==== TABLE OF CONTENTS ====
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//main.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/01-introduction.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/02-specialization.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/03-lie.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/04-hecke.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/05-two-hecke.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/06-slices.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/07-support.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/08-consequences.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/09-coefficients.tex
The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//refs.bib
===========================
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//main.tex ====
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\title{The Restricted Geometric Langlands Equivalence in Positive Characteristic}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the $\overline{\mathbb Q}_\ell$-linear restricted geometric
Langlands equivalence for smooth projective connected curves and connected
reductive groups in characteristic $p>0$, with $\ell\ne p$, in two regimes.
Over $\overline{\mathbb F}_q$, we assume the four characteristic conditions
of the restricted theory stated below. Over arbitrary algebraically closed
fields, we assume these conditions and additionally that $p$ is very good
and does not divide the Weyl-group order.
This proves Gaitsgory--Raskin's full-support conjecture
\cite[Conjecture~1.3.10]{GR} in these regimes.
\end{abstract}
\begingroup\small\tableofcontents\endgroup
\input{sections/01-introduction}
\input{sections/02-specialization}
\input{sections/03-lie}
\input{sections/04-hecke}
\input{sections/05-two-hecke}
\input{sections/06-slices}
\input{sections/07-support}
\input{sections/08-consequences}
\input{sections/09-coefficients}
\begingroup\small
\bibliographystyle{plain}
\bibliography{refs}
\endgroup
\end{document}
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//main.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
Restricted geometric Langlands theory compares automorphic sheaves with
sheaves on a stack of local systems whose variation is restricted in the
sense of Arinkin--Gaitsgory--Kazhdan--Raskin--Rozenblyum--Varshavsky
\cite{AGKRRV}. In positive characteristic, Gaitsgory--Raskin construct an
equivalence onto an open-and-closed union of spectral components
\cite[Main Theorem~1.3.9(i)]{GR}. The remaining support question asks whether
any component can be absent. We prove that none is absent in the two
precise settings below.
The geometric comparison grew from the construction of individual Hecke
eigensheaves. Over a finite field, unramified automorphic functions can be
viewed as functions on isomorphism classes of bundles; Frobenius traces
turn sheaves on the bundle stack into such functions. For general linear
groups, the constructions of Drinfeld and Laumon led to the question of
attaching a Hecke eigensheaf to each irreducible local system.
Frenkel--Gaitsgory--Vilonen reduced this existence problem to a vanishing
conjecture, which they established over finite fields using Lafforgue's
work \cite[\S0.1 and \S\S1--2]{FrenkelGaitsgoryVilonen}.
Gaitsgory subsequently proved the vanishing conjecture geometrically
\cite[\S0.1 and \S2]{GaitsgoryVanishing}.
The categorical formulation of Beilinson--Drinfeld seeks a comparison
of the whole automorphic category with a category varying over the stack
of local systems \cite[\S0.1]{AGKRRV}.
The choice of spectral category is essential. In characteristic zero,
Arinkin--Gaitsgory explained why quasi-coherent sheaves on the local-system
stack do not suffice for a general reductive group, and formulated the
comparison using ind-coherent sheaves with nilpotent singular support
\cite[\S\S1.1.1--1.1.6]{ArinkinGaitsgory}.
The extra singular direction is a nilpotent horizontal section of the
adjoint local system, reflecting the additional datum in an Arthur parameter.
Their formulation also satisfies the compatibility tests supplied by
geometric Satake and Eisenstein series. On the automorphic side,
Beilinson's theory supplies singular support for constructible sheaves
in arbitrary characteristic \cite[\S1.3]{BeilinsonSingularSupport}.
These two support theories concern different categories and play
different roles in the correspondence.
To formulate a categorical comparison for \(\ell\)-adic sheaves,
Arinkin--Gaitsgory--Kazhdan--Raskin--Rozenblyum--Varshavsky introduced
the restricted local-system stack \cite[\S\S0.1--0.2 and \S0.5.4]{AGKRRV}.
Its families are defined through the tensor category of local systems
in the chosen sheaf theory, rather than through a de Rham moduli problem.
The connected components are indexed by semisimple local systems,
while each component retains extensions and derived deformation data
\cite[Proposition~3.7.2 and Corollary~3.7.4]{AGKRRV}.
They constructed the coherent spectral action on automorphic sheaves
with nilpotent singular support and proved, under their characteristic
hypotheses, that Hecke eigensheaves have nilpotent singular support,
as predicted by Laumon
\cite[\S0.2.3 and Theorems~14.3.2 and~14.4.3]{AGKRRV}.
This supplies both the categories and the Hecke-compatible action used
in the present paper.
The characteristic-zero geometric Langlands theorem was established in
the five-part project of Arinkin, Beraldo, Campbell, Chen, Faergeman,
Gaitsgory, Lin, Raskin, and Rozenblyum. Gaitsgory--Raskin's construction
of the functor and their multiplicity-one theorem are the first and
final parts of that project \cite{GLCI,GLCV}.
Their subsequent work \cite{GR} uses geometric specialization to relate
the characteristic-zero and positive-characteristic \(\ell\)-adic theories.
It proves the primed equivalence used here and the full result for
general linear groups. Our input from characteristic zero is precisely
\cite[Main Theorem~1.3.9(ii)]{GR}.
An equivalence onto a union of components does not yet give an
automorphic category at a parameter outside that union. The issue is
therefore existence on the omitted components, even after the
categorical comparison has been constructed. Our proof addresses
this issue by showing that the geometric specialization functor
cannot annihilate the nonzero spectral summand attached to such a
component. The local tests that establish this fact are described
after the main statements.
\subsection{The spectral stack and the two hypothesis regimes}
Let $k$ be an algebraically closed field of characteristic $p>0$, let
$\ell\ne p$ be a prime, and set $E=\overline{\mathbb Q}_\ell$.
Let $X/k$ be a smooth projective connected curve and $G/k$ a connected
reductive group. Write $\check G/E$ for the Langlands dual group and
$\mathfrak g=\Lie(G)$. Such a group $G$ is split over $k$.
We use the $E$-linear sheaf categories of \cite{AGKRRV,GR}.
In particular, $\Shv(X)$ is the ind-constructible category, and
$\QLisse_E(X)$ denotes the full dg subcategory of $\Shv(X)$ whose ordinary
cohomology sheaves are filtered colimits of finite-rank lisse $E$-sheaves
\cite[\S1.2.5 and Definition~1.2.6]{AGKRRV}. Its distinction from the
ind-completion $\IndLisse_E(X)$ of constructible lisse complexes will
matter in Section~\ref{sec:specialization}. The restricted spectral
prestack $Y=LS^{\mathrm{restr}}_{\check G}(X)$ is defined on derived affine
$E$-schemes $T$ by
\begin{equation}\label{intro:restricted-stack}
Y(T)=\Fun^{\otimes,\mathrm{r.t.ex}}
\bigl(\Rep_E(\check G),\QCoh(T)\otimes_E\QLisse_E(X)\bigr).
\end{equation}
Here the superscript denotes right-t-exact symmetric monoidal functors,
with the conventions of \cite{AGKRRV}. A parameter is an $E$-point of
$Y$, hence such a tensor functor with $T=\Spec E$.
Write $\Bun_G(X)$ for the stack of principal $G$-bundles on $X$.
Its global nilpotent cone consists of cotangent Higgs fields with
nilpotent values. It defines the full dg subcategory
$\Shv_{\Nilp}(\Bun_G(X))$. On $Y$ we use the spectral nilpotent
singular-support condition and the category $\IndCoh_{\Nilp}(Y)$ of
\cite[\S\S21.2.1--21.2.5]{AGKRRV}. These are the established categories; the
new assertion concerns the spectral components on which the equivalence
is supported.
We state the characteristic assumptions individually. A Levi subgroup
always means a Levi factor of a parabolic subgroup. Semisimple Lie
elements are geometrically conjugate into the Lie algebra of a maximal
torus; nilpotent elements have zero in their geometric adjoint-orbit
closure.
\begin{hypothesis}[Common characteristic assumptions]\label{hyp:common}
The following conditions hold for $G/k$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item There is a nondegenerate $G$-invariant symmetric bilinear form
$\kappa$ on $\mathfrak g$ whose restriction to the center of
$\Lie(M)$ is nondegenerate for every Levi subgroup $M$.
\item For every Levi subgroup $M$, including $G$, and a maximal torus
$T_M\subset M$, Chevalley restriction is an isomorphism
\[
k[\Lie(M)]^M\xrightarrow{\ \sim\ }
k[\Lie(T_M)]^{W_M},\qquad W_M=N_M(T_M)/T_M.
\]
\item The scheme-theoretic centralizer in $G$ of every semisimple element of
$\mathfrak g$ is a Levi subgroup.
\item For every field extension $k'/k$, every nilpotent element of
$\mathfrak g\otimes_k k'$ belongs to $\Lie(R_u(P))$ for some
parabolic subgroup $P\subset G_{k'}$ defined over $k'$.
\end{enumerate}
\end{hypothesis}
The first three conditions are those of \cite[\S14.4.1]{AGKRRV}; the
last is the additional condition of \cite[\S D.1.1]{AGKRRV}.
These are the assumptions referenced by \cite[\S0.1.9]{GR}.
The center in condition~(i) is the center of a Lie algebra. We will
justify its relation to central cocharacters where that relation is used.
\begin{hypothesis}[Regime A: finite-field descent]\label{hyp:A}
There are $q=p^r$, a smooth projective geometrically connected curve
$X_0/\mathbb F_q$, and a split connected reductive group
$G_0/\mathbb F_q$ such that
\[
k=\overline{\mathbb F}_q,\qquad
X=X_0\times_{\mathbb F_q}k,\qquad
G=G_0\times_{\mathbb F_q}k.
\]
Hypothesis~\ref{hyp:common} holds. No additional restriction on the
order of the Weyl group is imposed in this regime.
\end{hypothesis}
\begin{hypothesis}[Regime B: arbitrary algebraically closed base]
\label{hyp:B}
The field $k$ is any algebraically closed field of characteristic $p>0$.
Hypothesis~\ref{hyp:common} holds, $p$ is very good for $G$, and
$p\nmid |W_G|$. No finite-field descent of $X$ is assumed.
\end{hypothesis}
Here very good has its usual root-system meaning: for type $A_n$,
$p\nmid n+1$; for $B_n,C_n,D_n$, $p\ne2$; for
$G_2,F_4,E_6,E_7$, $p\ne2,3$; and for $E_8$, $p\ne2,3,5$,
on every irreducible factor. A torus has no such exclusions.
We retain all the conditions in Hypothesis~\ref{hyp:B}, even where
some are redundant for a particular step. The two regimes have distinct
proofs of the needed Lie-theoretic statement; neither is used as a
substitute for the other.
\subsection{Full support}
Under Hypothesis~\ref{hyp:common}, the restricted Langlands functor
of \cite{GR} factors as
\begin{equation}\label{intro:primed}
\mathbb L_G^{\mathrm{restr}}:
\Shv_{\Nilp}(\Bun_G(X))
\xrightarrow{\ \sim\ }\IndCoh_{\Nilp}(Y')
\hookrightarrow\IndCoh_{\Nilp}(Y),
\end{equation}
where $Y'\subset Y$ is an open-and-closed union of connected
components. The functor is $\QCoh(Y)$-linear
\cite[Lemma~1.3.7 and Main Theorem~1.3.9]{GR}.
\begin{theorem}[Full support]\label{thm:full-support}
Let $k,\ell,E,X,G$ be as above. Under either
Hypothesis~\ref{hyp:A} or Hypothesis~\ref{hyp:B}, the inclusion
$Y'\subset Y$ is equality as spectral prestacks. Consequently the
given functor is a $\QCoh(Y)$-linear equivalence
\[
\mathbb L_G^{\mathrm{restr}}:
\Shv_{\Nilp}(\Bun_G(X))\xrightarrow{\ \sim\ }
\IndCoh_{\Nilp}(Y).
\]
\end{theorem}
This proves Conjecture~1.3.10 of \cite{GR} in the two displayed
regimes. The positive-characteristic primed equivalence, the full
characteristic-zero equivalence, and the full result for general linear
groups are established inputs from Main Theorem~1.3.9 of that paper.
The present theorem concerns the restricted theory with its stated
characteristic hypotheses.
Section~\ref{sec:consequences} records three consequences. Localizing
the given functor at any spectral component gives the corresponding
nonzero ind-coherent category. In regime A, the known primed trace
formula becomes the full arithmetic formula using derived geometric
Frobenius fixed points. For an arbitrary parameter, spectral linearity
produces a nonzero Hecke eigenobject with coherent tensor and fusion
compatibilities. Its asserted category is ind-constructible and
nilpotent; no bounded constructibility or perversity assertion is needed.
\subsection{Rational coefficients}
There is also a coefficient form of the support question. Put
$E_0=\mathbb Q_\ell$, retaining $E=\overline{\mathbb Q}_\ell$.
For the rational support statement let $k=\overline{\mathbb F}_q$, and let $X/k$ be
smooth projective connected and $G/k$ connected reductive satisfying
Hypothesis~\ref{hyp:common}. Let $\check G_0/E_0$ be the split dual group,
with $\check G=\check G_0\times_{E_0}E$. Define $Y_0$ by
\eqref{intro:restricted-stack} with $E_0$ in place of $E$, and put
$\mathcal C_0=\Shv_{\Nilp,E_0}(\Bun_G)$, using the rational
ind-constructible sheaf category and the same nilpotent condition.
These coefficient changes leave the geometric field $k$ unchanged.
Universal evaluation sends a representation to its local system in the
universal family on $Y_0$; for a finite set of legs it uses the exterior
products of these local systems. The rational spectral-action datum
consists of a continuous
$\QCoh(Y_0)$-action on $\mathcal C_0$ whose restriction along universal
evaluation is the usual multi-leg Hecke system, with its tensor,
permutation and collision compatibilities. We take this datum as part
of the rational support problem. The rational geometric Satake and
derived Hecke constructions are recalled in Section~\ref{sec:coefficients}.
We distinguish the given rational action and support from a construction
of a normalized rational Langlands equivalence: the cited theorem of
Gaitsgory--Raskin is used over $E$.
\begin{theorem}[Rational spectral support]\label{thm:rational-support}
For the rational datum just specified, every nonempty open-and-closed
substack $U\subset Y_0$ has
\[
\QCoh(U)\underset{\QCoh(Y_0)}\otimes\mathcal C_0\ne0.
\]
Consequently, if $Y'_0\subset Y_0$ is an open-and-closed support union
whose complementary structure idempotent acts by zero on
$\mathcal C_0$, then $Y'_0=Y_0$
as spectral prestacks. In particular this applies to the support of
any given $\QCoh(Y_0)$-linear primed equivalence.
\end{theorem}
The conclusion concerns the entire rational spectral prestack, without
identifying its components with rational points. A curve over
$\overline{\mathbb F}_q$ descends to a finite extension of
$\mathbb F_q$ by finite presentation; a split form of $G$ does as well.
Thus Theorem~\ref{thm:full-support} in regime A supplies the geometric
input without adding a Weyl-order restriction. No rational extension
of regime B is asserted here.
Section~\ref{sec:coefficients} proves the required coefficient comparison
on sheaf categories, affine families and the coherent spectral action.
It then descends the absence of a clopen summand acting by zero.
The rational arithmetic statement has a further input: a canonical
primed trace formula over $E_0$ with the stated Frobenius and
ind-coherent conventions. We prove unpriming conditional on that input;
we do not construct the rational primed comparison map. The
unconditional arithmetic formula above remains over $E$.
\subsection{The specialization and slicing argument}
We first outline the proof of Theorem~\ref{thm:full-support} over $E$.
A missing spectral component leads to a contradiction as follows.
Lift $X$ to a smooth proper curve $\mathscr X$ over the Witt trait,
lift $G$, and let $\mathcal B$ be the relative bundle stack. Write
$K$ for an algebraic closure of the fraction field and $\Psi$ for
geometric nearby cycles with monodromy forgotten. The established
specialization formalism and characteristic-zero equivalence produce,
from a missing component, a nonzero compact sheaf $F$ on
$\mathcal B_K$ with $\Psi F=0$.
Section~\ref{sec:specialization} strengthens this vanishing to Hecke
transforms pulled back along maps whose bundle projection is smooth.
The leg map may be arbitrary. This extra freedom is needed after
cutting a Hecke correspondence by a hypersurface.
On a smooth chart $U\to\mathcal B$, a special-fiber cotangent vector
outside the nilpotent cone gives a test function $f$. If that covector
comes from a Higgs field, Sections~\ref{sec:lie} and~\ref{sec:hecke}
construct a modification with a nonzero leg covector and transverse
zeros on both fixed-side Hecke fibers. The two characteristic regimes
enter only in constructing the Levi and central cocharacter required
for this calculation.
Section~\ref{sec:twohecke} takes two modifications with common output
and common leg. Proper pushforward and the reverse transversality
isolate the stalk at the diagonal. The forward transversality then
gives an actual \'{e}tale local equation
\[
\sum_{i=1}^m u_i v_i+f=0,
\]
with the sheaf pulled back from $U$. Here $m$ is the Hecke-orbit
dimension and $(u_i,v_i)$ are local coordinates transverse to the
chart and leg. This is the point where the two correspondences are
both essential.
Section~\ref{sec:slices} compares the projective closure of this
quadratic equation with its hyperplane classes. The remaining derived
sheaf is a constant line supported on $f=0$. A projective incidence
construction then gives the same vanishing on higher-codimension
slices whose cotangent planes avoid the nilpotent cone.
Finally, Section~\ref{sec:support} uses the nilpotent dimension bound
to choose such a slice through the actual support of $F$. Its support
is finite over a henselian trait near the chosen point, so its nearby
cycles are a finite direct sum of stalks with at least one nonzero
summand. This contradicts slice vanishing.
The central-in-a-Levi modification and moving-leg calculation originate
in \cite[\S\S20.4--20.7]{AGKRRV}. The additional argument here is the
two-correspondence transversality and slicing mechanism, including
the exact local equation and derived comparison triangles. All
singular-support dimension bounds used in the proof are on the
special fiber; the nearby-cycle tests themselves are proved directly.
The arithmetic application combines the automorphic trace comparison
of \cite{AGKRRVTrace} with the spectral trace calculation of
Beraldo--Lin--Reeves \cite[\S\S6.4.12--6.4.13]{BeraldoLinReeves}, as
assembled in \cite[\S1.5]{GR}. The coefficient argument in
Section~\ref{sec:coefficients} uses constructible coefficient
continuity \cite{HemoRicharzScholbach} and the universal-local-acyclicity
criterion of \cite{HansenScholze}; the rational spectral action and
primed arithmetic map remain the given data stated above.
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/01-introduction.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/02-specialization.tex ====
\section{Specialization and a missing spectral component}
\label{sec:specialization}
We first translate a missing component into a geometric vanishing statement.
The object we obtain will be compact, hence constructible on finite-type
charts. More importantly, its Hecke transforms will have zero nearby cycles
after any pullback whose projection to the bundle stack is smooth. The map
recording the Hecke point need not be smooth. This last qualification is
what permits the hypersurface tests in Sections~\ref{sec:twohecke}
and~\ref{sec:slices}.
Fix data satisfying Hypothesis~\ref{hyp:A} or Hypothesis~\ref{hyp:B}.
Nothing in this section assumes the full-support conclusion of
Theorem~\ref{thm:full-support}.
\subsection{A trait and geometric specialization}
Initially set
\[
R=W(k),\qquad S=\Spec(R),\qquad
K=\overline{\operatorname{Frac}(R)}.
\]
We fix the algebraic closure $K$ throughout. Since $k$ is perfect, every
nonzero Witt vector is a power of $p$ times a unit. Thus $R$ is a complete
Noetherian discrete valuation ring with uniformizer $p$ and residue field
$k$. It is strictly henselian, and it is excellent because it is a complete
Noetherian local ring; see \cite[Tag~07QS]{Stacks}.
