64c3814
==== TABLE OF CONTENTS ====
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//macros.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//paper.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//references.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/01-introduction.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/02-inputs.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/03-components.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/04-orbits.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/05-traces.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/06-measures.tex
Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/07-descent.tex
===========================
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//macros.tex ====
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\E}{\overline{\mathbb Q}_{\ell}}
\newcommand{\Ghat}{\widehat G}
\newcommand{\ghat}{\widehat{\mathfrak g}}
\newcommand{\Nhat}{\widehat{\mathcal N}}
\newcommand{\cA}{\mathcal A}
\newcommand{\cC}{\mathcal C}
\newcommand{\cD}{\mathcal D}
\newcommand{\cF}{\mathcal F}
\newcommand{\cO}{\mathcal O}
\newcommand{\cN}{\mathcal N}
\newcommand{\cZ}{\mathcal Z}
\newcommand{\Gm}{\mathbb G_m}
\newcommand{\Ga}{\mathbb G_a}
\DeclareMathOperator{\Bun}{Bun}
\DeclareMathOperator{\LocSys}{LocSys}
\DeclareMathOperator{\IndCoh}{IndCoh}
\DeclareMathOperator{\QCoh}{QCoh}
\DeclareMathOperator{\Shv}{Shv}
\DeclareMathOperator{\Nilp}{Nilp}
\DeclareMathOperator{\Tr}{Tr}
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\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
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==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//macros.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//paper.tex ====
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
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\usepackage{enumitem,booktabs,array}
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\input{macros}
\hypersetup{pdftitle={Rationality of the Canonical Unramified Arthur Filtration},pdfauthor={OpenAI}}
\title{Rationality of the Canonical Unramified Arthur Filtration}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
Under the characteristic hypotheses of restricted geometric Langlands theory,
we prove that the canonical Arthur filtration on finitely supported
unramified automorphic functions is defined over $\Q$ for split connected
semisimple groups. The result includes the noncuspidal part and every closed
invariant nilpotent support, establishing the rationality conjecture of
Gaitsgory--Lafforgue--Raskin in this setting.
\end{abstract}
\begingroup
\footnotesize
\tableofcontents
\endgroup
\input{sections/01-introduction}
\input{sections/02-inputs}
\input{sections/03-components}
\input{sections/04-orbits}
\input{sections/05-traces}
\input{sections/06-measures}
\input{sections/07-descent}
\input{references}
\end{document}
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//paper.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//references.tex ====
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==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//references.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/01-introduction.tex ====
\section{Introduction}
The canonical Arthur filtration organizes unramified automorphic functions
by nilpotent orbits in the Langlands dual Lie algebra. Its definition uses
singular support in a category of $\ell$-adic sheaves, whereas the underlying
space of finitely supported functions has a natural rational form. The
rationality question asks whether these two structures are compatible.
Arthur's conjectures organize departures from temperedness by an
additional algebraic $\mathrm{SL}_2$ factor in an automorphic parameter
\cite[Sections~4 and~8]{Arthur1989}. In geometric Langlands, Arinkin--Gaitsgory
introduced nilpotent singular support to encode this Arthur direction
\cite[Sections~1.1.2--1.1.6]{ArinkinGaitsgorySingularSupport}.
The restricted local-system stack and nilpotent automorphic category of
Arinkin--Gaitsgory--Kazhdan--Raskin--Rozenblyum--Varshavsky provide the
corresponding setting over a finite field \cite{AGKRRV1}.
Gaitsgory--Lafforgue--Raskin formulate this question, jointly with Kazhdan,
as the assertion that their canonical filtration is defined over $\Q$
\cite[Section~2.7, Conjecture~2.7.3]{GLR}. The construction and its relation
to Arthur's and Ramanujan's conjectures are discussed in
\cite{GLR,Raskin}. Restricted geometric Langlands theory supplies the
spectral description of the support categories, and the trace-of-Frobenius
theorem identifies the ambient trace with automorphic functions
\cite{AGKRRV1,AGKRRV3,GR}. These results make rationality a precise question
about the images of supported categorical traces.
