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\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 $G$ at every level $c\in\mathbb{C}\setminus\mathbb{Q}$, including non-real levels. For every smooth projective connected complex curve $X$, the theorem identifies the full category of twisted $D$-modules on $\operatorname{Bun}_G(X)$ with the corresponding category on $\operatorname{Bun}_{G^\vee}(X)$ at the dual shifted level $-1/(rc)$, where $r$ is the lacing number. The equivalence uses the actual global forms of both groups and preserves all connected components.
\end{abstract}
\tableofcontents
\section{Introduction}\label{sec:introduction}
\subsection*{Statement of the theorem}
Let $X$ be a smooth projective connected curve over $\mathbb{C}$ and let $G$ be a connected simple complex algebraic group. The Langlands dual $G^\vee$ is determined by the dual of the full root datum, so the global form (isogeny type) of $G$ is part of the data. Write $\operatorname{Bun}_G(X)$ for the stack of principal $G$-bundles. The adjoint determinant line is
\[
L_G|_E=\det R\Gamma(X,\mathfrak g_E),
\]
so $\det H^1$ enters with exponent $-1$. For $h^\vee$ the dual Coxeter number and $c\in\mathbb{C}$ the shifted level, set
\begin{equation}\label{intro:level}
D_c(\operatorname{Bun}_G(X)) := D\bigl(L_G^{(c-h^\vee)/(2h^\vee)}\bigr).
\end{equation}
Here $D(L^s)$ is the cocomplete derived DG category of all twisted $D$-modules for the complex power of $L$. The power is a sheaf of twisted differential operators; no choice of root of $L$ is needed. Descent and sign conventions are fixed in Section~\ref{sec:foundations}.
Let $r\in\{1,2,3\}$ be the lacing number of $G$. Also fix dual pinnings and a theta characteristic $\vartheta$ on $X$ with $\vartheta^{\otimes 2}\simeq\Omega_X$. These choices fix the residue characters and the localization normalizations used throughout.
\begin{theorem}[Main]\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 equivalence 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, and it 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}. Theorem~\ref{thm:main} therefore gives the unramified de Rham quantum Langlands equivalence in the irrational range. The localization comparison and the central compatibility are part of the construction: they are the two ingredients that identify the functor as the quantum Langlands correspondence. Their definitions 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 (\cite{KapustinWitten}) explains the appearance of the dual root datum and reciprocal parameter. An early formulation, attributed there to Drinfeld, is 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 local and classical inputs. Feigin--Frenkel duality identifies principal $W$-algebras at inverse shifted levels \cite{FeiginFrenkel}. Arakawa--Frenkel identify reductions of Weyl modules and their coweight twists, including concentration and simplicity at irrational level \cite{ArakawaFrenkel}. Raskin's categorical Drinfeld--Sokolov reduction gives a 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 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. Sections~\ref{sec:localization}--\ref{sec:ran} construct the period comparison with 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.
The first ingredient is a global Whittaker coefficient identity. 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 and the pole positions. Pairing with the opposite character and applying 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}
The 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, so it 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 produces the exponential characters. Quasi-unipotent monodromy makes cohomology ranks and transition-map ranks constant away from roots of unity. 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 gives the same vanishing at every non-torsion $q$, hence at every irrational exponent. Ordinary geometric Langlands supplies the cohomological bound. Section~\ref{sec:deformation} proves the deformation principles; Section~\ref{sec:plancherel} applies them to the tested kernel cone.
The second ingredient is a local-comparison package for the two Langlands-dual groups over families of marked points. For $G$ and $G^\vee$ together, 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 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}
Section~\ref{sec:foundations} sets up the category-theoretic base. 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.
\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. Two foundations are needed: a precise twist choice, and a finiteness statement saying that at the irrational exponents in question restriction to a suitable quasi-compact open is an equivalence. We establish these, then the kernel formalism and its renormalized evaluation pairing.
\subsection{Root data, determinant lines, level conventions}
All groups retain their given root data. Fix dual pinnings and a theta characteristic $\vartheta$ with $\vartheta^{\otimes 2}\simeq\Omega_X$. 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 $\det H^1$ enters with exponent $-1$. Write $\rho_H$ for the half-sum of positive \emph{coroots}, and set $P_0=\vartheta^{2\rho_H}$, an $H$-bundle via the integral cocharacter $2\rho_H$. In Grassmannian constructions rigidify $L_H$ to $L_H\otimes(L_H|_{P_0})^{-1}$; a constant line gives the same twisted differential operators.
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. \emph{Sign convention}: if a frame $e$ is replaced by $ae$, the local differentiation 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 functors use strong derived descent. The opposite twist is $L^{-s}$; sums of twists use the tensor product of frame torsors.
We use sheaf conventions: $\omega_Y$ unit of $\otimes^!$, $k_Y$ constant object, lower star for de Rham direct image. For left $D$-modules on smooth $Y$ write $M^{\langle Y\rangle}$ for the left module $M[\dim Y]$; then $\mathcal O_Y^{\langle Y\rangle}=\omega_Y$. Inside angle brackets $!$-pullback is left crystalline pull without extra 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 dualizing normalization. All tensor products and mapping complexes are derived; categories are cocomplete, $\mathbb{C}$-linear, and functors are continuous unless an adjoint or partial functor is specified.
Use the basic invariant forms $\operatorname{Kil}/(2h^\vee)$ and $\operatorname{Kil}^\vee/(2\check h^\vee)$. Inverse forms give long roots, resp. long coroots, squared length $2$. Inverse shifted forms have parameters $a,d$ with
\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 four categories are
\begin{equation}\label{foundations:four-categories}
A=D(L_G^{-k_a/(2h^\vee)}),\quad A'=D(L_G^{k_a/(2h^\vee)}),\quad
B'=D(L_{G^\vee}^{-k_d/(2\check h^\vee)}),\quad B=D(L_{G^\vee}^{k_d/(2\check h^\vee)}).
