulting operation by \(V\mapsto
V[Q]\).  The same definition works at collisions.  To give precise local
trivializations, write \(D=\sum_i\Gamma_i\subset X\times S\)
for the sum of the marked graphs over the tuple scheme \(S\).
This is finite locally free over \(S\), including at collisions.
The section scheme
\[
 \operatorname{Res}_{D/S}(Q|_D)\longrightarrow S
\]
is \(\acute{e}\)tale and surjective: a geometric fiber admits a
section by choosing a lift on each of its clusters, and formal
\(\acute{e}\)taleness extends this choice across the nilpotents.
This section scheme need not be finite when clusters split.
After passing to it, formal \(\acute{e}\)taleness extends the
universal section uniquely to each thickening \(nD\), and hence
to the completed divisor.  Two such sections differ by a section
of \(Z\) on the completed divisor, locally constant on each
cluster.  Their transitions satisfy the cocycle identity because
they are differences of actual torsor sections.  For insertion or
comparison of tuples, use the sum of their divisors as a common
refinement and restrict its sections.  Unique extension through
thickenings then gives compatibility with the crystal structure,
further collisions, and all resulting coherence diagrams.

On localization charts compare at these chosen lifts of \(Q\).
The out Lie algebra, its differential operators, and the determinant
identifications use the unchanged adjoint bundle.  A change of lift
by \(z\) changes the translated frame by \(z\), precisely the
central transition used in defining \(V[Q]\).  Thus the chart
coinvariant formulas give
\[
 \tau_Q^!L(V)\simeq L(V[Q]).
\]
The comparison holds on bounded-module formulas and their morphisms,
then on their continuous extension; it commutes with insertion and
parameter operations.  The same argument applies to \(M'\) and
\(Q'\) for the dual group.

\emph{Transgression on Whittaker inputs.}
Pull \(u_Q\) to the pole model and tensor there.  Its two pullbacks
along a Whittaker arrow identify canonically.  Indeed the arrow is a
unipotent modification, whose gluing lifts uniquely through the
simply connected cover by the corresponding root subgroups.  Change
the local lift by this lifted modification and retain the lift away
from the marking.  This identifies the lifting gerbes, multiplicatively
under composition of arrows.  Hence tensor by the transgression
system preserves Whittaker equivariance.  Projection formula shows
that coefficient operations and their counits commute with it.

For the local comparison \(\alpha\) of \eqref{eq:L11}, the two
operations are related by
\begin{equation}\label{eq:G8}
 \alpha(V[Q])\simeq u_Q^{-1}\otimes\alpha(V).
\end{equation}
Here the equality includes all tuple-parameter structures.  To check
it, use the Grassmannian model.  Its bundle is identified away from
the markings with \(P_0=\vartheta^{2\rho}\), which has its fixed
lift to the simply connected group.  The lifting obstruction is
therefore relative, supported at the markings.  The chosen disk
sections of \(Q\) trivialize its transgression line: they nullhomotope
\([Q]\) near the support of the relative obstruction, hence nullhomotope
the integrated cup product.  This construction is functorial in the
relative lifting gerbe, so the trivialization holds on the entire
Grassmannian chart, not just on its labeled cells.  On the section
cover, therefore, both sides of \eqref{eq:G8} identify with
\(\alpha(V)\) for every object \(V\), and we use the identity
comparison.  Changing a section by \(z\) on a cluster of total label
\(\lambda\) changes the transgression trivialization by
\((-\lambda)(z)\).

We next check compatibility across collisions.  In an analytic chart of any finite
Grassmannian bound, extend the chosen disk sections to neighborhoods
of the graph unions.  Trivialize the cup product using the relative
obstruction and this section.  Excision makes the trivialization
independent of shrinking the neighborhoods.  Its change is cup
product with the degree-zero change of section followed by
integration, hence evaluation of that section change on the cluster
degree.  That degree is \(-\lambda\) by the obstruction convention.
Neighborhoods and their intersections give these comparisons across
collisions; excision and associativity give the comparisons for
further clustering.  Finite-coefficient comparison transports these
finite-system isomorphisms to the algebraic charts, compatibly with
pullback.

On the KL side the same cluster acts through its total central
character \(\lambda\).  On the Whittaker side \(u_Q^{-1}\) has
transition \(\lambda(z)\), by the calculation just made.  The
component calculation in the proof of
Proposition~\ref{equivalence:sectors} says that \(\alpha\) preserves
these decompositions.  On each finite central-character sector the
transition is a scalar natural transformation, so \(\mathbb C\)-linearity
identifies the two transitions on all objects and morphisms.
For a varying section difference \(z\), its distinct values partition
the completed divisor into open-and-closed pieces; markings with
different values cannot collide.  Use the disjoint-product compatibility
of \(\alpha\) between these pieces, and the single central scalar
on each piece, including all its internal collisions.
The two descent data therefore agree.  The pole--Grassmannian comparison transports the
identification to the Whittaker categories.  It can be checked on
underlying Grassmannian objects because Whittaker equivariance
defines a full subcategory.  This proves \eqref{eq:G8}, with its
coherent parameter compatibilities.  The same construction gives
its dual-group counterpart with \(u'_{Q'}\).