Choose a smooth projective lift $\mathscr X\to S$ of $X$.
For completeness, such a lift exists over $W(k)$ for every algebraically
closed $k$ in either hypothesis. At each infinitesimal lifting step the
obstruction for a smooth curve lies in $H^2(X,T_X)$, which vanishes.
An ample invertible sheaf also lifts, since its obstruction lies in
$H^2(X,\mathcal O_X)=0$. Algebraization of the resulting proper formal
curve with an ample invertible sheaf gives a projective lift. It is smooth
near the special fiber; the nonsmooth locus, being closed in a proper scheme
over the local base, would meet the special fiber if it were nonempty.
Hence the lift is smooth everywhere. These deformation and algebraization
statements can be found in \cite[Tags~0DZQ and~0E7R]{Stacks}; the same lift is
used in \cite[\S3.1.1]{GR}. This construction requires no finite-field
descent of $X$.
The root datum of the split group $G$ gives a split reductive group over
$\mathbb Z$ and hence over $S$
\cite[Theorems~6.1.16(2) and~6.1.17]{Conrad}; we retain the notation
$G$ for this lift.
Write
\[
\mathcal B=\Bun_G(\mathscr X/S),\qquad
s=\Spec(k),\qquad \eta=\Spec(K).
\]
The stack $\mathcal B=\Hom_S(\mathscr X,B_SG)$ is algebraic and locally
of finite presentation over $S$, with affine diagonal, by
\cite[Theorem~1.2(i)--(iii)]{HallRydh}. Indeed, $\mathscr X/S$ is proper,
flat and finitely presented, while the classifying stack $B_SG$ is
locally of finite presentation and quasi-separated, with affine
stabilizers and affine diagonal. The obstruction group for deforming
a bundle on a curve is the second cohomology of its adjoint bundle,
which vanishes. Thus $\mathcal B$ is smooth over $S$.
We will replace $R$ by its integral closure in a finite extension of
$\operatorname{Frac}(R)$ inside $K$ when necessary. Such an integral closure
is a finite complete discrete valuation ring with residue field $k$;
see \cite[Tags~0EXQ and~0323]{Stacks}. It remains excellent and strictly
henselian. We retain $R,S,\mathscr X$ and $\mathcal B$ for the resulting
models. All dimension arguments below take place over these Noetherian
traits, not over the integral closure in the entire field $K$.
For a finite-type scheme or algebraic space $T$ over $S$, let
\[
\Psi_T:\Shv(T_K)\longrightarrow\Shv(T_s)
\]
be geometric specialization, with monodromy forgotten. Here, on a
finite-type scheme over a field, $\Shv$ is the ind-completion of the
category of bounded constructible $E$-adic complexes. We use ordinary
$*$-pullbacks, ordinary tensor products and ordinary stalks. The same
notation applies to locally finite-type algebraic stacks by smooth descent.
In particular, the functor $\Psi_{\mathcal B}$ is defined on the entire
sheaf category.
We recall how the construction in \cite[\S\S4.1.1--4.1.5]{GR} fixes the
meaning of this functor. On a finite-type affine chart one first uses
geometric nearby cycles for finite coefficient rings at finite extensions
of the generic field. One then passes through the compatible coefficient
limits, inverts $\ell$, extends the coefficient field to $E$, and extends
continuously from constructible objects. Compatibility with smooth
pullback gives the stack version. In particular, $\Psi_T$ is an exact
functor of stable categories, is continuous, and preserves constructibility.
It commutes with smooth pullback and representable proper pushforward
\cite[\S4.1.6]{GR}.
These functors are unchanged by the allowed finite trait replacements,
using the identified geometric fibers. Indeed, after fixing one finite
extension, the finite subextensions of $K$ containing it form a cofinal
subsystem: enlarge any stage by taking its compositum with the fixed
extension. The models and torsion nearby-cycle functors at the remaining
stages are the same. All subsequent coefficient operations and
ind-extensions therefore agree. For $T=S$, the functor $\Psi_S$ is the
identity on the underlying complexes of geometric-point coefficients.
At no stage do we take inertia invariants.
We will also use the following tensor compatibility. If
$a:T\to\mathcal B$ is smooth and $\mathcal L$ is a finite-rank lisse sheaf
on $T$, then for any $D\in\Shv(\mathcal B_K)$,
\begin{equation}
\Psi_T\bigl(a_K^*D\otimes\mathcal L_K\bigr)
\simeq
a_s^*\Psi_{\mathcal B}(D)\otimes\mathcal L_s.
\label{eq:spec-lisse-tensor}
\end{equation}
A lisse sheaf on the source of a smooth morphism is universally locally
acyclic over the target. Thus \eqref{eq:spec-lisse-tensor} is the
universally locally acyclic tensor-pullback formula of
\cite[Lemma~4.1.9]{GR}. Equivalently, it follows from smooth base change
and the tensor formula for a lisse sheaf extending over the model.
Continuity makes the formula applicable to arbitrary $D$ in the indicated
ind-category.
\subsection{The spectral summand and compactness}
Let
\[
\mathcal C_s=\Shv_{\Nilp}(\mathcal B_s),\qquad
\mathcal C_K=\Shv_{\Nilp}(\mathcal B_K),\qquad
Y_K=LS^{\mathrm{restr}}_{\check G}(\mathscr X_K).
\]
Recall that $Y=LS^{\mathrm{restr}}_{\check G}(X)$ denotes the special-fiber
spectral stack. Restriction to the special fiber gives an equivalence
$\QLisse(\mathscr X)\simeq\QLisse(X)$
\cite[\S3.1.3]{GR}. Composing its inverse with restriction to the
geometric generic fiber defines the symmetric monoidal functor
\[
\rho:\QLisse(X)\longrightarrow\QLisse(\mathscr X_K).
\]
The induced map
\[
i:Y\hookrightarrow Y_K
\]
identifies $Y$ with an open-and-closed union of connected components
\cite[\S3.1.3]{GR}. In particular,
\begin{equation}
\mathcal D_Y
:=\QCoh(Y)\underset{\QCoh(Y_K)}\otimes\mathcal C_K
\label{eq:spec-DY}
\end{equation}
is a direct summand of $\mathcal C_K$.
We need the compatibility of \emph{actual} specialization with this
summand. Corollary~4.3.6 and \S4.3.9 of \cite{GR} show that the restriction
of the chartwise functor $\Psi_{\mathcal B}$ constructed above gives
\[
\Psi_{\mathcal B}:\mathcal D_Y\longrightarrow\mathcal C_s.
\]
It is $\QCoh(Y)$-linear by \cite[Theorem~3.1.6(B) and \S4.3.10]{GR}.
Thus the functor used here is not merely one obtained by transporting a
spectral functor across Langlands equivalences. Its values on smooth charts
are the geometric nearby cycles used later.
The characteristic-zero equivalence of \cite[Theorem~1.3.9(ii)]{GR} and
its spectral linearity identify
\begin{equation}
\mathcal D_Y\simeq\IndCoh_{\Nilp}(Y).
\label{eq:spec-generic-equivalence}
\end{equation}
In positive characteristic we use only the given equivalence onto the
primed union
\[
\mathcal C_s\simeq\IndCoh_{\Nilp}(Y'),\qquad Y'\subseteq Y.
\]
\begin{proposition}
\label{spec:missing}
Suppose a connected component $C\subset Y$ is disjoint from $Y'$.
Then
\[
\mathcal D_C
:=\QCoh(C)\underset{\QCoh(Y)}\otimes\mathcal D_Y
\]
is nonzero, and $\Psi_{\mathcal B}(D)=0$ for every
$D\in\mathcal D_C$. Moreover, there exists a nonzero
$F\in\mathcal D_C$ that is compact both in $\mathcal C_K$ and in
$\Shv(\mathcal B_K)$. For every finite-type smooth scheme chart
$b:U\to\mathcal B$, the complex $b_K^*F$ is bounded constructible.
\end{proposition}
\begin{proof}
Let $e_C\in\QCoh(Y)$ be the structure sheaf of $C$ extended by zero.
Its action is the projection onto $\mathcal D_C$. On $\mathcal C_s$ this
object acts by zero, since $C\cap Y'=\varnothing$ and the primed equivalence
is spectral-linear. Hence, for $D\in\mathcal D_C$,
\[
\Psi_{\mathcal B}(D)
\simeq\Psi_{\mathcal B}(e_C\cdot D)
\simeq e_C\cdot\Psi_{\mathcal B}(D)=0.
\]
By \eqref{eq:spec-generic-equivalence}, the category $\mathcal D_C$ is
identified with $\IndCoh_{\Nilp}(C)$. It is nonzero: the fully faithful
functor from $\QCoh(C)$ into $\IndCoh_{\Nilp}(C)$ sends the nonzero object
$\mathcal O_C$ to a nonzero object; see \cite[\S\S1.3.1--1.3.4]{GR}.
The category $\mathcal D_Y$ is compactly generated
\cite[\S3.1.7]{GR}. The binary decomposition
\[
\mathcal D_Y\simeq\mathcal D_C\times\mathcal D_{Y\setminus C}
\]
shows that projection onto $\mathcal D_C$ preserves compactness: its right
adjoint, the inclusion of that factor, is continuous. Project compact
generators of $\mathcal D_Y$. At least one projection is nonzero, because
otherwise their colimit closure would give $\mathcal D_C=0$. Choose such
an object $F$.
The inclusion of either direct factor also preserves compactness, since
its right adjoint is the continuous projection. Applying this first to
$\mathcal D_C\subset\mathcal D_Y$ and then to
$\mathcal D_Y\subset\mathcal C_K$ shows that $F$ is compact in
$\mathcal C_K$. The nilpotent inclusion
$\mathcal C_K\hookrightarrow\Shv(\mathcal B_K)$ preserves compactness by
\cite[Theorem~1.1.7]{GR}. Finally, a compact sheaf on an algebraic stack
is constructible on smooth finite-type charts
\cite[\S F.2.2]{AGKRRV}. This proves the last assertion.
\end{proof}
The argument uses one missing component and a binary direct-factor
decomposition. It does not require that $Y$ have finitely many components
or that the unit $e_C$ be compact. If $Y'\ne Y$, such a component $C$
exists because $Y'$ is a union of the open-and-closed components of $Y$.
For the rest of the geometric proof, fix $C$ and $F$ as in
Proposition~\ref{spec:missing}.
\subsection{Hecke vanishing with an arbitrary leg}
Let $V$ be a finite-dimensional representation of $\check G$, and let
\[
\mathrm H_V:\mathcal C_K\longrightarrow
\Shv(\mathcal B_K\times_K\mathscr X_K)
\]
be the one-leg Hecke functor, with the normalization giving its usual
spectral-action description. The correspondence and its kernel will be
specified in Section~\ref{sec:hecke}. Here we use the universal evaluation
objects
\[
\mathcal E_{V,s}\in\QCoh(Y)\otimes\QLisse(X),\qquad
\mathcal E_{V,K}\in\QCoh(Y_K)\otimes\QLisse(\mathscr X_K).
\]
For an affine test scheme mapping to a spectral stack, the restriction of
$\mathcal E_V$ is the family of local systems obtained by applying the
corresponding tensor functor to $V$. Acting by its first tensor factor and
then taking the exterior product gives the Hecke functor
\cite[\S14.3.3]{AGKRRV}.
The definition of $i$ implies
\begin{equation}
(i^*\otimes\id)(\mathcal E_{V,K})
\simeq (\id\otimes\rho)(\mathcal E_{V,s}).
\label{eq:spec-universal-family}
\end{equation}
Indeed, this identity holds on every affine test mapping to $Y$: both sides
are the local system obtained by extending the specified special-fiber
family. To pass from these tests to the global identity, note that
$\QLisse$ of a smooth curve is compactly generated and hence dualizable
as a presentable dg category
\cite[\S1.2.11 and \S\S E.2.6--E.2.7]{AGKRRV}.
Tensoring with a dualizable category preserves limits, so it commutes with
the descent limit defining $\QCoh(Y)$. This justifies the passage from the
affine identities to \eqref{eq:spec-universal-family}.
A second, distinct input is
\begin{equation}
\mathcal E_{V,s}\in\QCoh(Y)\otimes\IndLisse(X),
\qquad \IndLisse(X)=\operatorname{Ind}(\operatorname{Lisse}(X)),
\label{eq:spec-IndLisse}
\end{equation}
proved in \cite[Remark~14.3.9]{AGKRRV}. Here
$\operatorname{Lisse}(X)$ consists of bounded constructible complexes
whose ordinary cohomology sheaves are finite-rank lisse sheaves.
We do not identify $\IndLisse(X)$ with $\QLisse(X)$: that identification
fails for $X=\mathbb P^1$. The dualizability used in descent and the
stronger membership assertion \eqref{eq:spec-IndLisse} have different roles.
Applying the $\QCoh(Y)$-action in \eqref{eq:spec-IndLisse} to $F$ gives an
object of $\mathcal D_C\otimes\IndLisse(X)$. After applying
$\id\otimes\rho$ and the exterior-product functor, its image is $\mathrm H_V(F)$
by \eqref{eq:spec-universal-family}. Elementary tensors generate the tensor
product of presentable dg categories under colimits. Bounded truncations
also build every object of $\operatorname{Lisse}(X)$ from finite-rank
lisse sheaves. Consequently $\mathrm H_V(F)$ belongs to the stable colimit closure
of objects
\begin{equation}
D\boxtimes\mathcal L_K,\qquad D\in\mathcal D_C,
\label{eq:spec-exterior-generators}
\end{equation}
where $\mathcal L_K$ is the restriction of a finite-rank lisse sheaf
$\mathcal L_S$ on $\mathscr X$.
To see the extension assertion, proper henselian invariance identifies the
finite \emph{\'etale} covers of $\mathscr X$ with those of $X$
\cite[Tag~0A48]{Stacks}. Apply this to the compatible finite coefficient
local systems of a lisse lattice, then invert $\ell$ and extend coefficients.
The resulting extension is the one inducing $\rho$.
\begin{lemma}
\label{spec:vanishing}
Let $a:T\to\mathcal B$ be smooth, where $T$ is a scheme or algebraic space
of finite type over $S$, and let $\ell_T:T\to\mathscr X$ be any $S$-morphism.
For $F$ as in Proposition~\ref{spec:missing} and every finite-dimensional
representation $V$ of $\check G$, one has
\[
\Psi_T(a_K^*F)=0,
\qquad
\Psi_T\bigl((a,\ell_T)_K^*\mathrm H_V(F)\bigr)=0.
\]
The same statements hold on the smooth locus of $a$, and after any of the
finite trait replacements described above.
\end{lemma}
\begin{proof}
The first equality follows from smooth base change and
$\Psi_{\mathcal B}(F)=0$.
For an exterior product in \eqref{eq:spec-exterior-generators}, ordinary
pullback gives
\[
(a,\ell_T)_K^*(D\boxtimes\mathcal L_K)
=a_K^*D\otimes\ell_{T,K}^*\mathcal L_K.
\]
The sheaf $\ell_T^*\mathcal L_S$ is lisse on $T$, regardless of the
smoothness of $\ell_T$. Formula~\eqref{eq:spec-lisse-tensor} and
Proposition~\ref{spec:missing} therefore give
\[
\Psi_T\bigl(a_K^*D\otimes\ell_{T,K}^*\mathcal L_K\bigr)
\simeq a_s^*\Psi_{\mathcal B}(D)\otimes\ell_{T,s}^*\mathcal L_s=0.
\]
Both pullback and specialization are continuous, so the same vanishing
holds for the stable colimit closure of these exterior products, which
contains $\mathrm H_V(F)$. The calculation is local on the smooth locus, and the
finite-extension assertion follows from the invariance of geometric
specialization already proved.
\end{proof}
We have now obtained the two vanishings needed for the geometric argument.
Their proof uses smoothness only of the map to the bundle stack; it makes
no smooth-base-change assertion for the combined map $(a,\ell_T)$.
The next two sections construct modifications that turn a non-nilpotent
covector into a test to which Lemma~\ref{spec:vanishing} applies.
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/02-specialization.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/03-lie.tex ====
\section{A Levi reduction for a formal Higgs field}\label{sec:lie}
The Hecke modification in the next section must preserve a given Higgs
field and produce a nonzero covector in the direction of the moving point.
We obtain both properties from a cocharacter in the center of a Levi
subgroup. The following proposition is the precise Lie-theoretic input.
Its two characteristic arguments are kept separate.
For a cocharacter $\lambda:\mathbb G_m\to G$, write
$d\lambda\in\mathfrak g$ for the image of $1\in\Lie(\mathbb G_m)=k$.
We distinguish the group center $Z(M)$ from the Lie-algebra center
$\mathfrak z(\mathfrak m)$, where $\mathfrak m=\Lie(M)$.
Write $\chi:\mathfrak g\to\mathfrak g/\!/G$ for the adjoint quotient.
\begin{proposition}\label{lie:input}
Let $k$ and $G$ satisfy Hypothesis~\ref{hyp:common} and either
Hypothesis~\ref{hyp:A} or Hypothesis~\ref{hyp:B}, and let $\kappa$ be
the invariant symmetric form in those hypotheses.
Suppose that $a(t)\in\mathfrak g[[t]]$ satisfies
$\chi(a(0))\ne\chi(0)$. After a change of frame by an element of
$G(k[[t]])$, there are a Levi subgroup $M\subset G$ and a cocharacter
$\lambda:\mathbb G_m\to Z(M)$ such that
\begin{equation}\label{eq:lie-output}
\begin{gathered}
a(t)\in\mathfrak m[[t]],\qquad \kappa(a(0),d\lambda)\ne0,\\
\operatorname{ad}_{a(0)}:\mathfrak g/\mathfrak m\longrightarrow
\mathfrak g/\mathfrak m\ \text{is invertible}.
\end{gathered}
\end{equation}
Here $\operatorname{ad}_x(y)=[x,y]$. The subgroup $M$ may equal $G$, and the
invertibility assertion includes the zero vector space.
\end{proposition}
The construction of a central modification follows the method of
\cite[\S\S20.4--20.5]{AGKRRV}. We give the characteristic and lattice
arguments explicitly, since a vector in $\mathfrak z(\mathfrak m)$
alone does not specify a cocharacter of $Z(M)$.
\subsection{Root pairings and Jordan parts}
Fix a maximal torus $T\subset G$. Put
$\Lambda=X_*(T)$, $\Lambda^\vee=X^*(T)$ and
$\mathfrak t=\Lie(T)=\Lambda\otimes_{\mathbb Z}k$.
For a root $\alpha$, its differential is the linear form
$d\alpha:\mathfrak t\to k$, whereas $d\alpha^\vee$ is the vector
obtained by differentiating its coroot.
\begin{lemma}\label{lie:root-pairing}
The form $\kappa$ is nondegenerate on $\mathfrak t$. For every root
$\alpha$, there is a scalar $c_\alpha\in k^\times$ such that
\begin{equation}\label{eq:root-coroot-pairing}
\kappa(d\alpha^\vee,h)=c_\alpha\,d\alpha(h)
\qquad(h\in\mathfrak t).
\end{equation}
In particular, neither $d\alpha$ nor $d\alpha^\vee$ vanishes.
Every element of $\mathfrak g$ is conjugate into the Lie algebra of a
Borel subgroup containing $T$. Every semisimple element is conjugate
into $\mathfrak t$.
\end{lemma}
\begin{proof}
The $T$-weight decomposition is
$\mathfrak g=\mathfrak t\oplus\bigoplus_\alpha\mathfrak g_\alpha$.
Invariance under $T$ makes $\mathfrak t$ orthogonal to the root spaces
and makes $\mathfrak g_\alpha$ pair only with
$\mathfrak g_{-\alpha}$. Nondegeneracy of $\kappa$ therefore gives
nondegeneracy on $\mathfrak t$ and on each opposite-root pairing.