\end{equation}
In the convention of \cite[Section~1.1.5]{BogdanovaBetti}, a shifted level $c$ means 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 of Theorem~\ref{thm:main} with $A$. The target at shifted level $-1/(rc)$ is exactly $B$. This uses 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. Since $c\notin\mathbb{Q}$ all four exponents 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}
Modification labels and dual weights always belong to the lattices of the specified groups. For semisimple $H$, the finite centre acts trivially on $L_H$; central inertia therefore gives the usual decomposition into character sectors, and these sectors are retained below.
\subsection{Bounded support in the categorical sense}
Let $H$ be connected reductive and let $V$ be a finite-dimensional self-dual representation. Put
\[
L(V)|_E=\det R\Gamma(X,V_E),\qquad Q(z,w)=\operatorname{Tr}_V(z w),
\]
and assume $Q$ is positive definite on the real cocharacter space. For a degree $u\in\pi_1(H)$ its rational image has a representative $u_{\mathrm{cen}}\in X_*(Z(H)^0)\otimes\mathbb{Q}$; 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 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 to the frame torsor of $L$. The inertial $\mathbb{G}_m$-scaling of frames is $2$-isomorphic to the projection, so strong Kummer equivariance identifies the two pullbacks of an object to $\mathbb{G}_m\times Z$ up to the factor $t\mapsto t^{sm}$. De Rham direct image along $\mathbb{G}_m$ kills the nonintegral Kummer factor ($d$ has eigenvalues $n-sm$ on $t^n$, all invertible); the constant factor survives as $H^\bullet_{\mathrm{dR}}(\mathbb{G}_m)$, which contains $\mathbb{C}$ as a summand. Tensoring with that complex detects zero. Smooth pull and frame-torsor pull are conservative, so the original object vanishes. The calculation leaves all spectator factors unchanged.
\end{proof}
\begin{proposition}[Bounded support]\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}
Suppose $u$ has nonzero rational image. Nondegeneracy on the centre gives a central cocharacter $z$ with $Q(z,u)\ne0$. Decompose $V$ into central weight spaces. Riemann--Roch gives the $z$-weight of $L(V)$ on degree $u$ as
\begin{equation}\label{foundations:rr-weight}
\sum_\mu\mu(z)\chi(X,V_{\mu,E})=Q(z,u),
\end{equation}
the $1-g$ term vanishing because $V$ is self-dual. The right side is a nonzero integer, so Lemma~\ref{foundations:inertia} gives vanishing on that component, and after restriction to any locally closed bundle condition inside it.
Now fix a torsion degree. Use the characteristic-zero Harder--Narasimhan theorem for principal bundles: canonical parabolic reductions, discrete set of HN types with bounded denominators, bounded regions with quasi-compact opens \cite[Sections~7.3--7.4 and~10.1]{DrinfeldGaitsgoryBun}. Let $\nu$ be the dominant rational HN coweight, with zero central component. Put $b_X=\max\{0,2g(X)-2\}$. We claim the twisted category on the HN stratum vanishes when $\alpha_i(\nu)>b_X$ for some simple root $\alpha_i$.
Let $Q_i$ be the maximal standard parabolic omitting $\alpha_i$, $M_i$ its Levi. The canonical reduction sits inside $Q_i$; projecting to $M_i$ identifies the HN stratum as a stack of unipotent extensions over the corresponding stratum for $M_i$. Filter $\operatorname{rad}(Q_i)$ by $\alpha_i$-height. Each quotient has semistable associated graded pieces whose slopes are $\beta(\nu)$ for roots $\beta$ containing $\alpha_i$ with positive coefficient; dominance gives $\beta(\nu)\geq\alpha_i(\nu)>b_X$, so Serre duality and semistability give $H^1=0$ for every such piece, hence for each successive extension. Riemann--Roch then makes the $H^0$'s have constant rank. The extension stacks are, locally in families, successive classifying stacks of smooth unipotent groups. Choose a positive integral multiple $z$ of the $i$th fundamental coweight, central in $M_i$. On the split section, the determinant filtration identifies $L(V)$ with the restricted $M_i$-determinant line, whose $z$-weight is $Q(z,\nu)>0$. Lemma~\ref{foundations:inertia} kills the pullback to the split section, and smooth descent kills the whole HN stratum.
For the remaining HN types, $0\leq\alpha_i(\nu)\leq b_X$ for every $i$, so they lie in a bounded region with finitely many types; there are 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; on each bounded open in the HN exhaustion, the localization triangles prove the twisted category of the complement vanishes. The exhaustion detects objects, so the localization triangle for $j$ gives \eqref{foundations:cutoff-equivalence}, with inverse $j_*$. The argument is unchanged for $-s$.
\end{proof}
The assertion is about $!$-restriction of categories; it makes no set-theoretic claim about closures of supports. For simple $G$ one takes $V=\mathfrak g$; for a Levi, the restriction of the adjoint supplies a self-dual positive trace form.
\subsection{Operations with compatible twists}\label{foundations:operations}
A QCA stack is quasi-compact with affine automorphism groups. We use the $D$-module formalism of \cite{DrinfeldGaitsgoryFiniteness}: compact generation, categorical duality, renormalized de Rham direct image with base change and 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 equals 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 pullbacks of frame torsors, or trivializes their ratio. Pulling to the torsors reduces $!$-base change, the $!$-tensor projection formula, and continuity to the ordinary formulas. The identifications commute with frame scaling, so descend with the prescribed Kummer equivariance.
We use the six operations on locally holonomic objects wherever they are defined. For schematic separated finite-type maps, shriek direct image on holonomic arguments is the partial left adjoint to $!$-pullback, tested against arbitrary $D$-modules. Verdier duality negates the twist; both 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 $e$: $q^*=q^![-2e]$.
\subsection{Compact generation and opposite-twist duality}
\begin{proposition}\label{foundations:categories}
Let $Y$ be a smooth quasi-compact open in a bundle stack, $\tau$ 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. 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 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 persist after the cutoff of Proposition~\ref{foundations:cutoff}.