We prove the rationality assertion for split connected semisimple groups
under the explicit characteristic assumptions below. Thus the result
positively resolves Conjecture~2.7.3 of \cite{GLR} in this setting. It
applies to the entire space of finitely supported functions, including its
noncuspidal part.
\subsection{Hypotheses and the filtration}
Let $X$ be a smooth projective geometrically connected curve over
$\mathbb F_q$, and let $G/\mathbb F_q$ be split, connected and semisimple.
Put $k_0=\overline{\mathbb F}_q$, $H=G_{k_0}$ and
$\mathfrak h=\Lie(H)$.
\begin{assumption}\label{intro:characteristic}
The following four conditions hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The Lie algebra $\mathfrak h$ has an $H$-invariant nondegenerate
symmetric bilinear form whose restriction to the center of $\Lie(M)$ is
nondegenerate for every Levi subgroup $M$ of $H$.
\item For each Levi subgroup $M$ of $H$, including $H$ itself, and each
maximal torus $T_M\subset M$, restriction induces an isomorphism
\[
k_0[\mathfrak m]^M\xrightarrow{\ \sim\ }
k_0[\mathfrak t_M]^{W_M},\qquad
\mathfrak m=\Lie(M),\quad \mathfrak t_M=\Lie(T_M),\quad
W_M=N_M(T_M)/T_M.
\]
\item The scheme-theoretic centralizer in $H$ of every semisimple element
of $\mathfrak h$ is a Levi subgroup.
\item For every extension $K/k_0$ and every nilpotent
$n\in\mathfrak h\otimes_{k_0}K$, there is a parabolic subgroup
$P\subset H_K$, defined over $K$, such that
$n\in\Lie(R_u(P))$.
\end{enumerate}
Here a Levi subgroup is a Levi factor of a parabolic subgroup. Semisimple
means geometrically conjugate into the Lie algebra of a maximal torus;
nilpotent means that the geometric adjoint orbit closure contains zero.
\end{assumption}
Fix a prime $\ell\ne\operatorname{char}(\mathbb F_q)$ and set $E=\E$.
Write $S$ for the set of isomorphism classes of $G$-bundles on $X$ over
$\mathbb F_q$. For a characteristic-zero field $L$, let
\[
\cA_L=L^{(S)}
\]
be the space of finitely supported functions on $S$. Its distinguished
basis consists of the point functions. In particular
$\cA_E=E\otimes_\Q\cA_\Q$.
Fix the pinned split $\Q$-form of the Langlands dual group and write
$\Ghat/E$ for its scalar extension, with Lie algebra $\ghat$
and nilpotent cone $\Nhat$. All later coefficient identifications use
this same pinned form. On
$\overline X=X\times_{\mathbb F_q}k_0$ put
\[
\cC=\Shv_{\Nilp}(\Bun_G(\overline X),E),\qquad
\Phi=(\Frob_{\Bun})_*.
\]
The subscript denotes singular support in the global nilpotent cone, the
zero fiber of the Hitchin map. We use geometric Frobenius and its
pushforward on the automorphic category.
For a closed $\Ad(\Ghat)$-invariant subset $Y\subset\Nhat$, let
$\cC_Y\subset\cC$ be the derived-Satake support category. More explicitly,
the restricted geometric Langlands equivalence identifies
\begin{equation}\label{intro:restricted-equivalence}
\cC\simeq\IndCoh_{\Nhat}(Z),\qquad
Z={}^{\prime}\!\LocSys_{\Ghat}^{\restr}(\overline X),
\end{equation}
and $\cC_Y$ corresponds to $\IndCoh_Y(Z)$. The prime denotes the union of
connected components occurring in that equivalence; it is retained
throughout. A singular point is a pair $(\sigma,A)$ with $A$ a horizontal
section of the adjoint local system, after identifying $\ghat^*$ with
$\ghat$ by an invariant form. The support condition requires $A$ to take
values in $Y$. Thus $Y$ is a condition on singular directions, not on the
support of a function on $S$.