Choose compatible root vectors $e_\alpha,e_{-\alpha}$ with
$[e_\alpha,e_{-\alpha}]=d\alpha^\vee$. This identity follows by
differentiating the rank-one root homomorphism from $\mathrm{SL}_2$;
it is valid for every isogeny type; see the root-subgroup construction
in \cite[\S\S8.1.1--8.1.4]{Springer} and the bracket formula in
\cite[Corollary~5.1.12, Equation~(5.1.4)]{Conrad}. Invariance of the form gives
\eqref{eq:root-coroot-pairing} with
$c_\alpha=\kappa(e_\alpha,e_{-\alpha})\ne0$.
Thus one of $d\alpha,d\alpha^\vee$ vanishes if and only if the other
does. Their simultaneous vanishing would put
$\alpha\in p\Lambda^\vee$ and $\alpha^\vee\in p\Lambda$,
contradicting the integral identity
$\langle\alpha,\alpha^\vee\rangle=2$, since $p^2$ does not divide~$2$.
These arguments use torus characters, rather than just their
differentials, and also apply in characteristic two.
Let $B\supset T$ be a Borel subgroup with Lie algebra $\mathfrak b$.
The incidence morphism
\[
G\times^B\mathfrak b\longrightarrow\mathfrak g,
\qquad [g,x]\longmapsto\Ad(g)x,
\]
is proper: its source is closed in $(G/B)\times\mathfrak g$.
Choose $c\in\mathfrak t$ outside the root differential hyperplanes.
At $[1,c]$, the negative-root tangent directions map isomorphically
onto the negative root spaces, and $\mathfrak b$ supplies the remaining
directions. The morphism is consequently dominant, hence surjective.
For a semisimple element in $\mathfrak b$, write it as $c+u$, with
$c\in\mathfrak t$ and $u$ in the positive-root Lie algebra.
Conjugate by positive root groups in increasing root height. If
$d\alpha(c)\ne0$, conjugation by the $\alpha$-root group changes the
$\alpha$ coefficient by $-z\,d\alpha(c)$ and changes the other
positive-root terms only in greater height. It therefore eliminates
that coefficient. After all such steps the result is $c+u_0$ with
$[c,u_0]=0$. In a faithful representation, $c$ is semisimple and
$u_0$ is nilpotent. Since their sum is semisimple, the matrix Jordan
decomposition forces $u_0=0$.
\end{proof}
Choose a closed faithful representation $G\hookrightarrow\mathrm{GL}(V)$.
Its differential identifies $\mathfrak g$ with a restricted Lie
subalgebra of $\End(V)$: the operation $x\mapsto x^{[p]}$ is matrix
$p$th power. This compatibility can also be defined intrinsically
by the $p$th iterate of a left-invariant derivation. We use the
ordinary matrix Jordan decomposition, and below prove that its two
parts belong to $\mathfrak g$ in each regime; compare
\cite[\S4.4, Theorem~4.4.20]{Springer}.
\begin{lemma}\label{lie:constant-parts}
Suppose that $a_0=s+n\in\mathfrak g$, where $s,n\in\mathfrak g$,
$s$ is semisimple, $n$ is nilpotent in a faithful representation,
and $[s,n]=0$. Suppose also that $\chi(a_0)\ne\chi(0)$.
After conjugation, $s$ is a nonzero element of $\mathfrak t$ and
$M=Z_G(s)$ is a Levi subgroup. One has $a_0\in\mathfrak m$ and
$\operatorname{ad}_{a_0}$ is invertible on $\mathfrak g/\mathfrak m$.
For every $\lambda:\mathbb G_m\to Z(M)$,
\begin{equation}\label{eq:nilpotent-orthogonality}
\kappa(a_0,d\lambda)=\kappa(s,d\lambda).
\end{equation}
\end{lemma}
\begin{proof}
Lemma~\ref{lie:root-pairing} conjugates $s$ into $\mathfrak t$.
If $s=0$, apply the Borel assertion of that lemma to $n$.
The projection $\mathfrak b\to\mathfrak t$ preserves restricted
$p$-operations. Since $n$ is killed by an iterated $p$-operation
and that operation is injective on $\mathfrak t$, its toral
projection is zero. A strictly dominant cocharacter contracts the
positive-root Lie algebra to zero. Hence every invariant polynomial
has the same value on $n$ and on zero, a contradiction. Thus $s\ne0$.
By Hypothesis~\ref{hyp:common}, $M=Z_G(s)$ is a Levi subgroup.
Its Lie algebra is the kernel of $\operatorname{ad}_s$, and it contains $a_0$.
On $\mathfrak g/\mathfrak m$, the operator $\operatorname{ad}_s$ is diagonalizable
with nonzero eigenvalues. The commuting operator $\operatorname{ad}_n$ is
nilpotent, since
$\operatorname{ad}_n^{p^e}=\operatorname{ad}_{n^{[p^e]}}=0$ for large $e$.
On each eigenspace of $\operatorname{ad}_s$, the sum
$\operatorname{ad}_{a_0}=\operatorname{ad}_s+\operatorname{ad}_n$ is therefore invertible.
To prove \eqref{eq:nilpotent-orthogonality}, conjugate $n$ inside $M$
into the Lie algebra of a Borel subgroup of $M$ containing $T$.
The incidence argument of Lemma~\ref{lie:root-pairing} applies to
$M$: its roots have nonzero differentials, and $\kappa$ is
nondegenerate on $\mathfrak m$ by the same weight decomposition.
Restricted nilpotence again makes the toral projection of $n$ zero.
Thus the conjugated $n$ lies in the positive-root Lie algebra of $M$,
which is orthogonal to $\mathfrak t$. Since $d\lambda\in\mathfrak t$
is fixed by $M$, invariance of $\kappa$ gives
$\kappa(n,d\lambda)=0$ before conjugation as well.
\end{proof}
It remains to choose the central cocharacter with nonzero pairing.
In the finite-field regime we prove an integral lattice statement;
in the arbitrary-field regime we use the separately assumed
invertibility of the Weyl-group order.
\subsection{The finite-field regime}
The lattice statement below uses only the common invariant-form and
Levi-center assumptions. We will then combine it with the finite-field
construction of the Jordan parts.
\begin{lemma}\label{lie:lattice}
Assume the invariant-form and Levi-center conditions of
Hypothesis~\ref{hyp:common}. Every Levi Cartan determinant is prime
to $p$. For every Levi subgroup $M$ containing $T$,
\begin{equation}\label{eq:central-span}
\operatorname{span}_k\{d\lambda:
\lambda\in X_*(Z(M))\}=\mathfrak z(\mathfrak m).
\end{equation}
Here $X_*(Z(M))$ denotes actual cocharacters of the group center,
viewed in $X_*(T)$.
\end{lemma}
\begin{proof}
After choosing a positive system, let
$I=\{\alpha_1,\ldots,\alpha_r\}$ be the simple roots of $M$.
Set $J_I=\operatorname{span}_k\{d\alpha_i^\vee\}\subset\mathfrak t$.
Because every root differential is nonzero, a central element of
$\mathfrak m$ has no root-space component. Equation
\eqref{eq:root-coroot-pairing} therefore identifies
\[
\mathfrak z(\mathfrak m)
=\bigcap_i\ker(d\alpha_i)=J_I^\perp.
\]
The form is nondegenerate on this center by hypothesis, and on
$\mathfrak t$ by Lemma~\ref{lie:root-pairing}; it follows that it is
nondegenerate on $J_I$.
Define integral maps
\[
U:\Lambda\longrightarrow\mathbb Z^r,\quad
y\longmapsto(\langle\alpha_i,y\rangle)_i,
\qquad
V:\mathbb Z^r\longrightarrow\Lambda,\quad
e_j\longmapsto\alpha_j^\vee.
\]
Their composite is the Cartan matrix
$C_I=UV=(\langle\alpha_i,\alpha_j^\vee\rangle)_{i,j}$.
A subscript $k$ will mean reduction to $k$.
Equation~\eqref{eq:root-coroot-pairing} gives
$V_k^*\kappa=D U_k$, where
$D=\operatorname{diag}(c_{\alpha_1},\ldots,c_{\alpha_r})$ is invertible.
The radical of $V_k^*\kappa V_k$ is exactly $\ker V_k$, by
nondegeneracy on $J_I=\im V_k$. Consequently
\begin{equation}\label{eq:cartan-ranks}
\operatorname{rank}(C_{I,k})
=\operatorname{rank}(V_k)=\operatorname{rank}(U_k).
\end{equation}
If some Levi Cartan determinant were divisible by $p$, the ordinary
determinant list would give a further Levi of type $A_{p-1}$.
Indeed, the Cartan matrices of the classified Dynkin diagrams
\cite[Theorem~C.56 and \S24(c)]{Milne} have determinants
\[
\begin{array}{c|ccccccccc}
\text{type}&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2\\ \hline
\det&r+1&2&2&4&3&2&1&1&1.
\end{array}
\]
In type $A_r$, take a consecutive string of $p-1$ vertices.
In the determinant-two and determinant-four cases take one vertex,
since $p=2$. In type $E_6$ take two adjacent vertices, since $p=3$.
Such a simple-root subdiagram defines a Levi of $G$, so the preceding
rank calculation applies to it.
For this $A_{p-1}$ Levi, $r=p-1$ and $\det C_I=p$.
Choose dual bases of the full character and cocharacter lattices,
of rank $N_T$. The matrices of $U,V$ have sizes
$r\times N_T$ and $N_T\times r$. By
\eqref{eq:cartan-ranks}, both reductions have rank less than $r$,
so every maximal minor of each matrix is divisible by $p$.
The integral Cauchy--Binet identity gives
\[
\det C_I=
\sum_{\substack{J\subset\{1,\ldots,N_T\}\\ |J|=r}}
\det(U_{:,J})\det(V_{J,:}),
\]
which is divisible by $p^2$, contrary to $\det C_I=p$.
Using the full lattices here retains all central torus directions
and makes no assumption on the isogeny type.
For an arbitrary Levi, put $d_I=\det C_I$ and define an integral
endomorphism of $\Lambda$ by
\[
\Pi_I=d_I\id_\Lambda-V\operatorname{adj}(C_I)U.
\]
It satisfies $U\Pi_I=0$. The lattice $\ker U$ consists precisely of
cocharacters into $Z(M)$: their images commute with $T$ and with
every root group of $M$. For $z\in\ker U_k$,
$\Pi_{I,k}(z)=d_Iz$. Since $d_I$ is invertible in $k$, the
differentials of elements of $\ker U$ span
$\ker U_k=\mathfrak z(\mathfrak m)$, proving
\eqref{eq:central-span}. For $r=0$, take $d_I=1$ and
$\Pi_I=\id_\Lambda$; the same proof covers tori.
\end{proof}
We now assume Hypothesis~\ref{hyp:A}, so
$k=\overline{\mathbb F}_q$, and construct the constant-term data in
Proposition~\ref{lie:input}. In a faithful representation, write
$a_0=a(0)=s+n$ for its matrix Jordan decomposition. The eigenvalues
of $s$ lie in a finite field $\mathbb F_{p^h}$. Choose $e$ divisible
by $h$ and large enough to have $n^{p^e}=0$. Then
\[
a_0^{[p^e]}=a_0^{p^e}=s^{p^e}+n^{p^e}=s.
\]
Thus $s,n\in\mathfrak g$. Lemma~\ref{lie:constant-parts} supplies
a nonzero $s\in\mathfrak t$, the Levi $M=Z_G(s)$ and invertibility
of $\operatorname{ad}_{a_0}$ on $\mathfrak g/\mathfrak m$.
The element $s$ lies in $\mathfrak z(\mathfrak m)$, where $\kappa$
is nondegenerate. By Lemma~\ref{lie:lattice}, some actual
$\lambda\in X_*(Z(M))$ satisfies $\kappa(s,d\lambda)\ne0$.
Equation~\eqref{eq:nilpotent-orthogonality} gives the required
$\kappa(a_0,d\lambda)\ne0$. This construction has used no
Weyl-order assumption.
\subsection{The arbitrary algebraically closed regime}
We now assume Hypothesis~\ref{hyp:B}. The eigenvalues of a matrix
over $k$ need not be fixed by any positive power of Frobenius.
Instead, a finite additive-polynomial interpolation gives its
Jordan parts inside $\mathfrak g$.
Write $a_0=s+n$ in a faithful representation and choose $e$ with
$n^{p^e}=0$, so $a_0^{p^e}=s^{p^e}$. Let $V_0\subset k$ be the
finite-dimensional $\mathbb F_p$-span of the eigenvalues of $s$,
and let $V_e=\{v^{p^e}:v\in V_0\}$. The inverse of Frobenius
$V_e\to V_0$ is $\mathbb F_p$-linear. If $V_0=0$, then $s=0$ and
$a_0=n\in\mathfrak g$, contrary to
Lemma~\ref{lie:constant-parts} and $\chi(a_0)\ne\chi(0)$.
We may therefore assume that $r=\dim_{\mathbb F_p}V_e\ge1$.
If
$\beta_1,\ldots,\beta_r$ is an $\mathbb F_p$-basis of $V_e$, the
Moore matrix
\[
(\beta_i^{p^j})_{1\le i\le r,\ 0\le j<r}
\]
is invertible. Otherwise a nonzero polynomial
$\sum_{j=0}^{r-1}c_j z^{p^j}$ of degree at most $p^{r-1}$ would
vanish on all $p^r$ elements of $V_e$, a contradiction.
There is therefore an additive polynomial
$Q(z)=\sum_{j=0}^{r-1}c_jz^{p^j}$ agreeing with inverse Frobenius
on $V_e$. Applying it to the diagonalizable matrix $s^{p^e}$ gives
\[
s=Q(a_0^{p^e})=\sum_{j=0}^{r-1}c_j a_0^{[p^{e+j}]}\in\mathfrak g.
\]
Thus $n\in\mathfrak g$ as well. Lemma~\ref{lie:constant-parts} gives
$s\in\mathfrak t\setminus\{0\}$, $M=Z_G(s)$ and the quotient
invertibility.
Let $W_M=N_M(T)/T$ be the Weyl group of $M$ and set $h_M=|W_M|$.
Since $W_M\subset W_G$, the integer $h_M$ is invertible in $k$.
The operator
\[
e_M=\frac1{h_M}\sum_{w\in W_M}w:\mathfrak t\longrightarrow\mathfrak t
\]
is a self-adjoint projection for $\kappa$.
Consequently $\kappa$ is nondegenerate on
$\mathfrak t^{W_M}=\im e_M$. This subspace contains $s$, since
$M$ centralizes $s$.
In fact $\mathfrak t^{W_M}=\mathfrak z(\mathfrak m)$:
the differential reflection formula is
$s_\alpha(h)=h-d\alpha(h)d\alpha^\vee$, and
$d\alpha^\vee\ne0$ by Lemma~\ref{lie:root-pairing}.
Thus, under the Weyl-order assumption, nondegeneracy on every Levi
Lie center also follows from the invariant form itself.
To obtain integral cocharacters, choose a basis
$\mu_1,\ldots,\mu_{N_T}$ of $\Lambda$ and form the integral sums
\[
\nu_i=\sum_{w\in W_M}w\mu_i\in\Lambda^{W_M}.
\]
Their differentials span $\mathfrak t^{W_M}$ because
$d\nu_i=h_Me_M(d\mu_i)$. The integral reflection formula
\[
s_\alpha(\lambda)=\lambda-
\langle\alpha,\lambda\rangle\alpha^\vee
\]
and torsion-freeness of $\Lambda$ show that
$\Lambda^{W_M}=X_*(Z(M))$: an invariant cocharacter has zero
integral pairing with every root of $M$, and conversely.
No division of a cocharacter by $h_M$ has been made.
Nondegeneracy on $\mathfrak t^{W_M}$ now gives an integral central
cocharacter $\lambda$ with $\kappa(s,d\lambda)\ne0$.
By \eqref{eq:nilpotent-orthogonality},
$\kappa(a_0,d\lambda)\ne0$ as required.
This argument applies to arbitrary algebraically closed $k$.
Its use of $p\nmid|W_G|$ is specific to Hypothesis~\ref{hyp:B};
the finite-field argument above obtained its cocharacter from the
Levi-center form condition instead.
\subsection{A single formal reduction}
Both regimes have now provided the same constant-term data.
The remaining step moves the entire formal Higgs coefficient into
the chosen Levi. It uses neither finite-field descent nor a
Weyl-group average.
\begin{lemma}\label{lie:formal}
Let $G$ be a connected reductive group over an algebraically closed
field $k$, let $M\subset G$ be a Levi subgroup, and let
$a(t)\in\mathfrak g[[t]]$ satisfy $a_0=a(0)\in\mathfrak m$.
If $\operatorname{ad}_{a_0}$ is invertible on $\mathfrak g/\mathfrak m$, then
there exists $g(t)\in G(k[[t]])$ with $g(0)=1$ such that
$\Ad(g(t))a(t)\in\mathfrak m[[t]]$.
\end{lemma}
\begin{proof}
Suppose that the current arc belongs to $\mathfrak m[[t]]$
modulo $t^r$, with $r\ge1$, and let $c_r\in\mathfrak g/\mathfrak m$
be the class of its coefficient of $t^r$. Choose $D\in\mathfrak g$
such that $c_r+[D,a_0]=0$ in the quotient.
Smoothness of $G$ gives an element $g_r(t)\in G(k[[t]])$ with
\[
g_r(t)\equiv1+t^rD\pmod {t^{r+1}}.
\]
Here the displayed expression specifies a point in the tangent
kernel for the square-zero ideal $(t^r)/(t^{r+1})$; formal
smoothness lifts it successively to every higher order.
Conjugation by $g_r(t)$ changes the coefficient of $t^r$ by
$[D,a_0]$ and leaves all lower coefficients unchanged.
It therefore places the arc in $\mathfrak m[[t]]$ modulo $t^{r+1}$.
Iterate this construction. Since $g_r(t)\equiv1\pmod {t^r}$,
the successive products converge in the $t$-adic topology.
The affine group $G$ takes this compatible system of points modulo
$t^j$ to a point $g(t)\in G(k[[t]])$ with $g(0)=1$.
Its conjugation sends the arc into $\mathfrak m[[t]]$.
No exponential series is used.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{lie:input}]
Under Hypothesis~\ref{hyp:A}, use the restricted-power construction
and Lemma~\ref{lie:lattice}. Under Hypothesis~\ref{hyp:B}, use the
additive-polynomial construction and integral Weyl-group averaging.
In each case Lemma~\ref{lie:constant-parts} supplies the Levi and
quotient invertibility, with
$\kappa(a_0,d\lambda)\ne0$. Apply Lemma~\ref{lie:formal} after the
initial constant conjugation. The formal change of frame fixes
$a_0$, so all three assertions in \eqref{eq:lie-output} hold.
\end{proof}
\begin{remark}\label{lie:zero-orbit}
Centrality of $\lambda$ in $M$ makes
$\Ad(\lambda(t))a(t)=a(t)$. Moreover, every nonzero
$\lambda$-weight space of $\mathfrak g$ injects into
$\mathfrak g/\mathfrak m$, so the adjoint action is invertible on
the truncated nonzero-weight modules used below.
We do not need $Z_G(\lambda)=M$. If $\lambda$ is central in $G$,
the Hecke orbit has dimension zero; the nonzero pairing in
\eqref{eq:lie-output} still supplies the moving-point covector.
In particular, tori are included in every allowed characteristic.
\end{remark}
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/03-lie.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/04-hecke.tex ====
\section{Hecke modifications and transverse cotangent sections}
\label{sec:hecke}
We now turn the Lie-theoretic data of Proposition~\ref{lie:input} into a
modification of a bundle and its Higgs field. The moving point of the
modification produces a nonzero covector on the curve. Two further
properties will be needed: at fixed input, and at fixed output, the
corresponding relative cotangent section has an invertible derivative.