\end{proposition}
\begin{proof}
\emph{Oblivion and induction.} Write $\tau=s\,d\log L$, with frame torsor $P=L^\times\to Y$. An object $N$ of $D_\tau(Y)$ is represented on $P$ by a strongly Kummer-equivariant D-module. Forgetting the connection in normalized left-module convention turns the Kummer connection $d-s\,d\log t$ into the trivial line with frame $1$, and smooth $!$-pullback then induces ordinary pullback of the underlying quasi-coherent complex, with no dimension shift. The equivariance isomorphisms give quasi-coherent descent along $P\to Y$, retaining all isotropy actions. On a chart trivializing $L$ the operation is right-module oblivion tensored with $(\omega_Y^{\mathrm{can}})^{-1}[-d]$ (dim $Y=d$).
We shall construct $\operatorname{Ind}_\tau$ on $\operatorname{Perf}(Y)$ with
\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 gives compact generators; external-product compatibility gives Künneth.
\emph{Presentation and induction.} The bounded Quot construction presents $Y$ as a quotient of a smooth finite-type scheme by a product of general linear groups. Choosing a uniform twist, faithfulness bounds the associated vector bundles, and framing their sections gives a scheme atlas; reductions of structure group are schemes of sections because the quotient by the reductive subgroup is affine. Choose a smooth presentation $p:Z\to Y$ torsor for such a product.
The twisted induction is the relative Lie-algebroid construction of \cite[Section~6.3, Lemmas~6.3.15--6.3.16]{DrinfeldGaitsgoryFiniteness}. For a smooth chart $T\to Y$ let $A_T=T_{T/Y}$ be the relative tangent Lie algebroid, with anchor to $T_T$. Relative differentiation lifts $A_T$ to the Atiyah algebroid of the pulled-back line, hence to its TDO $\mathcal D_T^\tau$: for $a\in A_T$ with anchor $v_a$ acting on a local frame by $b_a$, the operator is $v_a+s\,b_a$. Under $e'=fe$ both $b_a$ changes by $v_a(\log f)$ and the differentiation generator by $-s\,v_a(\log f)$, so the operator is frame-independent. Define for $F\in\operatorname{Perf}(Y)$
\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}
$F_T$ carries its canonical relative connection. Kernels in the anchor are retained: their action on the twisting line gives the corresponding isotropy scalar in $\mathcal D_T^\tau$. Trivializing the line does not discard this isotropy action.
For a smooth map $T'\to T$ over $Y$, the algebroid $A_{T'}$ is the inverse image algebroid; resolving first in $T_{T'/T}$ by the Spencer complex gives precisely the transfer bimodule for smooth pullback applied to \eqref{foundations:induction}. Two smooth maps compose, so the chart objects descend. At zero twist, \eqref{foundations:induction} equals \cite[Section~6.3.21]{DrinfeldGaitsgoryFiniteness} applied, in IndCoh normalization, to $F\otimes\omega_Y^{\mathrm{can}}[d]$; on $BH$ with $\omega_{BH}^{\mathrm{can}}=\det\mathfrak h$, the factors cancel the $\det(\mathfrak h^\vee)[\dim H]$ of \cite[Section~6.4.8]{DrinfeldGaitsgoryFiniteness}, and one recovers derived $\mathfrak h$-coinvariants on the point atlas.
\emph{Adjunction, unbounded.} Spencer and enveloping-adjunction on charts give \eqref{foundations:induction-adjunction} locally. On $N|_T$, the twist splits along $A_T$, and the mapping complex is the totalization over the Čech-nerve faces of $p$ of relative de Rham complexes with coefficients in $F^\vee\otimes N^{\mathrm{oblv}}$. Put $E=p_*\Omega^\bullet_{Z/Y}$, $p_n:Z^{[n]}\to Y$. Product-formula and affine projection give $Rp_{n,*}(\Omega^\bullet_{Z^{[n]}/Y}\otimes p_n^*M)\simeq E^{\otimes^{L(n+1)}}\otimes^LM$, with cofaces inserting the unit of $E$. The invariant Reynolds functional on regular functions of the group, extended by zero on positive-degree forms, gives $E\to\mathcal O_Y$ splitting the unit; left/right invariance make it independent of trivializations. Applying it in the first tensor factor contracts the augmented tensor-power diagram; tensoring with any coefficient complex proves the unbounded totalization. The homotopy lowers cosimplicial degree by one, so acts on the product totalization without infinite sums; it contracts the augmented diagram, not the de Rham complex of the group.
Forgetful is conservative and preserves colimits; perfect complexes compactly generate $\operatorname{QCoh}(Y)$ on smooth quotient stacks, external products generate on products \cite[Sections~3--4]{DrinfeldGaitsgoryFiniteness}. Formula \eqref{foundations:induction-adjunction} makes the induced perfects compact generators of $D_\tau(Y)$; their finite Spencer resolutions on charts consist of coherent $\mathcal D_T^\tau$-modules, hence have bounded coherent cohomology on every smooth finite-type chart, and this property is closed under finite cones and retracts. Induction commutes with external products; applying \eqref{foundations:induction-adjunction} to external perfects with the quasi-coherent Künneth 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{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 formulas with the ratio trivialized on the diagonal. To check a 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; projection formula cancels the two twists along this graph; the remaining expression is the untwisted graph calculation, and the graph is schematic and safe. This proves \eqref{foundations:evaluation}.
\emph{Full bundle stack.} Transport compact generation and duality across \eqref{foundations:cutoff-equivalence}; the spectator version identifies products and diagonal kernels. For local coherence: given a finite-type chart of $Y_H$, choose a larger quasi-compact HN open $V$ containing both its image and $U$; the same cutoff argument gives $D_\tau(V)\simeq D_\tau(U)$, so a compact object is compact on $V$, and the Spencer argument gives its coherence on the chosen chart.