For a dualizable differential graded category $\cD$ and an endofunctor
$T$, write $\Tr(T,\cD)$ for its categorical trace, a complex of vector
spaces. It is Hochschild homology with coefficients in the graph bimodule
of $T$. The trace/functions isomorphism gives
$\Tr(\Phi,\cC)\simeq\cA_E$, with the latter in degree zero. Define
\begin{equation}\label{intro:filtration}
\cF_Y\cA_E=
\im\bigl[H^0\Tr(\Phi,\cC_Y)\longrightarrow\cA_E\bigr].
\end{equation}
This image definition does not assume concentration of the supported
trace. The required concentration and injectivity statements will be
proved in Section~\ref{tr:section}.
\begin{theorem}\label{intro:main}
Under Assumption~\ref{intro:characteristic}, for every prime
$\ell\ne\operatorname{char}(\mathbb F_q)$ and every closed invariant
$Y\subset\Nhat$, the natural map
\begin{equation}\label{intro:main-map}
E\otimes_\Q\bigl(\cA_\Q\cap\cF_Y\cA_E\bigr)
\xrightarrow{\ \sim\ }\cF_Y\cA_E
\end{equation}
is an isomorphism. The intersection is taken inside $\cA_E$.
\end{theorem}
No Hecke-finiteness or cuspidality condition is imposed. The theorem does
not require $X(\mathbb F_q)$ to be nonempty. These rational filtration
subspaces are also independent of $\ell$, with the closed invariant
supports matched through the pinned split dual group, as shown in
Section~\ref{desc:section}. On the unramified cuspidal subspace,
Corollary~\ref{desc:cuspidal-comparison} further identifies this filtration
with the sums of the rational cuspidal Arthur summands of
\cite[Theorem~1.1]{CuspidalArthur} whose orbit labels lie in $Y$.
\subsection{A bilinear detection criterion}
The proof gives a characterization after every abstract field
isomorphism $\iota:E\xrightarrow{\sim}\C$. Such isomorphisms are used
without prescribing their restriction to the algebraic numbers.
Transport all coefficient-linear categories and spaces by $\iota$.
Fix a closed point $v$ of $X$ of degree $d$, and put $r=q^d$. For a
nilpotent orbit $o\subset\ghat_\C$, choose an $\mathrm{SL}_2$-homomorphism
associated with a triple for $e\in o$, and write $\lambda_e$ for its
diagonal cocharacter, so $e$ has weight two. Define
\begin{equation}\label{intro:compactsets}
B_o=\bigl\{[\lambda_e(\sqrt r)s]:
s\in Z_{\Ghat_\C}(\mathrm{SL}_{2,e})_{\cpt}\bigr\}
\subset\Ghat_\C\sslash\Ghat_\C,
\qquad
B_Y=\bigcup_{o\subset Y_\C}B_o.
\end{equation}
Here $\sqrt r$ is positive, $[\ ]$ denotes semisimple conjugacy value,
and the compact form includes the components of the triple centralizer.
The sets $B_o$, as in \cite[Sections~3.1.1--3.1.3]{GLR}, do not depend
on these choices, are compact and pairwise disjoint. Their elementary properties are proved in
Lemma~\ref{desc:compactsets}.
For a finite-dimensional algebraic representation $D$ of $\Ghat_\C$,
let $T_D$ be the spherical Hecke operator at $v$, with the positive
square-root Satake normalization. For $f,g\in\cA_\C$, put
\begin{equation}\label{intro:pairing}
[f,g]=\sum_{b\in S}\frac{f(b)g(b)}{|\Aut(b)|}.