In the next section, the fixed-output derivative will isolate the
distinguished stalk. The fixed-input derivative will complete the
coordinates for its quadratic local model.
\subsection{Integral correspondences and the open Satake kernel}
Keep the trait $S$, curve $\mathscr X/S$, and bundle stack $\mathcal B$
from Section~\ref{sec:specialization}. A one-point Hecke modification is a
quadruple $(P,P',x,\alpha)$, where $x$ is a point of $\mathscr X$ and
$\alpha$ identifies $P'$ with $P$ away from its graph. Write
\[
p(P,P',x,\alpha)=P,\qquad o(P,P',x,\alpha)=P',\qquad
\ell_H(P,P',x,\alpha)=x.
\]
We fix the following relative-position convention. In a formal coordinate
$t$ at $x$ and an input frame, modification by a cocharacter $\lambda$
means that the output frame is given by $\lambda(t)$. Its relative
position depends only on the Weyl orbit of $\lambda$. Let
$\mu=\lambda^+$ be the dominant representative and put
\[
m=\langle 2\rho,\mu\rangle,
\qquad \mu^\vee=-w_0\mu,
\qquad \mathcal Z=\mathcal B\times_S\mathscr X,
\]
where $2\rho$ is the sum of the positive roots and $w_0$ is the longest
Weyl element. Denote the exact-position correspondence by $H=H^\lambda$,
and set
\[
\widetilde p=(p,\ell_H):H\longrightarrow\mathcal Z,
\qquad q=(o,\ell_H):H\longrightarrow\mathcal Z.
\]
\begin{lemma}[Hecke models and kernels]\label{hecke:models}
Both $\widetilde p$ and $q$ are representable, separated, and smooth of
relative dimension $m$. The correspondence $H$ is open in a bounded
correspondence $\overline H$ for which both projections to $\mathcal Z$
are representable and proper. On the output side the bound has dominant
label $\mu^\vee$.
For the irreducible dual-group representation $V$ corresponding to this
bound, with the input and output convention just specified, the generic
Hecke functor has the form
\begin{equation}\label{hecke:kernel}
\mathrm H_V(F)=\overline q_{K,*}\mathcal I_K.
\end{equation}
On $H_K$, the integrand $\mathcal I_K$ is $p_K^*F$ tensored with a fixed
invertible constant complex. On a fixed-leg affine-Grassmannian fiber,
the perverse intersection-cohomology normalization of that complex is
$E[m]$, with the constant Tate line if weight normalization is used.
\end{lemma}
\begin{proof}
Use the split integral affine Grassmannian, whose loops are
$G(A((t)))$ and whose positive loops are $G(A[[t]])$. Its integral
Schubert bound is projective, and the positive-loop orbit is smooth,
open, and fiberwise dense, of relative dimension $m$; see
\cite[\S\S1.2 and~2.1]{RicharzScholbach}. We use these models by base change from
$\mathbb Z$. The variable $t$ is the curve parameter, including when the
base trait has mixed characteristic.
A coordinate along the graph of $x$ and a frame on the formal disc
identify modifications of a fixed input bundle with the corresponding
Grassmannian spaces. These descriptions glue by
Beauville--Laszlo descent \cite{BeauvilleLaszlo}. More explicitly, the
graph is an effective Cartier divisor. The gluing theorem for modules
also glues the coordinate algebra, multiplication, and coaction of an
affine $G$-torsor; the torsor identity can be checked on the two gluing
pieces. A disc frame exists locally: lift a frame on the graph
successively through its infinitesimal neighborhoods using smoothness of
$G$. Changes of frame and formal coordinate preserve the orbit and its
bound. Consequently the properness, smoothness, and dimensions descend
to the input-and-leg projection.
Swapping the bundles and inverting $\alpha$ is an involution of the Hecke
correspondence. On double cosets it replaces $\mu$ by $-\mu$, whose
dominant representative is $\mu^\vee$. It carries the bound to the
oppositely labelled bound. This proves the assertions for $q$, since
$\langle 2\rho,\mu^\vee\rangle=m$. Separatedness of the open
correspondence follows from that of the proper bound. This involution
is used on the correspondence, rather than as a purported inversion map
on the right-quotient affine Grassmannian.
The fibers here include an identification of the fixed bundle with the
specified bundle. They are relative-position spaces: an automorphism
of the moving bundle compatible with its identification off the graph
is the identity. In particular, no further quotient by automorphisms
of the fixed bundle is taken.
For the last assertion, geometric Satake gives the intersection complex
of the bound. Its restriction to the smooth open orbit is the shifted
constant sheaf, with the indicated optional Tate normalization; see
\cite[Proposition~4.1]{Richarz}. The Hecke construction tensors this kernel with the ordinary pullback
$p_K^*F$ and pushes forward along the proper output-and-leg projection;
see \cite[\S\S4.2.1--4.2.4]{GR}. Relabelling by $\mu^\vee$ if necessary
matches these projections with the chosen representation. A fixed
overall shift or Tate line in the Hecke normalization changes neither
this description nor any vanishing assertion below. Only the open-orbit
restriction is being identified; the kernel on the boundary remains its
intersection complex.
\end{proof}
\subsection{The transverse modification}
On the special fiber, a cotangent vector at $P\in\mathcal B_s(k)$ means
an element of $H^0$ of the cotangent-complex fiber. Deformation theory
and vector-bundle Serre duality \cite[Tag~0BS4]{Stacks}, followed by
the invariant form $\kappa$, identify this
space with
\[
T_P^*\mathcal B_s
=H^1(X,\mathfrak g_P)^*
=H^0(X,\mathfrak g_P\otimes\omega_X),
\]
where $\mathfrak g_P$ is the adjoint bundle. We call such a vector $A$
a Higgs field. For a representable smooth map, pullback on these
cotangent spaces is injective and its image is the kernel of passage to
the relative cotangent. This follows from the cotangent triangle; it
does not assert that the map to relative cotangents is surjective for
maps between stacks.
\begin{proposition}[Transverse modification]\label{hecke:transverse}
Assume the hypotheses of Proposition~\ref{lie:input}. Let
$P\in\mathcal B_s(k)$ and let $A\in T_P^*\mathcal B_s$ be non-nilpotent.
There exist a point $x\in X(k)$, a cocharacter $\lambda$, a modification
$h\in H^\lambda_s(k)$ from $P$ to a bundle $P'$, and a covector
\[
\theta=(A',\xi)\in T^*_{(P',x)}(\mathcal B_s\times X),
\qquad \xi\ne0,
\]
such that $A'$ agrees with $A$ away from $x$ and
\begin{equation}\label{hecke:cotangent-identity}
dp_h^*A=dq_h^*\theta.
\end{equation}
Moreover, form the fixed-side fibers
\[
H^{\mathrm{in}}_{P,x}=\widetilde p^{-1}(P,x),
\qquad H^{\mathrm{out}}_{P',x}=q^{-1}(P',x),
\]
with the fixed-bundle identifications retained. On the first fiber,
restriction of $p^*A$ to the $q$-relative tangent bundle defines a section
$S_A$ of $\Omega_{H_s/\mathcal Z_s,q}$. On the second fiber, restriction
of $o^*A'$ to the $\widetilde p$-relative tangent bundle similarly defines
$S_{A'}$. Both sections vanish at $h$, and both derivatives
\begin{align*}
(dS_A)_h &:T_hH^{\mathrm{in}}_{P,x}
\longrightarrow (T_hH^{\mathrm{out}}_{P',x})^*,\\
(dS_{A'})_h &:T_hH^{\mathrm{out}}_{P',x}
\longrightarrow (T_hH^{\mathrm{in}}_{P,x})^*
\end{align*}
are isomorphisms. In particular, each section has an isolated reduced
zero at $h$.
\end{proposition}
\begin{proof}
Since $A$ is non-nilpotent, choose $x\in X(k)$ where its value is
non-nilpotent. Choose a formal coordinate $t$ and a bundle frame at $x$,
and write $A=a(t)\,dt$. Chevalley restriction identifies the nilpotent
fiber with $\chi^{-1}(\chi(0))$, so $\chi(a(0))\ne\chi(0)$.
Proposition~\ref{lie:input} then supplies a formal change of frame,
a Levi subgroup $M$, and a cocharacter
$\lambda$ of $Z(M)$ such that
\begin{equation}\label{hecke:lie-data}
a(t)\in\mathfrak m[[t]],\qquad
\operatorname{ad}(a(0))
\text{ is invertible on }\mathfrak g/\mathfrak m,
\qquad \kappa(a(0),d\lambda)\ne0.
\end{equation}
Here $d\lambda$ is the value of the cocharacter differential on the
canonical generator of $\Lie(\mathbb G_m)$. After constant conjugation,
we regard $\lambda$ as an element of the fixed cocharacter lattice; it
therefore also labels the integral model of Lemma~\ref{hecke:models}.
Modify by $\lambda(t)$. Since $\lambda$ centralizes $M$, the transported
field is regular in the output frame and still has expression
$a(t)\,dt$. Together with $A$ away from $x$, it defines the global Higgs
field $A'$ on $P'$. This central-in-a-Levi construction and the resulting
leg covector are the modification used in
\cite[\S\S20.4--20.6]{AGKRRV}. We prove the derivative assertions
explicitly.
\smallskip\noindent
\emph{The two tangent spaces.}
In the input frame put
\[
L=\mathfrak g[[t]],\qquad
L'=\Ad(\lambda(t))L,\qquad I=L\cap L'.
\]
Let $\mathfrak g_j$ be the weight-$j$ space for the algebraic
$\mathbb G_m$-action defined by $\lambda$. The integers $j$ denote
characters of $\mathbb G_m$, without reduction modulo $p$. Then
\begin{equation}\label{hecke:weight-quotients}
\begin{split}
L'&=\bigoplus_j t^j\mathfrak g_j[[t]],\\
P_1:=L/I&=\bigoplus_{j>0}
\mathfrak g_j\otimes_k k[[t]]/(t^j),\\
P_2:=L'/I&=\bigoplus_{j<0}
\mathfrak g_j\otimes_k(t^jk[[t]]/k[[t]]).
\end{split}
\end{equation}
An infinitesimal disc automorphism $1+\epsilon D$, $D\in L$, acts on the
output lattice while fixing $(P,x)$. Its stabilizer tangent is $I$.
The orbit is smooth, and the root formula gives
$\dim_k(L/I)=m$, so this computes its full tangent. The same argument
with the output fixed gives
\begin{equation}\label{hecke:tangents}
T_hH^{\mathrm{in}}_{P,x}=P_1,
\qquad T_hH^{\mathrm{out}}_{P',x}=P_2.
\end{equation}
The notation $1+\epsilon D$ uses the canonical identification of the
kernel over dual numbers with the Lie algebra; no exponential map is
required.
There is a perfect pairing
\begin{equation}\label{hecke:residue-pairing}
\beta:P_1\times P_2\longrightarrow k,
\qquad \beta(D,C)=\Res_{t=0}\kappa(D,C)\,dt.
\end{equation}
Indeed, $L$ is its own annihilator for the residue pairing on
$\mathfrak g((t))$, and invariance of $\kappa$ gives the same assertion
for $L'$. Thus the pairing descends to the displayed quotients.
More explicitly, $\kappa$ pairs $\mathfrak g_j$ perfectly with
$\mathfrak g_{-j}$; the coefficient of $t^a$ in the first space, for
$0\le a<j$, pairs with the coefficient of $t^{-a-1}$ in the second.
This proves perfection, including when some $j$ is divisible by $p$.
\smallskip\noindent
\emph{Transport of the covector and motion of the leg.}
Choose the gluing sign convention in which an infinitesimal transition
$1+\epsilon C$ gives the boundary class of the principal part $C$.
Serre duality pairs this class with $A$ by
$\Res\kappa(a,C)\,dt$. For an output-fixed tangent $C\in L'$, this
residue is zero, since $a\in L'$ and $L'$ is self-annihilating.
Consequently $dp_h^*A$ vanishes in the $q$-relative cotangent, so it is
$dq_h^*\theta$ for a unique covector $\theta$.
Its bundle component is $A'$. To check this, choose a point $x'\ne x$
and vary both bundle transitions there while leaving the modification
at $x$ unchanged. Principal parts at $x'$ surject onto
$H^1(X,\mathfrak g_{P'})$: for sufficiently large $N$, Serre vanishing
gives $H^1(X,\mathfrak g_{P'}(Nx'))=0$. The pairings on these
directions agree because $A=A'$ near $x'$. Write therefore
$\theta=(A',\xi)$.
Now keep the input bundle fixed and move the leg from $t=0$ to
$t=\epsilon$. The output transition becomes $g_\epsilon=\lambda(t-\epsilon)$,
and its first-order displacement is
\[
(\delta g)g^{-1}=-\frac{d\lambda}{t}.
\]
Evaluating the cotangent identity on this motion yields
\[
0=-\kappa(a(0),d\lambda)+\xi(\partial_t),
\qquad
\xi=\kappa(a(0),d\lambda)\,dt\big|_x\ne0.
\]
This is also the residue formula of
\cite[Equation~(20.20)]{AGKRRV}. Reversing the gluing boundary convention
would reverse the common sign and would leave all subsequent assertions
unchanged.
\smallskip\noindent
\emph{The first relative derivative.}
The section $S_A$ is defined along the entire fixed-input fiber, using
its specified identification with $P$. Its value at $h$ is zero by the
preceding calculation. For $D\in L$, deform the output lattice to
\[
L'_\epsilon=\Ad(1+\epsilon D)L'.
\]
A test vector $C\in L'$ extends as
$C_\epsilon=\Ad(1+\epsilon D)C$ in the deformed lattice. The input
field remains the fixed $a(t)\,dt$. Differentiating the residue
pairing gives
\begin{equation}\label{hecke:derivative}
\begin{split}
\big\langle(dS_A)_h(D\bmod I),C\bmod I\big\rangle
&=\Res\kappa(a,[D,C])\,dt\\
&=\Res\kappa([a,D],C)\,dt
=\beta(\operatorname{ad}(a)D,C).
\end{split}
\end{equation}
Because $S_A(h)=0$, this derivative is independent of how the test
vector is extended. Also $\operatorname{ad}(a)$ preserves $L$, $L'$,
and $I$, so the formula is independent of representatives.
It remains to prove invertibility. The operator
$\operatorname{ad}(a(t))$ preserves every weight summand in
\eqref{hecke:weight-quotients}. Since $\lambda$ is central in $M$,
each nonzero weight space injects as an invariant summand into
$\mathfrak g/\mathfrak m$. Its endomorphism induced by
$\operatorname{ad}(a(0))$ is therefore invertible by
\eqref{hecke:lie-data}. The determinant of the corresponding
$k[[t]]$-linear block is a unit. It follows that
$\operatorname{ad}(a(t))$ is invertible on every truncated summand of
$P_1$ and $P_2$. Equation~\eqref{hecke:derivative}, together with the
perfect pairing~\eqref{hecke:residue-pairing}, now proves that
$(dS_A)_h:P_1\to P_2^*$ is an isomorphism. No regularity of $\lambda$
outside $M$ is required; any additional zero-weight directions do not
occur in these quotients.
\smallskip\noindent
\emph{The reverse derivative.}
The value of $S_{A'}$ at $h$ is zero because $a\in L$ and $L$ is
self-annihilating. Keeping the output fixed, vary the input lattice by
$C\in L'$ and extend a test vector $D\in L$ to
$\Ad(1+\epsilon C)D$. The same calculation, with the same gluing
convention, gives
\[
\big\langle(dS_{A'})_h(C\bmod I),D\bmod I\big\rangle
=\Res\kappa(a,[C,D])\,dt
=\Res\kappa([a,C],D)\,dt.
\]
Invertibility on $P_2$ and residue duality prove the second assertion.
Both sections are sections of rank-$m$ vector bundles on smooth
$m$-dimensional fibers. Their invertible derivatives make their zeros
at $h$ reduced and isolated.
When $m=0$, both tangent spaces are zero and both derivatives are
isomorphisms between zero spaces. The smooth zero-dimensional fibers
already isolate $h$, and the moving-leg computation still applies.
Thus the conclusion includes central modifications and torus factors.
\end{proof}
The proposition gives both kinds of control needed below: the nonzero
leg covector makes a suitable hypersurface smooth over the output
bundle, while the two relative derivatives isolate one point and produce
the quadratic coordinates around it.
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/04-hecke.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/05-two-hecke.tex ====
\section{Two Hecke correspondences and a quadratic test}
\label{sec:twohecke}
We now convert the Hecke vanishing of Lemma~\ref{spec:vanishing}
into a local test for a non-nilpotent cotangent direction.
The first Hecke correspondence supplies a proper pushforward.
The second makes its pullback vanish, while the two transverse
sections of Proposition~\ref{hecke:transverse} isolate one stalk
and identify its neighborhood with a split quadratic equation.
The next section will use a projective quadric to remove the
extra variables in this equation.
\begin{proposition}[Vanishing at the quadratic vertex]
\label{two:vertex}
Let $F\in\Shv(\mathcal B_K)$ be the object of
Proposition~\ref{spec:missing}, with the vanishings of
Lemma~\ref{spec:vanishing}. Let $b:U\to\mathcal B$ be a smooth
chart, where $U$ is a scheme of finite type over $S$.
Let $f\in\mathcal O(U)$ and $u\in U_s(k)$ satisfy $f(u)=0$.
Suppose
\[
df_u=b^*A
\quad\text{for a non-nilpotent }A\in T^*_{b(u)}\mathcal B_s.
\]
Then there exist $x\in\mathscr X_s(k)$ and an integer $m\geq0$
with the following property. Put $T=U\times_S\mathscr X$,
let $F_T$ be the pullback of $F_U=b_K^*F$, and set
\[
Q^a=\left\{\sum_{i=1}^m u_i v_i+f=0\right\}
\subset T\times_S\mathbb A_S^{2m},
\qquad \tau:Q^a\longrightarrow T.
\]
Here $u_i,v_i$ are affine coordinates, distinct from the point $u$,
and $f$ is pulled back from $U$. At the point
$\nu=(u,x,0,0)$ one has
\begin{equation}
\label{eq:two-vertex}
\bigl(\Psi_{Q^a}(\tau_K^*F_T)\bigr)_\nu=0.
\end{equation}
When $m=0$, this is the vanishing on
$Q^a=V(f)\times_S\mathscr X$.
\end{proposition}
\begin{proof}
Choose the modification $h$, its leg $x$, its orbit dimension $m$,
and $\theta=(A',\xi)$ from Proposition~\ref{hecke:transverse}.
Thus
\begin{equation}
\label{eq:two-cotangent}
p^*A=q^*\theta\quad\text{at }h,
\qquad \xi\ne0.
\end{equation}
We use the exact-position correspondence $H=H^\lambda$, its proper
bound $\overline H$, and its generic Satake integrand
$\mathcal I_K$ from Lemma~\ref{hecke:models}.
On $H_K$ this integrand is $p_K^*F$ tensored with a fixed invertible
shift and Tate line. Such a factor does not affect vanishing;
we suppress it after restricting to the open orbit.
\paragraph{A proper fiber with zero cohomology.}
Write $D=V(f)$. Take two copies $H_1,H_2$ of $H$ and form
\[
W_0=H_2\times_{p_2,\mathcal B,b}U,
\qquad W=H_2\times_{\mathcal B}D\subset W_0.
\]
Both map to $\mathcal Z=\mathcal B\times_S\mathscr X$ through
$q_2=(o_2,\ell_{H_2})$. The map $W_0\to\mathcal Z$ is smooth:
its projection to $H_2$ is the base change of $b$, and $q_2$ is
smooth. At $w=(h,u)$ the differential of the equation cutting
out $W$ is the pullback of $\theta$ by
\eqref{eq:two-cotangent}. Its relative differential over the
output bundle stack is nonzero, because the leg component is
$\xi\ne0$. The smooth hypersurface criterion therefore makes
$W\to\mathcal B$ by the output bundle smooth near $w$.