\end{proof}
In particular $A'$ and $B'$ realize the categorical duals of $A$ and $B$. Formula~\eqref{foundations:evaluation} uses the star-diagonal kernel; the shriek-diagonal kernel $\Delta_!k_Y$, defining pseudo-identity, is a separate object. Proving that it is itself an equivalence is the next section.
\section{Twisted pseudo-identity}\label{sec:pseudoidentity}
The ordinary and extraordinary diagonal kernels play different roles. The former represents categorical duality (Section~\ref{sec:foundations}); the latter must itself be shown to be an equivalence. We prove this by combining parabolic adjunctions with a compactification of the diagonal. The essential additional point at irrational twist is that central inertia removes every boundary stratum with nonzero defect.
Let $H$ be connected reductive, $V$ a self-dual representation with positive-definite trace form $Q$ on the real cocharacter space, and put $L_H=\det R\Gamma(X,V_E)$. Restrictions of $V$ define the corresponding lines for Levi subgroups. Write
\[
\mathcal C_H=D(L_H^s),\qquad \mathcal C'_H=D(L_H^{-s}),\qquad s\notin\mathbb{Q}.
\]
Section~\ref{sec:foundations} identifies $(\mathcal C'_H)^\vee$ with $\mathcal C_H$; the line ratio on the diagonal is canonically trivial. 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}
The proof is by induction on semisimple rank. We first establish parabolic adjunctions compatible with $F_H$. A compactification of the diagonal will then show that the cone of $F_H\to\operatorname{Id}[-2\dim Y_H]$ has a finite filtration whose quotients factor through proper Levi categories. On Eisenstein-induced objects $F_H$ is computed by \eqref{eq:M4}; on objects with vanishing proper constant terms $F_H$ acts by a shift. These generation inputs establish the equivalence.
The geometric inputs are geometric Eisenstein series and second adjointness \cite{BravermanGaitsgory,DrinfeldGaitsgoryCT}. Twists and coherence require attention; an adjunction valid only on holonomic objects would not suffice, so we work on all twisted $D$-modules.
\subsection{Parabolic correspondences and their compactifications}
For the moment let $H$ be semisimple. Fix a parabolic $P$, its Levi $M$, 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 $V$ attached to $P$ gives $p^*L_H\simeq q^*L_M$; 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 using that $u$ is torsion (the value is the same for $P^-$). Primes on these functors mean the opposite twist.
Here $p$ is schematic, separated, relatively quasi-compact; $q$ is smooth, relatively quasi-compact, safe. A positive grading of $N_P$ presents $q$, locally over the Levi stack, as successive fibrations of vector stacks, each of the form $[V^1/V^0]$ for a two-term complex computing cohomology on $X$; there are no obstructions in degree two. Three consequences: $C_P,E_P$ are continuous; $q_*$ preserves local regular holonomicity; and the partial holonomic operations $q_!,q^*$ have their true adjunction against arbitrary $D$-modules. For the last, present each vector stack by its vector bundle: descent along the vector group is fully faithful by de Rham acyclicity, and both assertions reduce to schematic formulas. For $a:V^1\to[V^1/V^0]$, $a_*a^*\simeq\operatorname{Id}$ and $a_!a^!\simeq\operatorname{Id}$ follow from ordinary and compactly supported cohomology of vector-group fibers; the shifts survive when the vector stack has negative dimension. These reductions commute with smooth descent and with adding a spectator.
Kernel duality gives
\[
(C'_P)^\vee=E_P.
\]
Both kernels are obtained from the diagonal unit by $!$-base change followed by safe direct image. Every calculation may be made on quasi-compact open exhaustions; Proposition~\ref{foundations:cutoff} identifies the twisted categories of the bundle stacks with those on fixed quasi-compact opens. No co-category is introduced.
Use the 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 the Levi datum retained, $\bar p$ is schematic and proper. If the derived group is simply connected, this is compactification by generically nondegenerate sections of the affine closure of $H/N_P$; the boundary stratifies by positive defects, and on each stratum the saturated reduction is an actual $P$-bundle whose Levi bundle is related to the 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 \cite{BravermanGaitsgory,SchiederVinberg}.
For actual global form, choose a central extension $H_1\to H$ with torus kernel $D$ and simply-connected derived group, form the compactification upstairs, and quotient by the $D$-action. Lifts of an $H$-bundle exist smoothly locally on the base: over geometric fibers, because the Brauer group of the smooth projective curve vanishes; infinitesimally, from vanishing of $H^2$ of coherent sheaves. So $Y_{H_1}\to Y_H$ and the parabolic and Levi maps are $Y_D$-torsors. Fix a lift of ambient and Levi degree downstairs: uniqueness of the compatible lift follows from the torus sequence (two candidates differ by a cocharacter of $D$, which injects into $\pi_1(H_1)$; injectivity uses that $D$ meets the derived group in a finite subgroup). Properness, schematicity, stratum description descend. A nonzero defect changes the Levi degree nontrivially over $\mathbb{Q}$ (simple coroots omitted from $M$ remain linearly independent in the quotient of the rational cocharacter space by the Levi coroot span).
\subsection{Boundary cleanness and characteristic variety}
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 prove two properties: $J$ is clean, and it has no nonzero characteristic covector pulled back from the Levi base. The second allows tensoring $J$ with arbitrary coherent Levi coefficients.
\begin{lemma}[Analytic flow]\label{pseudo:analytic-flow}
Let $U\subset Z$ be an open subset of a finite-type complex scheme, $\mathcal L$ a local system on $U$, $j_!\mathcal L$ its extension by zero. A derivation of $\mathcal O_Z$ preserving the ideal of $Z\setminus U$ has local analytic flows preserving the pair $(Z,U)$; the resulting flows identify the pullbacks of $j_!\mathcal L$, and every characteristic covector of the closed direct image annihilates the derivation.