\end{equation}
This is a bilinear pairing. The automorphism groups over the finite field
are finite, and the sum is finite.
The key result, Theorem~\ref{meas:detector}, says that for each $f,g$
there is a unique finite complex Borel measure $\nu_{f,g}$ on
$B=\bigcup_oB_o$ such that
\begin{equation}\label{intro:moments}
[T_Df,g]=\int_B\chi_D\,d\nu_{f,g}
\quad\text{for every }D,
\end{equation}
and that
\begin{equation}\label{intro:detector}
f\in\iota_*(\cF_Y\cA_E)
\quad\Longleftrightarrow\quad
\supp(\nu_{f,g})\subseteq B_Y
\quad\text{for every }g\in\cA_\C.
\end{equation}
A finite complex measure means one of finite total variation. No positivity
of these measures is asserted or needed.
The right side of \eqref{intro:detector} uses the same complex function
space, point-counting Hecke operations, rational stack weights and compact
sets for every $\iota$. It therefore gives invariance of the filtration
under all coefficient automorphisms over $\Q$. Finite-coordinate linear
algebra then proves \eqref{intro:main-map}.
\subsection{The local arguments}
The supported trace concentration and injection statements in
\cite[Corollaries~2.1.10 and~2.1.13]{GLR} are conditional on
Conjecture~2.1.5 there, as stipulated in its Section~2.1.9.
Section~\ref{tr:section} proves the concentration and injection needed
here from the component and orbit calculations, and constructs actual
compact Weil representatives of the supported trace classes.
The central task is to detect every nonzero top support class by a
finitely supported function test despite possible cancellation from its
boundary.
Our contribution is the bilinear criterion
\eqref{intro:moments}--\eqref{intro:detector} on the full space of
finitely supported functions, together with the local calculations and
compact representatives that establish it. First, on a retained fixed
component of the local-system stack, a pure Weil lift supplies separated
Frobenius weights. Finite covariant tests recover the component from a
closed-orbit slice while preserving completion along its invariant base.
Nonresonance then gives a quadratic moment-map model compatible with
evaluation bundles and singular support.
A deformation-theoretic antecedent is the quadratic-germ theorem of
Goldman--Millson for compact K\"ahler manifolds and compact-image
representations \cite[Theorem~1]{GoldmanMillson}. Pridham obtains
Frobenius-compatible classical deformation hulls and cup-product
quadratic equations from purity for smooth proper varieties
\cite[Theorem~2.11 and Corollary~2.12]{Pridham}. Our finite-jet
nonresonance argument retains the derived equivariant model, its formal
invariant base, evaluation bundles and singular-support map.
Second, a graded Koszul description reduces each nilpotent-orbit
quotient to equivariant Clifford modules together with strictly
contracting Hom directions. Clifford algebras and graded matrix
factorizations also enter the spectral Whittaker construction of
Ben-Zvi, Raskin and Venkatesh described in \cite[Section~4.5]{Raskin},
which develops an idea of V.~Lafforgue; the even-nilpotent construction
appears in \cite[Section~18.5]{BZSV}. Here a twisted-bar contraction
proves trace concentration. Proper orbit-closure extensions and finite Koszul lifts
produce actual compact Weil classes, so the result describes the image
in \eqref{intro:filtration}, not only an abstract associated graded.
The passage from equivariant localization to a cofiber sequence of
traces follows the formalism of Hoyois--Scherotzke--Sibilla
\cite[Theorem~3.4 and Proposition~5.4]{HSS}.
Third, nonstandard automorphic duality turns these classes into the
bilinear tests \eqref{intro:pairing}. Localization with supports gives
decaying normal-direction contributions and hence finite complex
measures. On a top orbit the common density is invertible; finite
character approximations remove it and leave a nondegenerate coefficient
pairing. A strict drop in ambient adjoint rank separates every proper
boundary contribution. These two facts prevent cancellation from hiding
a nonzero top class.