The arbitrary-leg assertion of Lemma~\ref{spec:vanishing} gives
\begin{equation}
\label{eq:two-hecke-zero}
\bigl(\Psi_W(q_{2,K}^*\mathrm H_V(F))\bigr)_w=0.
\end{equation}
This uses smoothness only of the projection to the bundle stack.
The two fiber products used below are
\[
P=H_1\times_{\mathcal Z}W_0,
\qquad
P_D=H_1\times_{\mathcal Z}W
\subset\overline P_D=\overline H_1\times_{\mathcal Z}W.
\]
The following squares are cartesian; the second vertical map
$\pi$ is proper. In the first square, $\Delta$ repeats the
modification in the two factors.
\[
\begin{tikzcd}[column sep=large,row sep=large]
P \arrow[r] \arrow[d] & H_1 \arrow[d,"q_1"] \\
W_0 \arrow[r,"q_2"'] \arrow[u,bend left=38,"\Delta"]
& \mathcal Z
\end{tikzcd}
\qquad
\begin{tikzcd}[column sep=large,row sep=large]
\overline P_D \arrow[r] \arrow[d,"\pi"']
& \overline H_1 \arrow[d,"\overline q_1"] \\
W \arrow[r,"q_2"'] & \mathcal Z.
\end{tikzcd}
\]
Write $c:P\to U$ for the map through $W_0$, and
$a:P\to\mathcal B$ for the input map $p_1$.
Thus $a$ records the first input, from which the sheaf is pulled back,
while $c$ records the second input in $U$, on which we impose $f=0$.
The distinguished point is $z=(h,h,u)\in P_D(k)$.
All these fiber products are algebraic spaces, because the
fixed-side Hecke maps are representable.
Let $\overline{\mathcal I}_K$ be the pullback of the Satake
integrand to $(\overline P_D)_K$. Proper base change on the
generic fiber identifies
\[
R\pi_{K,*}\overline{\mathcal I}_K
\simeq q_{2,K}^*\mathrm H_V(F).
\]
Apply proper compatibility of specialization and then proper
base change at $w$ to \eqref{eq:two-hecke-zero}. With
\[
Z_w=\pi^{-1}(w),\qquad
\mathcal K=
\bigl(\Psi_{\overline P_D}\overline{\mathcal I}_K\bigr)|_{Z_w},
\]
we obtain
\begin{equation}
\label{eq:two-fiber-zero}
R\Gamma(Z_w,\mathcal K)=0.
\end{equation}
The space $Z_w$ is the proper bounded Hecke fiber with output
bundle and leg fixed, including their identifications.
\paragraph{Isolating the distinguished stalk.}
The map $a:P\to\mathcal B$ is smooth: $P\to H_1$ is the base
change of $W_0\to\mathcal Z$, and $p_1$ is smooth.
At a point $(h_1,w)$ in the exact-position part of $Z_w$, the
differential of $f(c)$ is the pullback of $q_1^*\theta$.
If its relative differential over $\mathcal B$ is zero, smooth
cotangent injectivity for $P\to H_1$ implies that
$q_1^*\theta$ comes from the input cotangent under $p_1$.
Pass further to the relative cotangent for
$\widetilde p_1=(p_1,\ell_{H_1})$. The leg term disappears,
so the resulting necessary condition is
\[
S_{A'}(h_1)=0.
\]
By Proposition~\ref{hecke:transverse}, this section has an isolated
zero at $h$ on the fixed-output-and-leg fiber. Consequently
$P_D\to\mathcal B$ by $a$ is smooth at every point of a punctured
neighborhood of $z$ in $Z_w$.
On the exact-position open, the generic integrand is $a_K^*F$.
Lemma~\ref{spec:vanishing} therefore gives an open neighborhood
$O$ of $z$ in $Z_w$ such that
\[
\mathcal K|_{O\setminus\{z\}}=0.
\]
This local vanishing prevents cancellation with the boundary of
the Hecke bound. Indeed, put $C_w=Z_w\setminus O$ with its reduced
structure. The closed subsets $\{z\}$ and $C_w$ are disjoint.
Open--closed localization identifies $\mathcal K$ with its
restriction to their union, extended by zero. Hence
\[
\mathcal K\simeq
i_{z,*}\mathcal K_z\oplus i_{C_w,*}i_{C_w}^*\mathcal K,
\qquad
R\Gamma(Z_w,\mathcal K)\simeq
\mathcal K_z\oplus R\Gamma(C_w,i_{C_w}^*\mathcal K).
\]
The residue field of $z$ is algebraically closed, so the first
cohomology summand is exactly its stalk complex.
Equation~\eqref{eq:two-fiber-zero} forces that summand to vanish.
Removing the invertible Satake normalization gives
\begin{equation}
\label{eq:two-isolated-zero}
\bigl(\Psi_{P_D}(a_K^*F)\bigr)_z=0.
\end{equation}
No formula for the Satake kernel on the boundary has been used.
\paragraph{A chart lift along the whole diagonal.}
To interpret \eqref{eq:two-isolated-zero}, we next identify both
the local space and its sheaf. The diagonal
$\Delta\simeq W_0\subset P$ is a section of the separated smooth
map $P\to W_0$ of relative dimension $m$.
It is therefore a closed regular immersion near $z$.
On $\Delta$ there is a specified isomorphism $bc\simeq a$,
coming from the two identical modifications.
We claim that, after an \emph{étale} replacement of $P$ near $z$,
there is a map $r:P\to U$ with an isomorphism
\begin{equation}
\label{eq:two-chart-lift}
br\simeq a,
\qquad r|_\Delta=c|_\Delta,
\end{equation}
whose restriction to $\Delta$ is the specified isomorphism.
For this, form the smooth space
$E_P=P\times_{a,\mathcal B,b}U\to P$.
The given diagonal data define a section over $\Delta$.
First take an étale scheme neighborhood in $P$.
Since $\mathcal B$ has affine diagonal by the representability
statement in Section~\ref{sec:specialization}, $E_P\to P\times_S U$
is affine, so $E_P$ is a scheme as well.
Near the section point, smooth coordinates give an étale map
$E_P\to\mathbb A_P^e$
\cite[Tag~039P]{Stacks}.
After shrinking, the section over the closed diagonal lands in
this coordinate neighborhood. Extend its coordinate functions
from $\Delta$ to $P$, and pull back the étale map along the
extended tuple. This gives an étale neighborhood of $P$ with
a map to $E_P$. Its inverse image of $\Delta$ contains the
open branch specified by the original section.
Remove the closed complement of this branch in that inverse
image. The retained diagonal is the prescribed one, and the
map to $E_P$ proves \eqref{eq:two-chart-lift}, including its
isomorphism of bundles. We retain the notation $P,\Delta,z$.
\paragraph{Exact quadratic coordinates.}
After shrinking $U$, suppose that $U/S$ has constant relative
dimension $N$. The map
\[
\Delta\longrightarrow T=U\times_S\mathscr X,
\qquad (c,\text{leg}),
\]
is smooth of relative dimension $m$. Thus $T$, $\Delta$, and
$P$ are smooth over $S$ of relative dimensions
\[
N+1,\qquad N+1+m,\qquad N+1+2m,
\]
respectively. Choose generators $v_1,\ldots,v_m$ of the
diagonal ideal whose classes form a conormal frame.
Under the identification with
$\Omega_{P/W_0}|_\Delta$, the classes $dv_i$ form a basis.
Because $r=c$ on $\Delta$, there exist regular functions
$u_1,\ldots,u_m$ with the \emph{exact} identity
\begin{equation}
\label{eq:two-exact-quadratic}
f(c)-f(r)=\sum_{i=1}^m u_i v_i.
\end{equation}
The restrictions $u_i|_\Delta$ are determined by the conormal
class of the left side.
Consider the entire local diagonal fiber
$\Delta_{(u,x)}$ on the special fiber.
It parametrizes modifications with the input bundle
$b(u)$, its identification, and the leg $x$ fixed.
The relative differential of $f(c)$ for $P\to W_0$ is zero.
On this diagonal fiber, the relative differential of $f(r)$
is induced by $df_u=b^*A$: the restriction $r|_{\Delta_{(u,x)}}$
is constant with value $u$, and the bundle isomorphism in
\eqref{eq:two-chart-lift} is the canonical one along the whole
diagonal.
The conormal coefficient section in
\eqref{eq:two-exact-quadratic} is therefore
\begin{equation}
\label{eq:two-coefficients}
\sum_i(u_i|_\Delta)[dv_i]=-S_A
\quad\text{on }\Delta_{(u,x)}.
\end{equation}
In particular, $u_i(z)=0$, and the derivative of the tuple
$(u_i|_\Delta)$ is an isomorphism on
$T_z\Delta_{(u,x)}$, by Proposition~\ref{hecke:transverse}.
The equality on the whole fiber in
\eqref{eq:two-coefficients} is what supplies this derivative;
equality only at $z$ would not suffice.
Now define
\[
\Phi=(r,\text{leg},u_1,\ldots,u_m,v_1,\ldots,v_m):
P\longrightarrow T\times_S\mathbb A_S^{2m}.
\]
Its relative differential at $z$ is an isomorphism.
Indeed, a tangent vector in its kernel is first killed by every
$dv_i$, so lies in $T_z\Delta$. Its $(dr,d\text{leg})$ component
then places it in $T_z\Delta_{(u,x)}$. Finally the derivative
isomorphism in \eqref{eq:two-coefficients} forces the vector
to be zero. The source and target are smooth over $S$ with
the same relative dimension $N+1+2m$, so this injective
differential is an isomorphism. The relative Jacobian
criterion makes $\Phi$ étale at $z$.
By \eqref{eq:two-exact-quadratic}, the cut $P_D=V(f(c))$ is
the exact inverse image of $Q^a$ under $\Phi$. Therefore the
induced map $\Phi_D:P_D\to Q^a$ is étale near $z$, taking $z$ to $\nu$.
Moreover, \eqref{eq:two-chart-lift} identifies
\[
a_K^*F\simeq r_K^*F_U
\simeq \Phi_{D,K}^*\tau_K^*F_T
\quad\text{on }(P_D)_K.
\]
Thus this coordinate map identifies the actual generic sheaf
as well as the equation. Étale base change transforms
\eqref{eq:two-isolated-zero} into \eqref{eq:two-vertex}.
If $m=0$, the diagonal is open near $z$, the sum in
\eqref{eq:two-exact-quadratic} is empty, and the same argument
gives $Q^a=D\times_S\mathscr X$. In that case smooth base
change along $D\times_S\mathscr X\to D$ already yields
$(\Psi_D(F_U|_{D_K}))_u=0$.
\end{proof}
The argument produced exact étale coordinates over the trait.
It used no division by $2$ and no assertion about singular
support in mixed characteristic. Its output for $m>0$ is a
vanishing on the quadratic test space; we next recover the
vanishing on $D$ itself.
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/05-two-hecke.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/06-slices.tex ====
\section{From quadratic models to transverse slices}\label{sec:slices}
The local model in Proposition~\ref{two:vertex} gives a vanishing statement
at the vertex of an affine quadric. We first recover the corresponding
hypersurface test by a proper compactification. The cone of the
hyperplane-class map is a shifted constant Tate line supported on $f=0$.
After tensoring this comparison with the ambient sheaf, the hyperplane
terms have zero nearby cycles by the smooth-chart vanishing. Vanishing
on the whole proper quadric fiber then forces vanishing on the
hypersurface. A second proper comparison, using projective incidence,
gives tests of arbitrary codimension.
\subsection{The projective quadric comparison}
\begin{lemma}[Quadric comparison]\label{slice:quadric}
Let $K_0$ be an algebraically closed field of characteristic different
from $\ell$, let $E=\overline{\mathbb Q}_{\ell}$, and let $B$ be a
finite-type $K_0$-scheme. For $f\in\Gamma(B,\mathcal O_B)$ put
$i:D=V(f)\hookrightarrow B$. If $m\geq1$, consider
\[
g:Q_f=\left\{\sum_{a=1}^{m}u_av_a+fw^2=0\right\}
\subset\mathbb P_B^{2m}\longrightarrow B.
\]
The powers of the hyperplane class define a distinguished triangle
\begin{equation}\label{eq:slice-quadric-triangle}
\bigoplus_{j=0}^{2m-1}E_B(-j)[-2j]
\longrightarrow Rg_*E_{Q_f}
\longrightarrow i_*E_D(-m)[-2m]
\longrightarrow
\left(\bigoplus_{j=0}^{2m-1}E_B(-j)[-2j]\right)[1].
\end{equation}
The identification of its third term with the displayed Tate line
requires only a choice of generator of a fixed one-dimensional vector
space. Neither $B$ nor $D$ is required to be smooth or reduced.
\end{lemma}
\begin{proof}
Write $h=c_1(\mathcal O_{Q_f}(1))\in H^2(Q_f,E(1))$.
Adjunction applied to each $h^j$ gives the displayed morphism from
$E_B(-j)[-2j]$ to $Rg_*E_{Q_f}$. Denote the cone of their direct sum
by $\mathcal K$, and write $j_B:B\setminus D\hookrightarrow B$.
We recall the elementary cohomology calculation that controls this map.
We use the projective-space and projective-bundle calculations of
\cite[Example~16.3 and Theorem~23.2]{MilneEtale}, together with localization.
A split smooth projective quadric of dimension $n$ has no odd
cohomology, and has one copy of $E(-r)$ in degree $2r$ for
$0\leq r\leq n$, with an additional copy when $n$ is even and
$r=n/2$. One obtains the calculation inductively by writing its
equation as $x_0x_1+q'(x_2,\ldots)=0$. The locus $x_0\ne0$ is
$\mathbb A^n$; its complement is the projective cone over the split
quadric of dimension $n-2$. Removing the vertex of that cone gives a
line bundle over the smaller quadric. This yields a filtration by
affine cells, with one cell of every dimension $0,\ldots,n$ and a
second middle-dimensional cell when $n$ is even. The initial cases
are a conic, isomorphic to $\mathbb P^1$, and the split
zero-dimensional quadric, consisting of two points. Localization and
$R\Gamma_c(\mathbb A^r,E)=E(-r)[-2r]$ give the asserted groups.
On a smooth quadric of positive dimension, the top hyperplane power
has degree $2$, so is nonzero in $E$. All lower hyperplane powers are
therefore nonzero as well.
Over $B\setminus D$, the geometric fibers of $g$ are smooth quadrics
of odd dimension $2m-1$. The preceding calculation shows that the
hyperplane-power map is an isomorphism on their cohomology. Proper
base change therefore gives $j_B^*\mathcal K=0$.
Over $D$ the family, together with its hyperplane line bundle, is the
product with
\[
V_m=\left\{\sum_{a=1}^{m}u_av_a=0\right\}
\subset\mathbb P_{K_0}^{2m}.
\]
Let $e=[0:\cdots:0:1]$ be its vertex. Projection away from $e$
identifies $V_m\setminus\{e\}$ with the total space of a line bundle
over the split smooth quadric
\[
C_m=\left\{\sum_{a=1}^{m}u_av_a=0\right\}
\subset\mathbb P_{K_0}^{2m-1}.
\]
Compactly supported cohomology of this line bundle and localization at
the vertex give
\begin{equation}\label{eq:slice-cone-cohomology}
H^0(V_m,E)=E,\qquad H^1(V_m,E)=0,\qquad
H^r(V_m,E)=H^{r-2}(C_m,E)(-1)\quad(r\geq2).
\end{equation}
Thus $V_m$ has the cohomology of $\mathbb P^{2m-1}$ with one extra
copy of $E(-m)$ in degree $2m$.
The hyperplane powers remain nonzero on $V_m$. To check the top one
without a smoothness assertion about $V_m$, resolve the vertex by the
projective line bundle
\[
\pi:\widetilde V_m
=\mathbb P_{\mathrm{lines},C_m}
(\mathcal O_{C_m}(-1)\oplus\mathcal O_{C_m})
\longrightarrow C_m.
\]
Here the subscript specifies that the projective bundle parameterizes
lines. The inclusion of $\mathcal O_{C_m}(-1)\oplus\mathcal O_{C_m}$
into the trivial vector bundle with fiber $K_0^{2m}\oplus K_0$
defines $\widetilde V_m\to V_m$; its fiber over a point of $C_m$
parameterizes the projective line joining that point to $e$.
If $\zeta$ is the pullback
of the hyperplane class and $h_C=c_1(\mathcal O_{C_m}(1))$, the
projective bundle relation is $\zeta^2=\pi^*h_C\,\zeta$
\cite[\S23]{MilneEtale}. Hence
\[
\int_{\widetilde V_m}\zeta^{2m-1}
=\int_{C_m}h_C^{2m-2}=2.
\]
It follows that the top hyperplane power on $V_m$ is nonzero.
Multiplication gives the same conclusion for every lower power.
This includes $m=1$: $C_1$ consists of two points, $V_1$ consists of
two projective lines meeting at $e$, and the resolution is the
disjoint union of those lines. The hyperplane class has degree $1$
on each component.
Consequently the cone of the vector-space morphism
\[
\bigoplus_{j=0}^{2m-1}E(-j)[-2j]
\longrightarrow R\Gamma(V_m,E)
\]
has exactly one nonzero cohomology group, a copy of $E(-m)$ in degree
$2m$. Injectivity in every degree, established by the nonvanishing of
the hyperplane powers, rules out any additional cohomology group in
the cone. The cone is therefore isomorphic to $E(-m)[-2m]$.
Proper base change identifies $i^*\mathcal K$ with the constant
pullback of this vector-space cone: both the family over $D$ and
its hyperplane-power morphism are pulled back from $V_m$.
Finally, open--closed localization gives
\[
(j_B)_!j_B^*\mathcal K\longrightarrow\mathcal K
\longrightarrow i_*i^*\mathcal K\longrightarrow .
\]
Since its first term vanishes, this identifies
$\mathcal K\simeq i_*E_D(-m)[-2m]$, proving
\eqref{eq:slice-quadric-triangle}. In particular, the third term is
constant along $D$; no middle-cohomology local system is left
unidentified. Localization and proper base change hold with finite
coefficients \cite[Tags 095L and 0DDE]{Stacks} and in their adic forms
\cite[Tags 09AH and 09C9]{Stacks}; we use these compatibilities after
inverting $\ell$ and extending coefficients to $E$.
\end{proof}
\begin{remark}\label{rem:slice-rational-coefficients}
Lemma~\ref{slice:quadric} uses rational coefficients, including when
$\ell=2$. Its hyperplane-power comparison need not be an isomorphism
off $D$ with $\mathbb Z/2^a$ coefficients: on a smooth conic the
hyperplane class has degree $2$. We use finite coefficients to
construct the usual cohomological compatibilities, then use
\eqref{eq:slice-quadric-triangle} after rationalization. This imposes
no restriction on the residue characteristic. The case $m=0$ will
be handled directly below; formally the projective model is then
$Q_f=D\subset\mathbb P_B^0$ and the hyperplane sum is empty.
\end{remark}
\subsection{The hypersurface test}
We return to the trait $S$, the bundle stack $\mathcal B$, and the
compact constructible object $F\in\mathcal D_C$ of
Proposition~\ref{spec:missing}. For a smooth finite-type chart
$b:U\to\mathcal B$, put $F_U=b_K^*F$. Denote by
$\mathcal N_U\subset T^*U_s$ the image of the global nilpotent cone
under smooth cotangent pullback.
\begin{proposition}[Hypersurface test]\label{slice:hypersurface}
Let $b:U\to\mathcal B$ be smooth, with $U$ a finite-type $S$-scheme.