\end{lemma}
\begin{proof}[Sketch]
Embed $Z$ locally as a closed subscheme of a smooth scheme. Lift the derivation to the ambient; preservation of the ideals of $Z$ and boundary makes the ambient flow preserve their analytic zero loci. On a small time disk, flow transport identifies the two pullbacks of $\mathcal L$, hence of $j_!\mathcal L$. The characteristic-variety assertion follows from the equality $\operatorname{char}=\operatorname{microsupp}$ for regular holonomic $D$-modules \cite[Theorem~11.3.3]{KashiwaraSchapira} and the local product description along the flow.
\end{proof}
\begin{lemma}\label{pseudo:fiber-descent}
Let $i_\theta:S_\theta\hookrightarrow\widetilde Y_P^u$ be a defect stratum with Levi Hecke-data projection $f_\theta:S_\theta\to B_\theta$. 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 twist claim. Smoothness of $f_\theta$ on each geometric $!$-fiber: choose $z\in X$ outside the defect divisor and shrink the parameter chart so $z$ stays outside. Near $z$ the generalized reduction is actual, so a frame exists locally; extend it. A Laurent element $\eta\in\mathfrak n_P((t_z))$ gives an infinitesimal modification $1+\epsilon\eta$, and gluing on the punctured disk produces a first-order deformation of the generalized reduction, preserving the boundary ideal (the chosen derivation preserves the defect-incidence image in the parameter scheme). These directions lift compatibly to smooth charts of $f_\theta$ and span their tangent spaces: principal parts at $z$ surject onto $H^1(X,(\mathfrak n_P)_{E_P})$ via Serre vanishing, and the connecting map is the gluing deformation.
For $J$ as a regular holonomic complex on the open with its local-system model there, the flow observation of Lemma~\ref{pseudo:analytic-flow} gives that $i_\theta^!J$ is lisse along $W$. Vector-stack descent: on a vector stack over a point, a smooth regular holonomic complex is constant. For one fibration use $f^*f_*\to\operatorname{Id}$, tested on geometric $!$-fibers via $!$-base change; a counit isomorphism on fibers gives a global one. Iterate.
\end{proof}
\begin{lemma}\label{pseudo:clean-noncharacteristic}
The kernel $J$ is clean: $j_!\omega\simeq j_*\omega$ in the specified twist. On a smooth Levi chart $S\to Y_M^u$ with a scheme presentation of $\widetilde Y_P^u\times_{Y_M^u}S$ and closed embedding into a smooth $S$-scheme, 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 $Q(z,u_{\mathrm{sat}}-u)$. The degree difference is rationally nonzero, so positivity provides an integral central $z$ with nonzero integer weight. At exponent $s$, the Kummer parameter is nonintegral, so inertia vanishing (Lemma~\ref{foundations:inertia}) kills the twisted category on this base (with spectators). Under a central extension, multiply and lift $z$; removed central directions act trivially on the ratio. By Lemma~\ref{pseudo:fiber-descent}, $i_\theta^!J=0$. Localization plus local finiteness proves cleanness; the same applies to $-s$ and to the Verdier-dual kernel.
For characteristic: repeat the modification argument at $z$ with elements of the Levi loop Lie algebra, prescribed first in $S$ then in the presentation of the compactification. The derivations preserve the open and the boundary, principal parts in the adjoint Levi bundle span $H^1$, and the relative chart directions complete their images to $T_sS$. Extend in the smooth ambient. The flow observation makes every characteristic covector annihilate these derivations; a horizontal covector is pulled back from $T_s^*S$, and killing fields whose projections span $T_sS$ forces it to be zero.
\end{proof}
\subsection{First adjunction}
\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}
Let $E\in\mathcal C_M^u$ be compact, hence locally coherent by $\S\ref{sec:foundations}$, and set $J_E=J\otimes^!\bar q^!E$. In the charts of Lemma~\ref{pseudo:clean-noncharacteristic} the second factor has horizontal characteristic covectors; the absence of nonzero horizontal covectors for $J$ is exactly the condition making the diagonal noncharacteristic for the exterior product. The noncharacteristic inverse-image theorem, including duality \cite[Theorems~2.4.6 and~2.7.1(ii)]{HottaTakeuchiTanisaki}, proves $J_E$ locally coherent and that Verdier duality commutes with this tensor up to smooth shift; the same calculation applies after closed direct image from a possibly singular presentation, by projection formula.
$J_E$ and its Verdier dual have zero $!$-restriction to the boundary, so $J_E$ is simultaneously the star and the extraordinary extension of its open restriction. Extraordinary extension (partial adjoint meaning: no maps from $J_E$ to an arbitrary object pushed from the boundary) is verified for arbitrary targets via compact-embedding and descent. On every affine chart of the compactification embed as closed subscheme of a smooth affine; the closed direct image of $J_E$ is bounded coherent and compact in the D-category of that smooth affine scheme. By Kashiwara an arbitrary boundary object is a colimit of bounded coherent boundary-supported D-modules; Verdier duality and vanishing of the boundary $!$-restriction of $J_E^\vee$ kill maps to each, hence by compactness to their colimits. Apply on each level of a smooth hypercover; the stack mapping complex is a limit of zeros. No global compactness of $J_E$ is needed.
Push by proper $\bar p$: using $q^*=q^![-2e]$, the simultaneous extension gives $p_!q^*E\simeq p_*q^!E[-2e]=L_PE$ with its adjunction against every target. Compact objects generate the Levi category; $L_P$ is continuous, so the adjunction extends by colimits.
\end{proof}
\subsection{Second adjointness with the determinant twist}
\begin{proposition}\label{pseudo:second-adjunction}
On the same torsion component,
\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 $\mathbb A^1$, general fiber $(g,t^{-\lambda}gt^\lambda)$, special fiber $P^-\times_M P$ \cite[Section~2.4 and Proposition~2.5.2]{DrinfeldGaitsgoryCT}. This interpolation of bundle stacks is smooth over $\mathbb A^1$, its map to $\mathbb A^1\times Y\times Y$ schematic of finite type. Let $T$ be the open substack retaining degree $u$ at the special fiber and all of the general fiber.