The supported local-cohomology and orbit-distribution strategy is inspired
by Kazhdan--Okounkov's treatment of the unramified Eisenstein spectrum
\cite[Sections~2.6.1--2.6.4 and~3.3--3.4]{KazhdanOkounkov}.
The all-function bilinear detector, compact Weil lifts, and
noncancellation and rational-descent arguments are developed here;
that reference does not establish the rationality theorem above.
\begin{figure}[ht]
\centering
\begin{tikzcd}[row sep=2em]
\text{quadratic component model}\quad
\text{\footnotesize(Section~\ref{mod:section})}
\arrow[d,"\text{orbit traces}"] \\
\text{compact Weil trace classes}\quad
\text{\footnotesize(Sections~\ref{orb:section}--\ref{tr:section})}
\arrow[d,"\text{supported Hom pairings}"] \\
\text{bilinear measure criterion}\quad
\text{\footnotesize(Section~\ref{meas:section})}
\arrow[d,"\text{coefficient descent}"] \\
\text{rational form of }\cF_Y\cA_E\quad
\text{\footnotesize(Section~\ref{desc:section})}
\end{tikzcd}
\caption{Main line of the proof. The compact-set properties used by
the measure criterion are proved independently in
Section~\ref{desc:section}.}
\label{intro:diagram}
\end{figure}
Section~\ref{inp:section} records the established geometric inputs and
their precise scope. Sections~\ref{mod:section}--\ref{tr:section} prove the
local model and trace calculations. Section~\ref{meas:section} establishes
the measure criterion, and Section~\ref{desc:section} completes the
rational descent.
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/01-introduction.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/02-inputs.tex ====
\section{Geometric inputs and Frobenius conventions}
\label{inp:section}
We use the constructible $\ell$-adic geometric Langlands framework. The
results recalled here apply under Assumption~\ref{intro:characteristic}:
its first three conditions are those of \cite[\S14.4.1]{AGKRRV1}, and its
fourth is the additional condition of \cite[\S D.1.1]{AGKRRV1}.
All categories are $\E$-linear. A compact object means an object whose
Hom functor commutes with filtered colimits; compactness is stronger than
constructibility on an algebraic stack.
\subsection{Restricted Langlands and functions}
\begin{theorem}[Restricted geometric Langlands]\label{inp:langlands}
There is a Frobenius-invariant union $Z$ of connected components of
$\LocSys^{\mathrm{restr}}_{\Ghat}(\overline X)$ and an equivalence
\[
\mathbb L_G^{\mathrm{restr}}:\cC
\xrightarrow{\sim}\IndCoh_{\Nhat}(Z).
\]
It identifies $\cC_Y$ with $\IndCoh_Y(Z)$ for every closed
$\Ghat$-invariant $Y\subset\Nhat$ and intertwines Hecke functors with
tensoring by the corresponding evaluation bundles. If
$f:Z\to Z$ sends a local system to its pullback along geometric
Frobenius of $\overline X$, the functor corresponding to
$\Phi=(\operatorname{Frob}_{\Bun})_*$ is $f^!$.
The embedding $\cC\hookrightarrow\Shv(\Bun_G(\overline X),\E)$
preserves compact objects.
\end{theorem}
The equivalence and the last assertion are
\cite[Theorems~1.3.9(i) and~1.1.7]{GR}; Frobenius invariance of the
retained union is \cite[\S1.5.4]{GR}.
The Hecke action is compatible with the spectral linear structure
\cite[Lemma~1.3.7]{GR}, and the support compatibility is
\cite[Theorem~4.1.1 and Corollary~4.2.1]{Raskin}.
The stated Frobenius convention is exactly \cite[\S1.3.4]{GLR}.
Thus $Z$ is the prime union of the introduction throughout this paper.
In particular, this theorem does not require identifying that union with
the entire restricted stack.