Let $f$ be a regular function on $U$, let $D=V(f)$, and let
$u\in D_s(k)$. If
\[
d(f_s)_u\notin(\mathcal N_U)_u,
\]
then
\[
\bigl(\Psi_D(F_U|_{D_K})\bigr)_u=0.
\]
\end{proposition}
\begin{proof}
If $d(f_s)_u$ is not in the image of $T^*_{b(u)}\mathcal B_s$, its
image in the relative cotangent space for $b$ is nonzero. The smooth
hypersurface criterion shows that $D\to\mathcal B$ is smooth near
$u$. Lemma~\ref{spec:vanishing} then gives the assertion.
Otherwise $d(f_s)_u=b^*A$ for a non-nilpotent covector
$A\in T^*_{b(u)}\mathcal B_s$. Proposition~\ref{two:vertex}
provides a point $x\in\mathscr X_s(k)$ and an integer $m\geq0$ with
the following property. Set
\[
T=U\times_S\mathscr X,\qquad t_0=(u,x),\qquad
D_T=D\times_S\mathscr X,
\]
and let $F_T$ be the pullback of $F_U$ to $T_K$. For the affine
quadric over $T$
\[
Q^a=\left\{\sum_{a=1}^m u_av_a+f=0\right\}
\subset T\times_S\mathbb A_S^{2m},
\]
the nearby cycles of the pullback of $F_T$ vanish at
$(t_0,0,0)$. The symbols $u_a$ here are affine coordinates, distinct
from the fixed point $u\in U_s$.
If $m=0$, this affine model is $D_T$ itself. Smooth compatibility
for $D_T\to D$ immediately gives the desired vanishing.
Suppose $m\geq1$ and compactify to
\[
g:Q_f=\left\{\sum_{a=1}^m u_av_a+fw^2=0\right\}
\subset\mathbb P_T^{2m}\longrightarrow T.
\]
The fiber over $t_0$ is the split cone of
Lemma~\ref{slice:quadric}. At every point of this fiber other than
its vertex, some $u_a$ or $v_a$ is nonzero. In the chart $u_a=1$,
for example, its equation solves uniquely for $v_a$, so $g$ is
smooth there. This argument works also in characteristic $2$.
Lemma~\ref{spec:vanishing} gives $\Psi_TF_T=0$, and smooth
compatibility therefore gives vanishing of the nearby cycles of
$g_K^*F_T$ at all these nonvertex points. The vertex lies in the
chart $w=1$, where its vanishing was supplied by
Proposition~\ref{two:vertex}.
The nearby-cycle complex restricts to zero on the entire proper
fiber. Proper compatibility and the projection formula give
\begin{equation}\label{eq:slice-quadric-pushforward-zero}
\left(\Psi_T\bigl(F_T\otimes Rg_{K*}E\bigr)\right)_{t_0}=0.
\end{equation}
Apply Lemma~\ref{slice:quadric} on $T_K$ and tensor its triangle
with $F_T$. The projection formula identifies the third term with
\[
i_{K*}(F_T|_{(D_T)_K})(-m)[-2m],
\qquad i:D_T\hookrightarrow T.
\]
After applying $\Psi_T$ and taking the stalk at $t_0$, the first
term vanishes because $\Psi_TF_T=0$, and the second vanishes by
\eqref{eq:slice-quadric-pushforward-zero}. Proper compatibility for
the closed immersion $i$ identifies the third term with
\[
\bigl(\Psi_{D_T}(F_T|_{(D_T)_K})\bigr)_{t_0}
(-m)[-2m].
\]
It too vanishes. Smooth compatibility for $D_T\to D$ completes
the proof.
\end{proof}
The passage through the projective quadric uses no purity theorem for
the possibly singular hypersurface $D$. It uses the actual
closed-support term of the comparison triangle. We now use the same
principle to impose several equations simultaneously.
\subsection{Projective incidence and higher codimension}
\begin{lemma}[Incidence comparison]\label{slice:incidence}
Let $B$ be a finite-type scheme over an algebraically closed field of
characteristic different from $\ell$. Let $f_1,\ldots,f_d$ be
regular functions on $B$, where $d\geq1$, and set
$i:J=V(f_1,\ldots,f_d)\hookrightarrow B$. For
\[
r:I=\left\{\sum_{a=1}^d c_af_a=0\right\}
\subset B\times\mathbb P^{d-1}\longrightarrow B,
\]
the hyperplane powers give a distinguished triangle
\begin{equation}\label{eq:slice-incidence-triangle}
\bigoplus_{j=0}^{d-2}E_B(-j)[-2j]
\longrightarrow Rr_*E_I
\longrightarrow i_*E_J(-(d-1))[-2(d-1)]
\longrightarrow
\left(\bigoplus_{j=0}^{d-2}E_B(-j)[-2j]\right)[1].
\end{equation}
For $d=1$ the first term is zero and $I=J$.
\end{lemma}
\begin{proof}
If $d=1$, the equation in $B\times\mathbb P^0$ is $f_1=0$,
so the stated triangle is immediate. Suppose $d\geq2$.
Off $J$, the map $\mathcal O_B^d\to\mathcal O_B$ with coefficients
$(f_1,\ldots,f_d)$ is surjective. Its kernel is a vector bundle of
rank $d-1$, and $I$ is its projective bundle of lines. The powers of
the restricted hyperplane class identify its direct image with the
first term of \eqref{eq:slice-incidence-triangle}. This can also be
checked on geometric fibers by proper base change and the cohomology
of $\mathbb P^{d-2}$.
On $J$, the family is exactly $J\times\mathbb P^{d-1}$ and its
hyperplane class comes from the second factor. The restricted cone
of the displayed morphism is therefore the constant top class
$E_J(-(d-1))[-2(d-1)]$. The global cone vanishes off $J$, so
open--closed localization identifies it with the pushforward of this
restricted cone, proving the triangle.
\end{proof}
\begin{proposition}[Higher-codimension test]\label{slice:higher}
Let $b:U\to\mathcal B$ be smooth, with $U$ a finite-type $S$-scheme,
and let $F_U=b_K^*F$ as above. Suppose that $u\in U_s(k)$ and
$f_1,\ldots,f_d$ are regular functions vanishing at $u$. Assume
that their special-fiber differentials at $u$ are linearly independent
and that
\[
\operatorname{span}_k\{d(f_{1,s})_u,\ldots,d(f_{d,s})_u\}
\cap(\mathcal N_U)_u=\{0\}.
\]
Then, for $J=V(f_1,\ldots,f_d)$,
\[
(\Psi_J(F_U|_{J_K}))_u=0.
\]
The assertion includes $d=0$, with $J=U$.
\end{proposition}
\begin{proof}
For $d=0$ the assertion is Lemma~\ref{spec:vanishing}.
For $d\geq1$ form the proper incidence morphism
\[
r:I=\left\{\sum_{a=1}^d c_af_a=0\right\}
\subset U\times_S\mathbb P_S^{d-1}\longrightarrow U.
\]
Each affine projective-space chart in
$U\times_S\mathbb P_S^{d-1}$ is still smooth over $\mathcal B$.
On the chart $c_a=1$, the hypersurface equation is
$f_a+\sum_{b\ne a}c_bf_b=0$. At a point $(u,[c])$ its
special-fiber differential is
\[
\sum_{b=1}^d c_b\,d(f_{b,s})_u.
\]
Indeed its projective-parameter component is zero since every
$f_b(u)$ is zero. The displayed combination is nonzero by linear
independence, and lies outside $(\mathcal N_U)_u$ by the hypothesis.
The nilpotent cone on the product chart is the pullback of
$\mathcal N_U$ with zero parameter component. Proposition~\ref{slice:hypersurface}
therefore gives vanishing of the nearby cycles of $r_K^*F_U$ at
every closed point over $u$.
The restriction of this nearby-cycle complex to the projective fiber
is constructible. A nonzero constructible cohomology sheaf on a
finite-type scheme over $k$ has a nonzero stalk at a closed
$k$-point. Thus the complex vanishes on the entire fiber. Proper
compatibility gives
\[
(\Psi_U Rr_{K*}r_K^*F_U)_u=0.
\]
Tensor the triangle of Lemma~\ref{slice:incidence} on $U_K$ with
$F_U$ and apply the projection formula. The first term is a finite
sum of shifts and twists of $F_U$, the second is
$Rr_{K*}r_K^*F_U$, and the third is
\[
i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)],
\qquad i:J\hookrightarrow U.
\]
After specialization and taking the stalk at $u$, the first two
terms vanish. Proper compatibility for $i$ shows that the third
is the corresponding shift and twist of
$(\Psi_J(F_U|_{J_K}))_u$, proving the result. For $d=1$ the first
sum is empty, so the same argument remains valid.
\end{proof}
We have obtained all finite transverse-slice tests while applying the
hypersurface argument only on schemes smooth over $\mathcal B$.
The remaining task is to find one such slice on which the nonzero
constructible object $F$ has a nonzero nearby-cycle stalk.
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/06-slices.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/07-support.tex ====
\section{A finite slice of the support}
\label{sec:support}
We complete the proof by applying the slicing theorem to the support of
$F$. First we arrange that a nonzero generic stalk specializes to the
closed fiber. We then cut its support to a finite scheme over the trait.
On such a scheme, geometric nearby cycles are a finite direct sum of stalk
complexes, so they cannot vanish if one of those stalks is nonzero.
\subsection{Bringing a generic stalk to the closed fiber}
We use the bounded Hecke spaces of Section~\ref{sec:hecke} to extend
bundles. The argument applies to connected reductive groups, including
those with a central torus.
\begin{lemma}
\label{support:extension}
Let $\mathscr X\to\Spec R$ be the smooth proper curve fixed in
Section~\ref{sec:specialization}, and let $K$ be the fixed algebraic
closure of $\operatorname{Frac}(R)$. Every $G$-bundle on $\mathscr X_K$
extends to a $G$-bundle on $\mathscr X_{R'}$ for some finite extension
$\operatorname{Frac}(R')/\operatorname{Frac}(R)$ inside $K$, where $R'$
is the integral closure of $R$ in that extension.
\end{lemma}
\begin{proof}
Let $\mathcal P_K$ be the bundle. By
\cite[Theorem~2 and Remark~2(b)]{DrinfeldSimpson}, it is trivial on a
nonempty open subset of $\mathscr X_K$. Here Theorem~2 applies to
connected reductive groups, and the base field $K$ is algebraically
closed. Choose a trivialization away from distinct points
$x_1,\ldots,x_r\in\mathscr X(K)$. Gluing at these points gives a chain
of one-point modifications
\[
\mathcal P_0\dashrightarrow\mathcal P_1\dashrightarrow\cdots
\dashrightarrow\mathcal P_r=\mathcal P_K,
\qquad \mathcal P_0\text{ trivial},
\]
whose $i$th modification has leg $x_i$. Choose a Schubert bound for each
of the finitely many relative positions in this chain.
Starting with the trivial bundle over $S=\Spec R$, form the space of
chains satisfying these bounds, allowing the legs to vary on
$\mathscr X$. At each stage the bounded Hecke space is representable
and proper over the preceding bundle and its new leg. Since
$\mathscr X\to S$ is proper, induction shows that the space of chains is
a proper algebraic space of finite type over $S$. It carries the
universal final bundle and contains the chosen chain as a $K$-point.
This point descends to a finite extension of $\operatorname{Frac}(R)$:
the space is of finite type and $K$ is algebraic over that field. The
valuative criterion extends the descended point over $R'$, and the
universal final bundle gives the required extension. The construction
allows the legs to coincide on the closed fiber, since it uses successive
one-point modifications.
\end{proof}
Suppose now that a spectral component is missing, and let $F$ be the
nonzero compact object supplied by Proposition~\ref{spec:missing}.
It is constructible on finite-type smooth charts. There is therefore a
$K$-valued bundle at which its stalk is nonzero: a nonzero constructible
complex on a finite-type scheme over $K$ has a nonzero stalk at a closed
$K$-point. Apply Lemma~\ref{support:extension} to this bundle and replace
$R$ by the resulting $R'$. All specialization statements from
Section~\ref{sec:specialization} remain valid with the same geometric
generic field $K$.
Choose a finite-type smooth affine chart $b:U\to\mathcal B$ through
the special value of the extended bundle. The fiber product of this
chart with its trait section is smooth over $S$ and has a $k$-point.
It has an $R$-point lifting that $k$-point. Indeed, after taking an
affine \'etale neighborhood if needed, formal smoothness gives
compatible lifts modulo every power of the uniformizer, and completeness
of $R$ gives an $R$-point of the affine scheme. Thus the trait section
lifts to $U$. After shrinking around its closed point, we may suppose
that $U/S$ has constant relative dimension $N$. The generic stalk of
$F_U=b_K^*F$ on the lifted section is nonzero, so its support has closure
meeting $U_s$.
\subsection{A slice with nonzero nearby cycles}
The remaining argument uses only the dimension of a cone in the
special-fiber cotangent bundle. We state it separately to make clear
that the support strata, rather than the sheaf, are the objects descended
to a finite field extension.
\begin{lemma}
\label{support:finite-slice}
Let $R$ be a complete discrete valuation ring with algebraically closed
residue field $k$ of characteristic different from $\ell$, let
$S=\Spec R$, and let $K$ be an algebraic closure of its fraction field. Let $U/S$ be a smooth affine
scheme of constant
relative dimension $N$, let $\mathcal F$ be a bounded constructible
$E$-complex on $U_K$, and let
$\Lambda\subset T^*U_s$ be a closed conical subset with
$\dim\Lambda\leq N$.
Suppose that the closure in $U$ of the nonzero-stalk locus of
$\mathcal F$ meets $U_s$.
Then, after a finite extension of the trait and shrinking $U$, there
exist $u\in U_s(k)$, an integer $0\leq d\leq N$, and functions
$f_1,\ldots,f_d$ vanishing at $u$ such that their special-fiber
differentials are independent,
\[
\operatorname{span}\{d(f_{1,s})_u,\ldots,d(f_{d,s})_u\}
\cap\Lambda_u\subseteq\{0\},
\]
and, for $J=V(f_1,\ldots,f_d)$,
\begin{equation}
\label{eq:support-nonzero}
\bigl(\Psi_J(\mathcal F|_{J_K})\bigr)_u\ne0.
\end{equation}
For $d=0$, the list of functions is empty and $J=U$.
\end{lemma}
\begin{proof}
\emph{Descend the finite support data.}
Let $A_1,\ldots,A_t$ be the irreducible components of the closure of
the nonzero-stalk locus in $U_K$. Bounded constructibility gives a
proper closed subset $B_i\subsetneq A_i$ such that every stalk on
$A_i\setminus B_i$ is nonzero. Give these closed sets their reduced
structures. Their finitely generated ideals descend to a common finite
extension of $\operatorname{Frac}(R)$; enlarge that extension to descend
all containments as well. Replace the trait accordingly. The new
support closure still meets
the special fiber: the finite map from the new model to the old one
maps its closure onto the old closure, since it is closed and agrees
on the geometric generic support. This step does not assert that
$\mathcal F$ descends.
Write $\overline A_i$ and $\overline B_i$ for their horizontal closures
in $U$, and put $e_i=\dim A_i$. Each $\overline A_i$ is integral and
dominates $S$. Every nonempty special fiber
$(\overline A_i)_s$ is pure of dimension $e_i$
\cite[Tag~0B2J]{Stacks}. Applying the same statement to the
irreducible components of $B_i$ shows that
\[
\dim(\overline B_i)_s<e_i
\quad\text{whenever }(\overline B_i)_s\ne\varnothing.
\]
The trait is excellent, and hence universally catenary: a complete
Noetherian local ring is excellent \cite[Tag~07QW]{Stacks}.
Consequently, at every closed point $v\in(\overline A_i)_s$, the
dimension formula gives
\begin{equation}
\label{eq:support-local-dimension}
\dim\mathcal O_{\overline A_i,v}=e_i+1
\qquad\text{\cite[Tag~02JT]{Stacks}}.
\end{equation}
\emph{Choose the special point.}
Let $d$ be maximal among the $e_i$ for which
$(\overline A_i)_s$ is nonempty. Such an index exists by the support
assumption. Choose a reduced irreducible component $T$ of dimension
$d$ in one of these special fibers. At a general closed point of $T$,
its reduced structure is smooth, no other distinct reduced component of
$\bigcup_i(\overline A_i)_s$ passes through the point, and none of the
$\overline B_i$ passes through it. To see the last assertion, a
component with empty special fiber contributes nothing, while every
remaining bad special locus has dimension strictly below $d$.
Different horizontal components $\overline A_i$ may have the same
reduced special component $T$; we retain all of them.
We can also arrange that
\begin{equation}
\label{eq:support-cone-fiber}
\dim\Lambda_u\leq N-d.
\end{equation}
Indeed, restrict $\Lambda$ to $T$. Its components that do not dominate
$T$ can be excluded over a proper closed subset of $T$. The generic
fiber of each remaining component has dimension at most $N-d$, since
$\dim\Lambda\leq N$. The same bound holds over a nonempty open subset
of $T$. Choose $u\in T(k)$ satisfying all these conditions.
\emph{Choose transverse equations.}
Let $V=T_u^*U_s$, so $\dim V=N$. If $d>0$, a general $d$-plane
$L\subset V$ satisfies $L\cap\Lambda_u\subseteq\{0\}$. Indeed,
\eqref{eq:support-cone-fiber} gives
$\dim\mathbb P(\Lambda_u)\leq N-d-1$, and the incidence of
$d$-planes meeting this projective set is a proper closed subset of
$\Gr(d,V)$. The condition that
\[
L\longrightarrow T_u^*T
\]
be an isomorphism is another nonempty open condition. These two open
subsets meet because the Grassmannian is irreducible. Choose such an
$L$ and a basis of it. For $d=0$, take $L=0$.
Lift the basis to functions $f_1,\ldots,f_d$ vanishing at $u$, on an
affine neighborhood in $U$. Their restrictions to the smooth
$d$-dimensional variety $T$ form a regular system of parameters at $u$.
Hence, for $J=V(f_1,\ldots,f_d)$ and
\[
Z=J\cap\bigcup_i\overline A_i,
\]
the point $u$ is isolated in $Z_s$ after shrinking. This assertion
concerns the union of all the support closures: its reduced special
fiber near $u$ is precisely $T$, even if several horizontal components
specialize to $T$.
We have obtained the required differentials. It remains to show that
the slice retains a nonzero generic stalk and that this stalk survives
specialization.
\emph{Find a generic generization in the slice.}
Choose an index $i$ for which $e_i=d$ and $T$ is a component of
$(\overline A_i)_s$. By \eqref{eq:support-local-dimension}, the local
ring of $\overline A_i$ at $u$ has dimension $d+1$. Quotienting by
$f_1,\ldots,f_d$ leaves dimension at least one. Its quotient by a
uniformizer $\pi$ has dimension zero, by the isolation just proved.
Thus some prime of the cut local ring does not contain $\pi$:
otherwise $\pi$ would be nilpotent and the ring would have dimension
zero. This gives a generic-fiber point $z$ specializing to $u$.
The morphism $Z\to S$ is quasi-finite at $u$, since $u$ is an isolated
closed point of its fiber \cite[Tag~01TH]{Stacks}. Its quasi-finite
locus is open, so it contains $z$. Thus $z$ has residue field finite
over $\operatorname{Frac}(R)$. Its geometric lifts to $K$ avoid
$B_i$: a point in the closed set $\overline B_i$ could not specialize
to $u\notin\overline B_i$. Each such lift therefore has a nonzero
stalk of $\mathcal F|_{J_K}$.
\emph{Compute nearby cycles on a finite neighborhood.}
Take an affine neighborhood of $u$ inside the quasi-finite locus of
$Z\to S$. The henselian decomposition of a finite-type algebra
\cite[Tag~04GJ]{Stacks} provides a finite local factor containing $u$.