The determinant-line ratio of the ends has canonical splitting for $t\ne0$. It extends across $t=0$: its order is locally constant, the special fiber is a tower of vector stacks over connected $Y_M^u$, and on split reductions the order is the $\lambda$-weighted sum of Euler characteristics, $\pm Q(\lambda,u)=0$. Smoothness then extends the section to a nowhere-vanishing one. Its value at zero equals the parabolic splitting (quotient is a unit, constant along each fiber, equals one on the split section). The splitting is equivariant under dilation, holding away from zero; hence everywhere.
Push $\omega_T$ to $\mathbb A^1\times Y^2$ with this splitting. The result is conic in the first factor; its $!$-fiber at one is the identity kernel, at zero is the kernel of $E_P C_{P^-}$. Cospecialization supplies
\[
\eta:\operatorname{Id}\longrightarrow E_P C_{P^-}.
\]
Specifically, on the conic axis use $i_{1,*}i_1^!\to\operatorname{Id}\leftarrow i_{0,*}i_0^!$ and apply compactly supported image along $\mathbb A^1$; contraction makes the second arrow invertible (checked holonomically by dualizing: the star extension from the punctured axis has zero compactly supported image). Continuous extension defines the construction.
Define the counit
\[
\varepsilon:C_{P^-}E_P\longrightarrow\operatorname{Id}
\]
by restricting to the open where reductions are transverse. This open is $Y_M^u$; restriction then star image gives the arrow; the line identification is the split-reduction one.
Triangular identities. Precompose $\eta$ with $E_P$. The family is $R=Y_P^u\times_Y T$, with $Y_P^u$ on the input end. Include the general fiber and the everywhere-transverse locus of its special fiber in an open $R^o$; this open is $\mathbb A^1\times Y_P^u$ (an extra reduction is a flag in $H/P$; open orbit under 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 conjugating conjugation gives the identification across zero. The maps to output bundle and input Levi are constant on this product, and so is the determinant splitting (unit at zero, constant on affine line). Restriction gives identity cospecialization.
Other identity: use $T\times_Y Y_{P^-}^u$ with the parabolic factor on the output end; second projection of $\widetilde H$ acts on the flags; the open orbit has stabilizer (first projection) identified with $P^-$; the open family is $\mathbb A^1\times Y_{P^-}^u$ with constant maps and splitting. Same calculation. All kernel operations are safe or holonomic partial; their extension with spectators and through open exhaustions is legitimate.
\end{proof}
The two adjunctions interact with the extraordinary diagonal as
\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 apply $C_{P^-}$ to the output factor of $\Delta_!k_Y$. In the calculation $p,q$ are the maps for $P$; $q_!p^*$ is the partial left adjoint of $E_P=p_*q^!$ on locally holonomic arguments, adjoint against arbitrary targets. Proposition~\ref{pseudo:second-adjunction} provides the genuine left adjoint $C_{P^-}$; uniqueness of left-adjoint values identifies its value with $q_!p^*$, and base change 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. Note $(L'_P)^\vee$ is the right adjoint of $E_P$; so \eqref{eq:M4} identifies actual continuous functors.
\subsection{The boundary of the diagonal}
We now identify the complement of $\Delta_!k_Y$. Let $\bar Y$ classify two $H$-bundles and a section of their Vinberg bundle which is generically nondegenerate on $X$, modulo $T_H$. The parameter $v$ is constant on the complete curve. Write $b:\bar Y\to Y^2$. Over $v\in T_{\mathrm{ad}}$ there is an open immersion $j_0$ and a finite étale $Z_H$-torsor $l:Y\to\operatorname{im}(j_0)$: fixing $v=1$ gives an isomorphism between the two bundles with residual central scaling, so $bj_0l=\Delta$.
The twisted complement of $\Delta_!k_Y$ inside $\Delta_*k_Y$ in the target twist $\boxtimes(-\tau)$ is $J^b$, the image of $(j_0)_!l_*k_Y$ under proper pushforward. It is locally holonomic, and kernel duality gives $(J^b)^\vee=J'^b$, the corresponding object at the opposite twist.
\begin{proposition}\label{pseudo:boundary-factorization}
There is a finite filtration of the cones $\Delta_!k_Y/J^b$ and $\Delta_!k_Y[2\dim Y]/J'^b$ whose successive quotients are supported on the collision divisor and, on each stratum, are continuous functors factoring as
\begin{equation}\label{eq:M5}
E_M\circ\operatorname{Id}\circ C_{M^-}
\qquad(\text{or}\ +\ \operatorname{Id})
\end{equation}
through a proper standard Levi subgroup $M\ne H$ with twisted constant term and Eisenstein functors for the corresponding parabolics.
\end{proposition}
\begin{proof}
Take the stratification by parabolic type; its strata lie in the collision divisor. Let $B$ be an $H$-bundle. The closed condition at the collision divisor gives a reduction to a parabolic $P$. On each stratum where the Vinberg matrices vanish at a single point, the two flags underlying a nondegenerate section are preserved: at the generic curve point the two flags are those of $B$ and $B'$; they extend by properness. Every matrix factors through its Levi block because it does so generically, and the remaining regular maps give a $\bar M$-modification. Concretely, take positive multiples $\mu_i$ of the fundamental weights for omitted roots; their Levi blocks are lines. Stratify by lengths of zero divisors of the generically rank-one matrix maps, flatten, and on each piece divide out the equations; the resulting lines recover the flags. Their ranks, Plücker relations, and 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 open conditions concern only the modification. Therefore on each stratum the classified data are exactly
Set $J^b=\Delta_!k_Y\cdot J$, $D^b=\operatorname{cone}(J^b\to(j_0)_*l_*k_Y)$. For a boundary-stratum inclusion $i$, the identity $i^!(j_0)_*=0$ gives $i^!D^b\simeq i^!J^b[1]$. So the boundary calculation above applies to the cone. Localization with $!$-restrictions and star extensions from strata filters $D^b$. Within a fixed parameter orbit, the $!$-restrictions on every nonzero-defect stratum vanish by the preceding argument (checked on quasi-compact pieces using the locally finite stratification). Thus the cone has a finite filtration involving only defect-zero strata for proper parabolics. Push by proper $b$. On a stratum over $B$ its contribution has the form
by $!$-base change from $B$, where $K$ is a kernel on $Y_M\times Y_M$ (the map forgetting the modification is quasi-compact and safe on fixed degrees; in the defect-zero case it is the isomorphism datum modulo central scaling, acting freely relative to the two Levi bundles). If $T_K$ denotes the functor represented by $K$, the displayed kernel represents $E_P\circ T_K\circ C_{P^-}$; a 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 $bj_0l=\Delta$, $b$ proper, and $k_Y=\omega_Y[-2\dim Y]$; the two pushed kernels are therefore $\Delta_!k_Y$ and $\Delta_*\omega_Y[-2\dim Y]$.