We now pass to the global functor.  Put \(P_B=e_B(S')^\vee\).
Tensoring one slot of the opposite-character pairing by a finite
local system moves that same tensor to its other slot; coefficients
commute with this tensor by the preceding argument.  In the shriek
diagonal kernel, tensor on one factor also moves to the other factor
with the same local system.  Consequently
\[
 P_B(u_Q^{-1}\otimes C)\simeq u_Q^{-1}\otimes P_B(C).
\]
Together with \eqref{eq:G5}, the KL twisting comparison, and
\eqref{eq:G8}, this proves
\(\Phi\tau_Q^!L\simeq u_Q^{-1}\otimes\Phi L\).
Precomposition with the localization \(L\) is fully faithful on
functor categories, so this comparison descends uniquely and gives
the first isomorphism of \eqref{eq:G7}.

For the second, apply the dual-group twisting comparison to
\(R\simeq(M')^\vee f\).  Let \(F\in A\) and \(E\in\mathcal D'\),
viewing \(E\) as a dual-group KL object through the constructed
identification.  Write \(E[Q']\) for the transported central-torsor
action.  The dual-group version of \eqref{eq:G8} gives
\(u'_{Q'}\otimes E\simeq E[(Q')^{-1}]\).  Evaluation on the
localized test \(M'E\) now gives
\[
 \begin{aligned}
 f(u'_{Q'}\otimes F)(M'E)
 &\simeq\bigl\langle R(u'_{Q'}\otimes F),E\bigr\rangle\\
 &\simeq\bigl\langle R(F),u'_{Q'}\otimes E\bigr\rangle\\
 &\simeq\bigl\langle R(F),E[(Q')^{-1}]\bigr\rangle\\
 &\simeq f(F)\bigl((\tau_{Q'}^{-1})^!M'E\bigr).
 \end{aligned}
\]
The first equality is \eqref{eq:G2}; the second is the coefficient
projection formula and the pairing's tensor symmetry; the third is
the dual-group comparison; and the last is the KL localization
compatibility with torsor twisting.  Since \(M'\) is a localization,
these evaluations descend to the natural equality
\[
 f(u'_{Q'}\otimes F)
 \simeq f(F)\circ(\tau_{Q'}^{-1})^!.
\]
To check the variance after applying \(e_B\), let
\(K_B\in B'\otimes B\) be its shifted shriek diagonal kernel,
and put \(\tau=\tau_{Q'}\).  Simultaneous translation preserves
\(K_B\), so
\[
 ((\tau^{-1})^!\otimes\operatorname{Id})(K_B)
   \simeq(\operatorname{Id}\otimes\tau^!)(K_B).
\]
For any continuous functional \(h\) on \(B'\), evaluation of
its first factor consequently gives
\[
 e_B\bigl(h\circ(\tau^{-1})^!\bigr)
   \simeq\tau^!e_B(h).
\]
Taking \(h=f(F)\) proves the second isomorphism.

All maps used in the construction are maps on torsor groupoids:
section descent, the cup product, excision, projection formula, and
diagonal invariance preserve their composition and tensor laws.
The descended comparisons retain these coherences by full
faithfulness.  One can also read the two signs directly on torsor
automorphisms.  An automorphism \(z\in Z_G\) of the trivial
\(Q\) acts on a source object of character \(\eta\) by
\(\eta(z)\).  Its image has component \(-\eta\), so
\(u_Q^{-1}\) has exactly this automorphism.  An automorphism
\(z'\in Z_{G^\vee}\) of the trivial \(Q'\) acts on the
target by \(p(z')\); on the source component \(p\),
\(u'_{Q'}\) has the same automorphism.  Thus the full groupoid
formulas agree with both parts of \eqref{eq:G6}.

Finally, tensoring by \(L_G\) commutes with central
translation, including torsor automorphisms, because this determinant
is formed from the adjoint bundle.  The same is true of central
inertia.  This proves the last assertion and completes the global
unramified correspondence.
\end{proof}
==== END Global-quantum-geometric-Langlands-at-irrational-level//13-equivalence.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Global quantum geometric Langlands at irrational level}
\author{OpenAI}
\date{October 4, 2026}
\hypersetup{pdftitle={Global quantum geometric Langlands at irrational level},
  pdfauthor={OpenAI}}
\begin{document}
\maketitle
\begin{abstract}
We prove the unramified quantum geometric Langlands equivalence for every
connected simple complex algebraic group $G$ and every level
$c\in\mathbb C\setminus\mathbb Q$, including non-real levels. For every smooth
projective connected complex curve $X$, it identifies the full twisted
$D$-module categories on $\operatorname{Bun}_G(X)$ and
$\operatorname{Bun}_{G^\vee}(X)$ at the dual shifted levels $c$ and
$-1/(rc)$, where $r$ is the lacing number. The equivalence uses the given
global forms and includes all connected components.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-foundations}
\input{sections/03-pseudoidentity}
\input{sections/04-whittaker}
\input{sections/05-deformation}
\input{sections/06-plancherel}
\input{sections/07-loop-categories}
\input{sections/08-w-algebras}
\input{sections/09-local-comparison}
\input{sections/10-localization}
\input{sections/11-unipotent-period}
\input{sections/12-ran-period}
\input{sections/13-equivalence}
\input{references}
\end{document}
==== END Global-quantum-geometric-Langlands-at-irrational-level//main.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//preamble.tex ====
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\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{mathrsfs}
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==== END Global-quantum-geometric-Langlands-at-irrational-level//preamble.tex ====

==== BEGIN Global-quantum-geometric-Langlands-at-irrational-level//references.tex ====
\begin{thebibliography}{99}
\raggedright

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\end{thebibliography}
==== END Global-quantum-geometric-Langlands-at-irrational-level//references.tex ====