It defines an open neighborhood $Q$ of $u$ in $Z$, finite over $S$,
whose special fiber has only the point $u$. Equivalently, the
separated form of Zariski's Main Theorem embeds the quasi-finite
neighborhood into a finite $S$-scheme; its henselian local factor at
$u$ lies in the original open because an open subset of a local
spectrum containing the closed point is the entire spectrum.
Every generization of $u$, including $z$, lies in $Q$.
The restriction $\mathcal F|_{J_K}$ is supported on the closed subset
$Z_K$. To use the finite neighborhood, remove $Z\setminus Q$ from
$J$. We realize this removal in the ambient chart: the complement is
closed in $J$, hence in $U$, so remove it from $U$ and then choose an
affine neighborhood of $u$. This neighborhood contains all of $Q$,
since every point of the finite local scheme $Q$ specializes to $u$.
The new $J$ is still cut out by the same functions in a smooth affine
chart; the change leaves its nearby-cycle stalk at $u$ unchanged and
makes $Q$ closed in $J$. If $\iota:Q\hookrightarrow J$ denotes this
closed immersion and $\mathcal H=\mathcal F|_{Q_K}$, localization gives
\[
\mathcal F|_{J_K}\simeq\iota_{K,*}\mathcal H.
\]
Proper compatibility for $\iota$ therefore reduces
\eqref{eq:support-nonzero} to the stalk of $\Psi_Q\mathcal H$ at $u$.
Let $a:Q\to S$ be the finite structural map. Its special fiber has
only the point $u$, so proper compatibility gives
\[
(\Psi_Q\mathcal H)_u
\simeq \Psi_S\bigl(\RGamma(Q_K,\mathcal H)\bigr)
\simeq \RGamma(Q_K,\mathcal H).
\]
The second identification uses geometric specialization on the trait,
which is the identity on coefficient complexes. The finite $K$-scheme
$Q_K$ has the \'etale topos of a finite discrete set. Consequently
\[
\RGamma(Q_K,\mathcal H)
\simeq\bigoplus_{x\in|Q_K|}\mathcal H_x.
\]
At least one summand is nonzero, as proved above, so this complex is
nonzero. This proves \eqref{eq:support-nonzero}.
These are geometric nearby cycles with the monodromy action forgotten;
no inertia invariants are taken. The displayed finite direct-sum
calculation is compatible with finite coefficients, their
$\ell$-adic limit, inversion of $\ell$, and extension to $E$.
It therefore applies to the specialization functor of
Section~\ref{sec:specialization} without requiring a field of descent
for $\mathcal F$.
\end{proof}
\subsection{Completion of the full-support proof}
\begin{proof}[Proof of Theorem~\ref{thm:full-support}]
If $Y'\ne Y$, Proposition~\ref{spec:missing} supplies the nonzero
compact object $F$ used above. Lemma~\ref{support:extension} and the
subsequent chart construction give a smooth affine $U\to\mathcal B$
of constant relative dimension $N$ for which the closure of the
nonzero-stalk locus of $F_U$ meets $U_s$.
Let $\mathcal N_U\subset T^*U_s$ be the pullback of the global
nilpotent cone. Under either displayed hypothesis regime, the
nilpotent-parabolic assumption and invariant form give the
half-dimensionality theorem of
\cite[Appendix~D.1.1--D.1.8]{AGKRRV}. In its smooth-chart formulation,
\cite[Appendix~D.1.2]{AGKRRV}, this says exactly that
\[
\dim\mathcal N_U\leq\dim U_s=N.
\]
Apply Lemma~\ref{support:finite-slice} with
$\mathcal F=F_U$ and $\Lambda=\mathcal N_U$. It gives a slice
$J=V(f_1,\ldots,f_d)$ and $u\in J_s(k)$ for which
\[
\bigl(\Psi_J(F_U|_{J_K})\bigr)_u\ne0,
\]
while the independent differentials of its equations span a plane
meeting $(\mathcal N_U)_u$ only at zero. The higher-codimension
vanishing of Proposition~\ref{slice:higher} applies to this same chart
and slice and says that this stalk is zero. This contradiction also
covers $d=0$, when the test is the original smooth-chart vanishing.
Thus the complementary union is empty. The primed equivalence recalled
in Section~\ref{sec:specialization} is consequently the full restricted
equivalence asserted in Theorem~\ref{thm:full-support}.
\end{proof}
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/07-support.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/08-consequences.tex ====
\section{Component categories and consequences}
\label{sec:consequences}
Throughout this section assume either Hypothesis~\ref{hyp:A} or
Hypothesis~\ref{hyp:B}. We apply Theorem~\ref{thm:full-support} to the spectral action.
First we identify each localized automorphic category. We then derive the
finite-field arithmetic formula and construct Hecke eigenobjects with
coherent structure.
Throughout this section put
\[
\mathcal C_s=\Shv_{\Nilp}(\Bun_G),\qquad
\mathcal A=\QCoh(Y),\qquad
\mathcal D=\IndCoh_{\Nilp}(Y),
\]
and denote the restricted Langlands equivalence by
\(\mathbb L_G^{\mathrm{restr}}:\mathcal C_s\xrightarrow{\sim}\mathcal D\).
Its \(\mathcal A\)-linear structure is the one furnished by
\cite[Lemma~1.3.7]{GR}.
\subsection{The category on a connected component}
For an open-and-closed component \(i:Z\hookrightarrow Y\), define
\[
\mathcal C_{s,Z}:=
\QCoh(Z)\underset{\QCoh(Y)}\otimes\mathcal C_s.
\]
Here the tensor product is the tensor product of presentable dg categories.
The idempotent quasi-coherent object \(e_Z=i_*\mathcal O_Z\) acts as the
projection to this summand.
\begin{corollary}\label{cor:components}
Under either Hypothesis~\ref{hyp:A} or Hypothesis~\ref{hyp:B}, every
connected component \(Z\subset Y\) has a nonzero localized automorphic
category. The restriction of the given spectral-linear Langlands functor
induces an equivalence
\[
\mathcal C_{s,Z}\xrightarrow{\sim}\IndCoh_{\Nilp}(Z).
\]
\end{corollary}
\begin{proof}
The decomposition \(Y=Z\sqcup(Y\setminus Z)\) decomposes quasi-coherent
and nilpotent ind-coherent sheaves into their two corresponding summands.
Consequently
\[
\QCoh(Z)\underset{\mathcal A}\otimes\mathcal D
\simeq\IndCoh_{\Nilp}(Z).
\]
Tensor the \(\mathcal A\)-linear equivalence of
Theorem~\ref{thm:full-support} with \(\QCoh(Z)\) over \(\mathcal A\).
This proves the asserted identification by the specified functor.
Every component contains a semisimple \(E\)-point
\cite[Proposition~3.7.2 and Corollary~3.7.4]{AGKRRV}.
Pullback to this point shows that \(\mathcal O_Z\ne0\).
The fully faithful embedding
\(\QCoh(Z)\hookrightarrow\IndCoh_{\Nilp}(Z)\)
therefore proves nonvanishing.
\end{proof}
The component index and the parameter itself have different roles.
Two geometric parameters lie in the same component precisely when their
semisimplifications are isomorphic
\cite[Proposition~3.7.2]{AGKRRV}.
The eigenobject construction below uses the actual parameter, including
its extension data.
\subsection{The arithmetic formula}
Assume Hypothesis~\ref{hyp:A}. The finite-field models \(X_0\) and
\(G_0\) determine geometric \(q\)-Frobenius. Its pullback action on
\(\QLisse(X)\) induces an automorphism \(\Frob\) of \(Y\).
Define its derived fixed-point stack by
\[
Y^{\mathrm{arithm}}:=Y^{\Frob}
=Y\underset{Y\times_EY}{\overset{\mathrm R}\times}Y,
\]
where the two maps to \(Y\times_EY\) are the diagonal and the graph of
\(\Frob\). In particular, an \(E\)-point includes a Weil structure on
the geometric local system. These are the conventions of
\cite[\S24.1]{AGKRRV} and \cite[\S1.5.1]{GR}.
\begin{corollary}\label{cor:arithmetic}
For the finite-field datum of Hypothesis~\ref{hyp:A}, there is a canonical
isomorphism
\[
\Funct_c\bigl(\Bun_{G_0}(\mathbb F_q),E\bigr)
\simeq
\Gamma^{\IndCoh}\bigl(Y^{\mathrm{arithm}},
\omega_{Y^{\mathrm{arithm}}}\bigr).
\]
The left side is the vector space of compactly supported functions on
isomorphism classes of rational bundles, regarded as a complex concentrated
in degree zero. The right side uses the ind-coherent dualizing object
and ind-coherent global sections on the derived fixed-point stack.
\end{corollary}
\begin{proof}
The primed union \(Y'\) is Frobenius-stable. The spectral trace
calculation of Beraldo--Lin--Reeves
\cite[\S\S6.4.12--6.4.13]{BeraldoLinReeves} enters the known primed
formula \cite[\S\S1.5.3--1.5.5 and Corollary~1.5.6]{GR}:
\[
\Funct_c\bigl(\Bun_{G_0}(\mathbb F_q),E\bigr)
\simeq
\Gamma^{\IndCoh}\bigl((Y')^{\Frob},
\omega_{(Y')^{\Frob}}\bigr).
\]
Its automorphic trace input is
\cite[Theorem~0.2.6]{AGKRRVTrace}, with the endofunctor given by
pushforward along geometric Frobenius on \(\Bun_G\).
By Theorem~\ref{thm:full-support}, the actual inclusion \(Y'\hookrightarrow
Y\) is equality. Taking derived fixed points identifies
\((Y')^{\Frob}\) with \(Y^{\mathrm{arithm}}\), and transports the
same dualizing object and global-sections functor. Substitution gives
the result.
\end{proof}
Keeping \(\Gamma^{\IndCoh}\) explicit avoids a convention in
\cite[\S1.5.3]{GR}: its ordinary-global-sections notation uses the image
of the dualizing object under the dual of
\(\Upsilon:\QCoh\to\IndCoh\).
No classical truncation of \(Y^{\mathrm{arithm}}\) or replacement of its
dualizing object by its structure sheaf is made here.
\subsection{Hecke eigenobjects with coherent structure}
We first record the categorical argument that supplies the eigenobject.
It will also give its tensor compatibilities.
\begin{lemma}\label{lem:parameter-adjoint}
Let \(\mathcal A_0\) be a compactly generated, presentable, stable,
\(E\)-linear symmetric monoidal category whose compact objects are
dualizable. Assume its tensor product preserves colimits separately.
Let \(a:\mathcal A_0\to\Vect_E\) be a continuous \(E\)-linear symmetric monoidal
functor, and give \(\Vect_E\) its \(\mathcal A_0\)-action through \(a\).
The right adjoint \(R_a\) of \(a\) is continuous and carries a coherent
\(\mathcal A_0\)-linear structure. Moreover, \(R_a(E)\ne0\).
\end{lemma}
\begin{proof}
Presentability gives the right adjoint. Every compact object of
\(\mathcal A_0\) is dualizable, so its image under \(a\) is a perfect
complex of \(E\)-vector spaces. Thus \(a\) preserves compact objects.
Testing against compact generators and using adjunction shows that
\(R_a\) preserves filtered colimits. The right adjoint is exact, since
both categories are stable. Exactness and preservation of filtered
colimits imply preservation of all colimits.
The adjunction has a canonical projection morphism
\[
b\otimes R_a(V)\longrightarrow R_a\bigl(a(b)\otimes V\bigr).
\]
For dualizable \(b\), its invertibility follows by testing maps out of
an arbitrary \(c\in\mathcal A_0\):
\begin{align*}
\RHom_{\mathcal A_0}(c,b\otimes R_a(V))
&\simeq\RHom_{\mathcal A_0}(b^\vee\otimes c,R_a(V))\\
&\simeq\RHom_E(a(b)^\vee\otimes a(c),V)\\
&\simeq\RHom_E(a(c),a(b)\otimes V).
\end{align*}
Both sides of the projection morphism preserve colimits in \(b\).
Compact generation therefore proves invertibility for every \(b\).
These morphisms are the adjunction mates of the module structure on
\(a\); hence their unit and associativity compatibilities hold before
invertibility is imposed. They now give a coherent module-functor
structure on \(R_a\).
Finally,
\[
\RHom_{\mathcal A_0}(\mathbf1,R_a(E))
\simeq\RHom_E(E,E)\simeq E,
\]
so \(R_a(E)\ne0\). This last argument does not require a compact unit.
\end{proof}
For our spectral stack, \(\mathcal A=\QCoh(Y)\) satisfies the hypotheses
of Lemma~\ref{lem:parameter-adjoint} by
\cite[\S7.9 and Corollary~C.4.4]{AGKRRV}.
On the disjoint union of components, the compact generators have finite
component support. We use the continuous, fully faithful \(\mathcal A\)-linear functor
\[
\jmath_Y:\QCoh(Y)
\xrightarrow[\sim]{\Upsilon_Y}\IndCoh_{\{0\}}(Y)
\hookrightarrow\IndCoh_{\Nilp}(Y).
\]
Here \(\Upsilon_Y(Q)=Q\otimes\omega_Y\), as in
\cite[\S1.3.1, Equation~(1.2)]{GR}.
Its essential image has zero singular support and therefore lies in the
nilpotent category.
For a finite set \(I\), put \(\mathcal Q_I=\QLisse(X)^{\otimes I}\).
Given \(V\in\Rep(\check G)^{\otimes I}\), let \(H_I(V,-)\) denote the
multi-leg Hecke functor and let
\(\mathcal E_V^I\in\mathcal A\otimes\mathcal Q_I\)
be the universal evaluation object. For a parameter
\(\sigma:\Spec(E)\to Y\), define its evaluation local system by
\[
\sigma^I(V):=(\sigma^*\otimes\id)(\mathcal E_V^I)
\in\mathcal Q_I.
\]
For a family of representations \((V_i)_{i\in I}\), this is the exterior
product of the local systems \(\sigma(V_i)\) on the corresponding factors
of \(X^I\).
\begin{definition}\label{def:eigenstructure}
A Hecke eigenobject of eigenvalue \(\sigma\) with coherent structure is an
object \(M\in\mathcal C_s\) equipped, for all finite sets \(I\), with
isomorphisms
\[
H_I(V,M)\simeq M\boxtimes\sigma^I(V)
\]
natural in \(V\in\Rep(\check G)^{\otimes I}\) and compatible with units,
tensor products, permutations of the legs and collisions of legs, with
all compatibilities imposed homotopy-coherently
\cite[\S15.2]{AGKRRV}.
\end{definition}
\begin{corollary}\label{cor:eigenobjects}
Under either Hypothesis~\ref{hyp:A} or Hypothesis~\ref{hyp:B}, every
parameter \(\sigma:\Spec(E)\to Y\) admits a nonzero object
\(M_\sigma\in\Shv_{\Nilp}(\Bun_G)\) with a coherent Hecke eigenstructure
of eigenvalue \(\sigma\). If \(\sigma\) belongs to a component \(Z\),
then \(M_\sigma\in\mathcal C_{s,Z}\).
The assertion concerns the ind-constructible dg category and includes
no boundedness, perversity or Weil structure.
\end{corollary}
\begin{proof}
Apply Lemma~\ref{lem:parameter-adjoint} to
\(a=\sigma^*:\mathcal A\to\Vect_E\), and write \(\sigma_*\) for its
continuous right adjoint. The composite
\[
\Phi_\sigma:
\Vect_\sigma\xrightarrow{\sigma_*}\mathcal A
\xrightarrow{\jmath_Y}\mathcal D
\xrightarrow{(\mathbb L_G^{\mathrm{restr}})^{-1}}\mathcal C_s
\]
is a continuous \(\mathcal A\)-linear functor. Here \(\Vect_\sigma\)
denotes \(\Vect_E\) with its action through \(\sigma^*\).
Set \(M_\sigma=\Phi_\sigma(E)\).
It is nonzero by the lemma, full faithfulness of
\(\jmath_Y\), and the Langlands equivalence.
If \(\sigma\) lies in \(Z\), the idempotent \(e_Z\) pulls back to \(E\),
while its complementary idempotent pulls back to zero.
Linearity therefore places \(M_\sigma\) in \(\mathcal C_{s,Z}\).
To verify the eigenstructure, fix a finite set \(I\).
The action of \(\mathcal E_V^I\) on the automorphic category is the
multi-leg Hecke functor \(H_I(V,-)\)
\cite[\S\S14.2--14.3]{AGKRRV}.
Extend \(\Phi_\sigma\) by the identity on \(\mathcal Q_I\).
Its module structure gives
\[
H_I(V,M_\sigma)
\simeq(\Phi_\sigma\otimes\id)\bigl(\sigma^I(V)\bigr)
\simeq M_\sigma\boxtimes\sigma^I(V).
\]
The second equivalence uses that a continuous \(E\)-linear functor from
\(\Vect_E\) is tensoring with its value on \(E\).
These equivalences are natural in \(V\). They preserve tensor products
and the unit by the coherent module structure of \(\Phi_\sigma\), and
are compatible with collisions of legs and maps between finite sets by
the universal symmetric monoidal evaluation system.
Thus they supply the homotopy-coherent Hecke eigenstructure of
\cite[\S15.2]{AGKRRV}. Equivalently, restricting \(\Phi_\sigma\) along
the symmetric monoidal Ran-to-spectral action gives the eigenobject in
\cite[\S15.1]{AGKRRV}.
The evaluation uses \(\sigma\) itself, so its eigenvalue is the actual
parameter, whether or not it is semisimple.
\end{proof}
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/08-consequences.tex ====
==== BEGIN The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/09-coefficients.tex ====
\section{Rational coefficients and spectral support}
\label{sec:coefficients}
Theorem~\ref{thm:rational-support} concerns the given rational spectral
action. If the structure idempotent of a nonempty open-and-closed
$U\subset Y_0$ acted by zero, its coefficient extension would also act
by zero. For $X\ne\mathbb P^1$, we will compare the sheaf categories,
spectral prestacks and actions, so that geometric full support rules
out this vanishing. Faithful flatness then descends the absence of a
missing support summand. For $X=\mathbb P^1$, connectedness of the
spectral prestack will suffice.
Throughout this section
\[
E_0=\mathbb Q_\ell,\qquad E=\overline{\mathbb Q}_\ell,
\qquad k=\overline{\mathbb F}_q.
\]
The curve and group satisfy the rational hypotheses in the introduction.
Put $\mathcal C_E=\Shv_{\Nilp,E}(\Bun_G)$ and retain
$\mathcal C_0=\Shv_{\Nilp,E_0}(\Bun_G)$ and $Y_0$ from there.
The symbol $Y$ continues to mean the spectral prestack over $E$.
All tensor products of categories below are presentable dg categorical
tensor products. In particular, extension of coefficients means
$\Vect_E\otimes_{E_0}(-)$; it does not mean extension of the geometric
base field.
\subsection{Coefficient categories, including their unbounded objects}
On a finite-type scheme $T/k$, $\Shv_L(T)$ is the ind-completion of
bounded constructible $L$-adic complexes, for $L=E_0$ or $E$.
For a smooth $T$ and a closed conical subset $N\subset T^*T$, write
$\mathcal P_{N,L}(T)$ for the category of constructible perverse sheaves
with singular support in $N$. The category $\Shv_{N,L}(T)$ consists of
the objects whose perverse cohomology belongs to
$\operatorname{Ind}(\mathcal P_{N,L}(T))$ in every degree.
This is the cohomological singular-support definition of
\cite[\S\S E.5.1--E.5.8]{AGKRRV}. It need not be the ind-completion
of support-bounded constructible complexes. The zero-cone case on a
smooth scheme is $\QLisse_L$, with its equivalent ordinary-cohomology
definition recalled in the introduction
\cite[\S\S1.2.5--1.2.11]{AGKRRV}.
\begin{lemma}[Coefficient comparison]\label{lem:coeff-sheaves}
Extension of coefficients induces equivalences
\begin{align}
\Vect_E\otimes_{E_0}\Shv_{E_0}(T)&\simeq\Shv_E(T),
\label{eq:coeff-ambient}\\
\Vect_E\otimes_{E_0}\QLisse_{E_0}(X)&\simeq\QLisse_E(X),
\label{eq:coeff-qlisse}\\
\Vect_E\otimes_{E_0}\mathcal C_0&\simeq\mathcal C_E.