\end{proof}
\subsection{Induction and the pseudo-identity equivalence}
Remaining: identify the two kernels. Write the stratified contribution of Proposition~\ref{pseudo:boundary-factorization} uniformly as
\[
\tau^{\bar q}(A)=\tau^{\bar p}A\otimes\tau_{\bar q}(K)\ \text{on}\ T^\bullet,
\]
where $p,q$ are the maps for a proper parabolic $P$, $A\in T^\bullet$ is the first Levi input, and the kernel is conic in the first factor. Here $\bar p:B=\widetilde Y_P\times_{Y_M}Y_{P^-}\to Y_P^-\to Y$ forgets the input reduction; $\bar q:B\to M^2$ forgets the modification and both parabolics.
The determinant-line splitting on $B$ at height one is from a modified gluing. For comparison with the moduli kernel, put $\Lambda=d_v\operatorname{ord}\mathfrak t$. Then $\bar q$ is smooth of relative dimension $2(\dim Z(H)+\dim N_P-\dim\Lambda) = 2\dim N_P$ (summing the dimension of the modification stack: $\dim\operatorname{PP}$ minus the kernel width). The twist on $B$ is the relative determinant between the modified lattices, giving $\overline K_{\mathrm{mod}}$.
For fixed Levi inputs both pushforwards of the constant object give the same kernel; smoothness of $\bar q$ and the base-change isomorphism between the modification-classifying stack and the sublocus $\operatorname{PP}_M(L^M)$ supply the identification. In \eqref{eq:M5}, since the twist splits along both factors, the correction through a standard Levi matches its correction through the opposite parity. Iterating leaves the form compatible with $F_H$; the stratifications use $g=2(\dim Y_M+\dim N_P)$ and the twists agree. Hence
\[
\Delta_!k_Y\simeq J^b+\sum_{i}c_iE_{P_i}K_iC_{P_i^-}
\]
(direct sum indices over strata); each successive quotient of the cone of $D^b\to\Delta_*k_Y$ is supported at the collision divisor where $I'$ has a $D$-submodule complementary to $I$. Here $I$ is the kernel of some $E_MK_ME_{M^-}$. So an inclusion
\[
\Delta_!k_Y\longrightarrow\Delta_*\omega_Y[-2\dim Y]
\]
is an isomorphism: on the open complement of the collision divisor both terms are the identity-modification kernel, and outside stratification the distinguished triangle splits.
The unit and counit of $F_H\ \dashv\ F_H'$ are then canonical. Being isomorphisms at the open, they are isomorphisms after pushforward. By induction on dimension of the stratum, each term of the cone factors through a proper Levi, both of which are killed by the unit-isomorphism induction hypothesis. Full faithfulness $F_H\simeq\operatorname{Id}$ follows.
\begin{theorem}[Twisted pseudo-identity, \cite{GLCIV}]\label{pseudo:equivalence}
For $H$ reductive, torsion $u\in\pi_1(H)$, $s$ with the chosen twist, the functor
\begin{equation}\label{eq:pseudo-1}
F_H=D^u(\Delta_{H,!}k_{Y_H})\ \in\ \operatorname{Funct}(\mathcal C^u,\mathcal C^u)
\end{equation}
admits the same untwisted kernel as adjoint and counit, is invertible, and coincides with the identity up to given identification.
\end{theorem}
\begin{proof}
Continuity of $F_H$ from properness of $b$ and monoidality from sheaf-theoretic construction. From Proposition~\ref{pseudo:boundary-factorization} and induction on Levi type the object $\Delta_!k_Y$ differs from $\Delta_*\omega_Y[-2\dim Y]$ by a sum of the form $c_iE_{M_i}K_iC_{M_i^-}$, using the split of the twist. Take any test object $A$; if $A$ is generated by an Eisenstein object then by (ii) or (iii) the conclusion depends on the induction hypothesis; if $c_{P^-}A=0$ then by (iii) the contribution vanishes. In both cases vanishing on a common generating set proves the identity.
\end{proof}
\section{Whittaker models}\label{sec:whittaker}
Fix a Borel $B=TN$ and the unipotent radical $N=[B,B]$, with $T$ the Cartan quotient. Let $T\subset X$ be a finite tuple of marked points. The notation \emph{positive-loop subgroup} means the part of $N(X-T)$ that extends across $T$; when it matters we specify the disk model. Characters are nondegenerate additive ones of the positive unipotent loops; on the full positive-loop group we always use one of them, fixed together with the theta characteristic $\vartheta$.
All statements below are relative to a smooth affine chart for the tuple $T$; the chart hides nothing, because the categories are crystals in $T$ by [Section~\S4, GLCII] and become global after the announced descent. Parameters will be written $S$, $R=\Gamma(S,\mathcal O_S)$, $E=z^{-1}(T)$ its Cartier divisor in the coordinate $z$.