\label{eq:coeff-nilp}
\end{align}
The first equivalence also holds on locally finite-type algebraic stacks
by smooth descent. The second is symmetric monoidal and respects the
t-structures used in the tensor-functor definition of $Y_0$ and $Y$.
On smooth charts, the third respects the cohomological nilpotent
condition. The free and forgetful functors in these comparisons are
t-exact for the ordinary and perverse t-structures whenever these are
used.
\end{lemma}
\begin{proof}
\emph{Constructible coefficient extension.}
We first prove the ambient statement. For a finite extension
$L/E_0$, coefficient extension and restriction preserve bounded
constructibles. On objects extended from $E_0$, the mapping complexes
are obtained by tensoring their original mapping complexes with $L$.
Moreover, every $L$-constructible $M$ is a retract of
$L\otimes_{E_0}\operatorname{Res}_{L/E_0}M$: the multiplication map
admits an $L$-linear section given by the separability idempotent of
$L/E_0$. Thus the induced compact objects have the required mapping
complexes and generate after taking retracts. Constructible
$E$-complexes and their morphisms are obtained by passage through the
finite coefficient fields. These coefficient conventions and the
compact-object description are supplied by
\cite[Proposition~5.2, Theorem~7.7, Lemma~7.9, Proposition~8.2 and
Corollary~8.3]{HemoRicharzScholbach}; see also
\cite[\S2]{HansenScholze}. The cohomological finiteness
condition~(8.1) of \cite{HemoRicharzScholbach} holds on every finite-type
chart over the separably closed field $k$, with either algebraic
$\mathbb Q_\ell$-coefficient field, by Lemma~8.6(1) there. Thus its
compactness statements apply to the charts used here.
Passing to ind-completions proves
\eqref{eq:coeff-ambient}.
\emph{Detecting the cohomological support condition.}
View the left side of \eqref{eq:coeff-ambient} as internal $E$-module
objects in $\Shv_{E_0}(T)$. We will show that such a module satisfies
the $E$-coefficient support condition exactly when its underlying
$E_0$-object satisfies the $E_0$-condition. This will identify the
required subcategories degree by degree, including their unbounded
objects.
Write $F$ for the free-module functor and $U$ for its forgetful
functor. Both are exact for ordinary and perverse cohomology. For $U$,
this assertion includes infinite coefficient
extension: if a constructible $E$-object has a form $M_L$ over a finite
extension $L/E_0$, its underlying $E_0$-object is
\begin{equation}\label{eq:coeff-forgetful}
U(M_L\otimes_L E)=
\mathop{\operatorname{colim}}_{L\subset L'\subset E,
\,[L':E_0]<\infty}
\operatorname{Res}_{L'/E_0}(M_L\otimes_L L').
\end{equation}
Finite extension and restriction are t-exact, and the ind-t-structures
commute with filtered colimits. This proves the assertion first on
constructibles, then on the ambient ind-categories.
For bounded constructibles, coefficient extension preserves and reflects
the singular-support bound. The defining test pairs express this bound
by universal local acyclicity. We use its dualizability criterion in
the algebraic $\mathbb Q_\ell$-coefficient setting of
\cite[Theorem~1.6]{HansenScholze}. The correspondence category in
\S3 of that paper has compositions formed from pullback, tensor
product and $f_!$. These operations commute with coefficient extension
by the coefficient formalism and projection formula of \S2 there.
Extending the duality data therefore preserves universal local
acyclicity.
For reflection, suppose a bounded constructible complex over a finite
extension $L/E_0$ becomes universally locally acyclic over $E$.
The relative dual of its $E$-extension is constructible by
\cite[Proposition~3.4(ii)]{HansenScholze}.
The dual, evaluation and coevaluation maps, and both triangle
identities are finite data on the fixed finite-type test schemes and
their products. By \cite[Lemma~7.9]{HemoRicharzScholbach}, they descend
to a common finite coefficient extension. Thus the complex is already
universally locally acyclic over that finite extension. Finite
extension reflects local acyclicity: after forgetting the extension,
the local-acyclicity test is a nonzero finite direct sum of the
original test. This proves reflection for each test pair and hence
for the singular-support bound. Finite restriction preserves the
same bound, since after a splitting coefficient extension it is a
finite direct sum of coefficient conjugates. The same arguments, or
the finite-rank local-system description, apply to lissity.
It follows that $F$ and $U$ preserve the relevant ind-perverse hearts.
For $U$, use \eqref{eq:coeff-forgetful} and filtered colimits; a finite
rank over $E$ is not being treated as a finite rank over $E_0$.
To check the converse explicitly, let $P$ be a perverse internal
$E$-module and suppose that $U(P)$ lies in the required ind-heart over
$E_0$. The counit
\[
F(U(P))\longrightarrow P
\]
is an epimorphism in the module heart: its underlying map splits by
$u\mapsto1\otimes u$. Its source belongs to the required ind-heart
over $E$. That heart is closed under quotients, by the Serre property
of the singular-support heart. Hence $P$ belongs to it as well.
Thus, in every degree, the cohomological support condition for an
$E$-module is precisely the condition on its underlying $E_0$-object.
For $\QLisse$, apply this argument to the zero-cone ind-perverse
heart and use its equivalence with the ordinary-cohomology definition
\cite[\S1.2.9]{AGKRRV}. Ordinary lisse sheaves need not form a
Serre subcategory of the full ordinary heart; no such assertion is
used.
\emph{Scalar extension and smooth descent.}
We have identified which ambient $E$-modules satisfy the support
condition. To realize them as the categorical scalar extension of
the corresponding $E_0$-subcategory, we use their free-module
resolutions.
The perverse Serre condition gives closure under finite cones.
Since perverse cohomology commutes with filtered colimits, these
subcategories are closed under filtered colimits and arbitrary direct
sums, hence under all colimits, including bar realizations. They are also
closed under scalar tensors. Internal modules in each
such subcategory are therefore exactly the ambient modules whose
underlying objects belong to it. The free-forgetful bar resolution
realizes every module from free modules within that subcategory.
This identifies these module categories with the corresponding categorical scalar
extension and proves \eqref{eq:coeff-qlisse} and the corresponding
singular-support comparison on smooth charts. Ordinary tensor products
and their structural maps commute with coefficient extension, giving
the monoidal assertion. Connective free modules generate the
connective module category under connective colimits, by the same bar
resolution, so the induced tensor t-structures agree.
Finally, take the smooth-chart limits defining sheaves on a stack.
Internal modules commute with these limits. Equivalently,
$\Vect_E$ is the module category of the continuous monad
$E\otimes_{E_0}(-)$ on $\Vect_{E_0}$ and is dualizable as a presentable
$E_0$-linear category \cite[Lemma~1.6.3]{GKRV}. Tensoring with a
dualizable category has tensoring with its dual as a left adjoint,
so it preserves limits. The smooth pullbacks defining these limits
are continuous. Applying the
chartwise support comparison proves \eqref{eq:coeff-nilp}.
\end{proof}
In particular, this argument does not exchange scalar extension with
an unspecified left completion. It treats the actual cohomological
categories. It applies to $\QLisse(\mathbb P^1)$ as well, without
identifying that category with $\IndLisse(\mathbb P^1)$.
\subsection{Affine families and the coherent Hecke action}
The sheaf comparison now applies to the affine families defining the
spectral prestacks. We retain the right-t-exact condition on their
tensor functors: a comparison of field-valued parameters would not
identify these prestacks.
\begin{lemma}[Spectral coefficient comparison]\label{lem:coeff-spectral}
There are natural identifications
\begin{equation}\label{eq:coeff-spectral}
Y_0\times_{\Spec E_0}\Spec E\simeq Y,
\qquad
\Vect_E\otimes_{E_0}\QCoh(Y_0)\simeq\QCoh(Y).
\end{equation}
The second is symmetric monoidal. These identifications compare the
universal evaluation systems for every finite set of legs.
\end{lemma}
\begin{proof}
Let $A$ be a connective commutative $E$-algebra and write
$\operatorname{Mod}_A=\QCoh(\Spec A)$. Lemma~\ref{lem:coeff-sheaves}
and associativity give a symmetric monoidal equivalence
\[
\operatorname{Mod}_A\otimes_{E_0}\QLisse_{E_0}(X)
\simeq
\operatorname{Mod}_A\otimes_E\QLisse_E(X).
\]
The tensor t-structures are generated by connective objects of the
factors; equivalently they are the module t-structures with t-exact
forgetful functor \cite[\S1.4.1]{AGKRRV}. They agree under this
identification by the connective assertion in
Lemma~\ref{lem:coeff-sheaves}.
For the split dual group, scalar extension identifies
$\Vect_E\otimes_{E_0}\Rep_{E_0}(\check G_0)$ with
$\Rep_E(\check G)$ as a symmetric monoidal category with t-structure.
In characteristic zero, finite-dimensional representations in the
heart generate its connective objects; over a split reductive group
the irreducible highest-weight representations are defined over
$E_0$ and remain irreducible after extension
\cite[Theorem~22.2, \S\S22.3--22.5 and Theorem~22.42]{Milne}.
The universal property of scalar extension therefore identifies
tensor functors into the displayed target, and right t-exactness is
equivalent before and after this extension. This proves the first
assertion on every connective affine $E$-algebra $A$, naturally in
$A$, using precisely the moduli definition
\cite[\S1.4.2]{AGKRRV}.
Present $Y_0$ as the colimit of its affine test schemes. Base change
presents the left side of \eqref{eq:coeff-spectral} by the base-changed
affines. Quasi-coherent categories form the corresponding limit.
On an affine, the coefficient comparison is the usual equivalence
$\Vect_E\otimes_{E_0}\operatorname{Mod}_B
\simeq\operatorname{Mod}_{B\otimes_{E_0}E}$.
The affine pullbacks in this diagram are continuous. The categorical
dualizability of $\Vect_E$, justified in
Lemma~\ref{lem:coeff-sheaves}, therefore lets tensoring commute with
this limit and proves the second assertion, including its monoidal
structure. No finite-dimensionality of $E$ over $E_0$ is required.
Evaluation of a representation at the tensor functor parametrized by
$\Spec A$ is unchanged by these identifications. The assertion is
natural in representations, affine tests and maps of finite sets.
It therefore identifies the entire universal evaluation system,
including tensor products and collisions of legs.
\end{proof}
It remains to compare the given rational action with the geometric
action. We establish this comparison for $X\ne\mathbb P^1$, the only
case needed below. The preceding coefficient comparisons hold for
every $X$; for $\mathbb P^1$ we will instead deduce support from
connectedness of the spectral prestack.
\begin{lemma}[Comparison of spectral actions]\label{lem:coeff-action}
Suppose $X\ne\mathbb P^1$ and equip $\mathcal C_0$ with the rational
spectral-action datum of Theorem~\ref{thm:rational-support}.
Under Lemmas~\ref{lem:coeff-sheaves} and~\ref{lem:coeff-spectral}, its
scalar extension is the established $\QCoh(Y)$-action on
$\mathcal C_E$.
\end{lemma}
\begin{proof}
First, $\QLisse_E(X)=\IndLisse_E(X)$ for this curve
\cite[\S1.3.3 and \S E.2.8]{AGKRRV}. The same equality holds over
$E_0$. To see this, extend an object of $\QLisse_{E_0}(X)$ and express
it using the constructible lisse generators over $E$. Forgetting
coefficients takes those generators into the colimit closure of the
$E_0$-constructible lisse objects, by
\eqref{eq:coeff-forgetful}. The original object is a retract of its
extension followed by forgetting: choose an $E_0$-linear retraction
$E\to E_0$ of the unit. Constructible lisse objects are compact in
the ambient ind-constructible category. Thus these $\QLisse$
categories are compactly generated and dualizable. Their exterior
product and affine-limit evaluation constructions consequently commute
with the coefficient comparisons above.
Fix the geometric Satake identification over $E_0$, available in
\cite[Theorem~14.1]{MirkovicVilonen}, and use its scalar extension
over $E$. Take the same dual-group identification, cohomological
shifts and geometric Tate-line conventions. We compare the whole
finite-set monoidal diagram giving the Hecke functors, not only its
objects indexed by highest weights. On each bounded Schubert support,
coefficient extension commutes with exterior products, pullbacks,
descent and proper convolution pushforwards. It is perverse t-exact,
so it also commutes with intermediate extension, expressed as the
image of the map from perverse extension by zero to perverse direct
image. Hence it takes the Satake IC objects, their convolution maps,
and the fusion diagrams with their unit and symmetry constraints to
the corresponding diagrams over $E$. The fiber and weight functors
in geometric Satake identify the same split dual group under this
operation \cite[\S\S4--6 and Theorem~14.1]{MirkovicVilonen}.
For completeness, the derived functors used here are the derived
extension of naive geometric Satake. In
\cite[\S\S B.3.5--B.3.8]{GKRV} the compatible monoidal diagram in
the shifted perverse hearts is extended using the bounded-derived
universal property. The induced right-lax monoidal structure is strict;
one then restricts to compact representations and ind-extends.
Apply this construction
to the entire coefficient-comparison diagram. Naturality gives the
comparison of the derived functors and all finite-set structural
maps; the monoidal equivalences persist under the exact coefficient
functor. The correspondence action of
\cite[\S\S B.2.5--B.2.10]{GKRV} then compares the continuous Hecke
systems, including the units, composition maps and base-change
compatibilities used in their action.
The factorization into nilpotent sheaves with $\QLisse$-valued legs
is through the fully faithful exterior-product functors
\cite[\S14.2.5]{AGKRRV}. Lemma~\ref{lem:coeff-sheaves} compares
these subcategories, so the comparison also holds after this
factorization. Lemma~\ref{lem:coeff-spectral} identifies universal
evaluation. By the assumed rational datum, restricting the extended
spectral action along evaluation therefore gives exactly the
established coherent Hecke system over $E$. The equivalence of spaces
of actions in \cite[Theorem~8.1.4]{AGKRRV}, applied over the
algebraically closed field $E$, identifies it with the established
spectral action of \cite[Theorem~14.3.2]{AGKRRV}.
\end{proof}
No Frobenius structure or normalized rational Langlands functor has
been chosen in this comparison. What it establishes is the comparison
of the given rational spectral action with the geometric action used
in Theorem~\ref{thm:full-support}.
\subsection{Descent of support and the arithmetic qualification}
For $X\ne\mathbb P^1$, we have now compared the sheaf categories,
the spectral prestacks and the coherent actions. The remaining support argument uses the
idempotent of the missing open-and-closed substack.
\begin{lemma}[Detection of a clopen support]\label{lem:coeff-idempotent}
Suppose the coefficient comparisons
\eqref{eq:coeff-nilp} and~\eqref{eq:coeff-spectral} identify the two
spectral actions. If an open-and-closed substack $U\subset Y_0$ acts
by zero on $\mathcal C_0$, then $U=\varnothing$.
\end{lemma}
\begin{proof}
Let $e_U$ denote $\mathcal O_U$ extended by zero in $\QCoh(Y_0)$.
Its extension is the analogous idempotent $e_{U_E}$, where
$U_E=U\times_{E_0}E$. Compatibility of the actions and generation
by scalar-extended objects show that $e_{U_E}$ acts by zero on all
of $\mathcal C_E$.
The curve and split group descend to a finite field, so
Theorem~\ref{thm:full-support} in regime A identifies this action
with the action on $\IndCoh_{\Nilp}(Y)$. Apply $e_{U_E}$ to the
object $\jmath_Y(\mathcal O_Y)$, with $\jmath_Y$ as defined in
Section~\ref{sec:consequences}. Its value is $\jmath_Y(e_{U_E})$.
Full faithfulness of $\jmath_Y$ implies $e_{U_E}=0$, hence
$U_E=\varnothing$.
To descend this equality, take any affine test $T=\Spec A\to Y_0$.
The inverse image of $U$ is an open-and-closed affine summand of $T$,
cut out by an idempotent of $\pi_0(A)$. Its base change to
$A\otimes_{E_0}E$ is empty. Faithful flatness of $E/E_0$ forces that
idempotent to vanish. This holds on every affine test, so
$U=\varnothing$. No rational-point description of components is used.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:rational-support}]
For $X\ne\mathbb P^1$, Lemma~\ref{lem:coeff-action} supplies the
hypothesis of Lemma~\ref{lem:coeff-idempotent}. An open-and-closed
localization is the image of its idempotent projection, so the
vanishing of the localized category is exactly zero action of that
idempotent. The lemma proves the assertion.
If $X=\mathbb P^1$, the geometric spectral prestack $Y$ has one
connected component: its components are indexed by semisimple
geometric local systems \cite[Proposition~3.7.2]{AGKRRV}, and the
geometric \'etale fundamental group of $\mathbb P^1$ is trivial.
Indeed, a connected finite \'etale cover $C\to\mathbb P^1$ of degree
$n$ satisfies $2g(C)-2=-2n$ by Riemann--Hurwitz
\cite[Tag~0C1B]{Stacks}, so $n=1$. By
Lemma~\ref{lem:coeff-spectral} and the faithful affine test in
Lemma~\ref{lem:coeff-idempotent}, $Y_0$ has no nonempty proper
open-and-closed substack. The category $\mathcal C_0$ is nonzero;
for instance its dualizing object has shifted constant restrictions
on smooth charts and has zero singular support. Thus its only
nonempty open-and-closed localization, the whole category, is
nonzero. This also proves the theorem in genus zero.
Finally, the stipulated complement of $Y'_0$ acts by zero, so it is
empty. In particular a given spectral-linear primed equivalence
has this property by linearity, and its specified inclusion is an
equality of prestacks.
\end{proof}
Here is the precise arithmetic implication of rational support.
Fix finite-field descent data $X_0,G_0$ and write $\Frob$ for geometric
Frobenius on $Y_0$.
\begin{corollary}[Conditional rational unpriming]
\label{cor:rational-arithmetic}
Let $Y'_0\subset Y_0$ be a Frobenius-stable open-and-closed
support union as in Theorem~\ref{thm:rational-support}.
In addition to the rational hypotheses, assume a canonical primed
trace isomorphism over $E_0$ is given:
\[
\Funct_c\bigl(\Bun_{G_0}(\mathbb F_q),E_0\bigr)
\xrightarrow{\ \sim\ }
\Gamma^{\IndCoh}\bigl((Y'_0)^{\Frob},
\omega_{(Y'_0)^{\Frob}}\bigr).
\]
Assume the given map intertwines the excursion and spherical Hecke
actions in the same Frobenius and Satake conventions.
Then the same canonical formula and compatibilities hold with
$Y_0^{\Frob}$ in place of $(Y'_0)^{\Frob}$.
\end{corollary}
\begin{proof}
Theorem~\ref{thm:rational-support} identifies the actual inclusion
$Y'_0\subset Y_0$ with equality. Its derived fixed-point stacks,
dualizing objects and ind-coherent global-sections functors are
therefore the same. Substitute this equality in the given map.
\end{proof}
The additional comparison map in this corollary is not constructed
here. The primed arithmetic theorem cited in
Corollary~\ref{cor:arithmetic} has coefficients $E$; its existence
does not by itself give a canonical map over $E_0$. An unconditional
rational arithmetic formula still requires that map and its
Frobenius, dualizing-object, ind-coherent global-sections and excursion
compatibilities. Equality of derived fixed-point prestacks alone
supplies none of these additional comparison data.
==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//sections/09-coefficients.tex ====
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==== END The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic//refs.bib ====