\subsection{Pole models and the Whittaker category}
Following \cite[Sections~1--2 and~6]{GaitsgoryWhit}, we use the \emph{pole model} for the bundle stack: a Plücker point of view with a closed substack $\mathcal Y_m\subseteq\mathcal Y$ bounded by a pole divisor $m\cdot E$. The construction uses a faithful representation; the quotient of the scheme atlas by the general linear group presents $\mathcal Y$ as an increasing union of the quasi-compact QCA substacks $\mathcal Y_m$; restriction of the twist to $\mathcal Y_m$ is a Kummer twist, and $\mathcal Y_m$ is schematic, separated, finite type over $\operatorname{Bun}_H\times S$.
For a coweight cocharacter $z\in X_*(T)$ write $\Omega_X(z\cdot E)$ for the line with divisors $z\cdot E$; it acts on the Plücker coordinates by the corresponding weight. For $v\in X_*(T_{\mathrm{ad}})$ a sum of positive combinations of simple coroots, write $v=z_{\mathrm{dom}}+\gamma$ with $\gamma\in\mathbb Z\Phi^\vee$; classes in the lattice then serve as modification labels. Set
\begin{equation}\label{eq:W1}
\psi_0=\sum_{\alpha\text{ simple}}\operatorname{Res}_E\langle\alpha,\cdot\rangle\in\mathcal O_{B_N^{\mathrm{lv}}},
\end{equation}
is the zero-defect phase. The Whittaker exponentiation uses $\psi_0$ coupled with the residue pairing, and operates on the additional pole bound; the categorical condition we construct below is the pole-model realiziation of the star condition \cite[Section~3]{GaitsgoryWhit}.
\begin{definition}[Whittaker category]\label{whittaker:category}
For $s\in\mathbb C$ the category $\mathcal W_{-s}^-(S)$ is the full subcategory of $D_{L_N^{-s}}(\operatorname{Bun}_{N_0}\times S)$ on objects which are smooth star-averaged with character $-\chi_0$ along the pole-model positive-loop group: equivalently, the strongly $N$-equivariant with the appropriate Kummer twist, on every \emph{relevant} (all $\lambda_y$ dominant) stratum of the pole model. $\mathcal W^+_s(S)$ is defined with $L_N^s$ and $+\chi_0$. At irrational exponent the total Whittaker category is, using $HLop$-truncation, $\mathcal W_{-s}^-(S)$.
\end{definition}
By the pole-model construction of \cite[Section~0.4]{GaitsgoryWhit} the two signs agree under the $T$-duality, and the conditions are preserved under the parameter descent. A Whittaker object of the opposite group is produced by inversion of the characteristic.
The pole model permits the construction of models with defect zero. Set
\[
D_{s}=D(L_G^s),\quad R_s:\operatorname{Bun}_{H}(X)\times S\longrightarrow D_{L_N^s}(\operatorname{Bun}_{N_0}\times S),
\]
where $R_s$ is the map to the pole model on the Whittaker side. At the level of the pair $(H,N_0)$, the eigenvalue of the coweight $\lambda$ in the Plücker model is $\langle 2\rho,\lambda\rangle$. For the opposite choice of pole bound the labels are replaced by their negatives.
\subsection{The functional realization and its adjoints}
The \emph{functional realization} of \cite[Theorem~2.6]{GaitsgoryWhit} identifies the Whittaker category (under relevant strata) with $D$-modules on the pole model supported at the pole bound. This identification is functorial in the tuple $T$ and commutes with parameter descent. Together with the pole-model factorization, we can use it on relevant strata only.
\begin{proposition}\label{whittaker:dual-twists}
At positive defect, i.e.\ where the phase vanishes, both conditions are vacuous, and the strata live on fibered products of actual reduction data. The resulting correspondence defines a star pairing
\begin{equation}\label{eq:W2}
(q_{0,*})\bigl(\operatorname{Exp}(-\psi_0)\otimes^!R_s(-)\bigr)[-2\delta_N]:D_s\times D_{-s}\longrightarrow D(S),
\end{equation}
where $q_0:\mathfrak{n}^{\mathrm{lv}}$ is the defect-zero stratum, $\delta_N=-\chi(X,\mathfrak n_0)$, and $R_{-s}$ is used in the second slot. It preserves the parameter descent, and it is functorial in $T$, with $!$ and $*$ pulls for insertion and factorization.
\end{proposition}
The proof is the incidence structure $\{([vn],v):v\in V,\ n\in N\}$ from \eqref{eq:P7}: pull by $!$ from $V/B^-$, tensor with the exponential of $\chi_0(\cdot)$, and star extend to $V$. The identification uses [Section~6.3]{GaitsgoryWhit}, and its normalization is fixed by the constant shift normalized so that $\mathsf F(\delta_{u=0})=\omega_t$ (Fourier normalization, $\S$\ref{sec:deformation}). The correctness of the twisting line is the $K$-invariant $Z_0$-version of the same argument.
\subsection{Star and pull properties}
The following diagram of adjunctions holds on quasi-compact opens:
\[
\begin{tikzcd}
D_{s} \arrow[rr,"R_s"] \arrow[dr,"q_0"] & & D_{L_N^{-s}}(\operatorname{Bun}_{N_0}\times S)\\
& D(S). \arrow[ur,dashed,"(q_0)_*"] &
\end{tikzcd}
\]
The dashed arrow is star direct image carrying exponential $-\psi_0$. The star-pull and star-push are the \emph{positive} star operations of construction; they are continuous and preserve the character descent.
\begin{proposition}\label{whittaker:strata-properties}
Star-pull and star-push commute with parameter change, collisions, and disjoint clusters. For a stratum $S'$ with distinct-mark pole configuration $c'=n-n'$, the star restriction of $(j_\alpha)_*\omega_{S_\alpha}$ has an upper ordinary degree $c'-2n$ and an upper complex-weight bound depending only on $c'$.
\end{proposition}
\begin{proof}
The commutation statements are the functoriality of the pole model from \cite[Sections~3.2--3.4]{GaitsgoryWhit}; the degree and weight bounds are from the arrangement calculation in $\S\ref{sec:plancherel}$, which applies identically to the Whittaker side. Only the identity of the phase needs checking, and that is the zero-defect $\psi_0$ construction.
\end{proof}