Under Assumption~\ref{intro:characteristic}, the companion full-support
theorem~\cite[Theorem~1.4, regime~A]{RGLCompanion} identifies
$Z=\LocSys^{\mathrm{restr}}_{\Ghat}(\overline X)$ as spectral prestacks
over the fixed coefficient field $E=\E$.
At a geometric point $x$, a representation $D$ of $\Ghat$ supplies the
vector bundle $\sigma\mapsto D_{\sigma,x}$ on $Z$. At a closed point of
degree $d$, the geometric points form a Frobenius cycle of length $d$.
The corresponding Hecke construction uses all $d$ evaluation bundles,
with cyclic transport. This observation will keep the degree of a
closed point visible in our trace calculation.
\begin{remark}[Theta normalization]\label{inp:theta}
The theta normalization of the equivalence can be made over $\mathbf F_q$.
Indeed, $X$ admits a line bundle $\vartheta$ defined over $\mathbf F_q$
with $\vartheta^{\otimes2}\simeq\omega_X$.
For odd characteristic this follows from
\cite[\S5, Remark~2 after Proposition~5.2, p.~61]{Atiyah}: there is a
Frobenius-invariant square-root class, and the Brauer group of the
finite field vanishes, so this class descends. In characteristic two, choose a rational
function $a\in\mathbf F_q(X)$ with $da\ne0$. The orders of $da$ are even,
as a Laurent expansion over each perfect residue field shows. Hence
$\mathcal O_X(\tfrac12\operatorname{div}(da))$ is such a square root
\cite[\S4, p.~191]{Mumford}. No rational point of $X$ is needed.
\end{remark}
\begin{theorem}[Trace and ordinary local terms]\label{inp:trace}
There is a canonical isomorphism of complexes
\[
\operatorname{LT}:\Tr(\Phi,\cC)\xrightarrow{\sim}\cA_{\E},
\]
with the right-hand side in degree zero. For a compact object $M$ and
a morphism $b:M\to\Phi M$, it sends the categorical class $[M,b]$ to
the ordinary alternating trace function of the adjoint morphism
$(\operatorname{Frob}_{\Bun})^*M\to M$. The function has finite support.
This comparison is compatible with Hecke operations and with the
sheaf--function operations of pullback, star tensor product, and
compactly supported pushforward.
\end{theorem}
The trace isomorphism is \cite[Theorem~0.2.6, equivalently
Corollary~4.1.4]{AGKRRV3}. Its identification with local terms is
\cite[Theorem~0.6.8, equivalently Theorem~5.2.3, and
Remark~0.6.5]{AGKRRV3}; the compact-embedding hypothesis used there is
provided by Theorem~\ref{inp:langlands}. The compatibilities and the
stack convention for summation are recalled in \cite[\S1.1]{GLR}.
This is a statement about all of $\cA_{\E}$. It does not restrict to
cuspidal or Hecke-finite functions. The stronger assertion that actual
invertible Weil structures span the supported traces will be proved below.
\subsection{Duality and finite evaluations}
\begin{theorem}[Nonstandard duality]\label{inp:duality}
The pairing
\begin{equation}\label{inp:duality-pairing}
\cC\otimes\cC\longrightarrow\operatorname{Vect}_{\E},\qquad
(M,N)\longmapsto
R\Gamma_c\bigl(\Bun_G(\overline X),M\overset{*}{\otimes}N\bigr)
\end{equation}
defines a self-duality of $\cC$. It is invariant under simultaneous
Frobenius. If $M,N$ are compact, its value is a perfect complex, also
after any finite Hecke representation is inserted in either variable.
\end{theorem}
The duality is \cite[Theorem~3.2.2]{AGKRRV2}.
Here is the finiteness deduction, which is useful when applying it.
By Theorem~\ref{inp:langlands}, the two objects are compact in ambient
sheaves, hence constructible \cite[Appendix~A.1.6]{AGKRRV2}.
Star tensoring an ambient compact with a constructible object
preserves compactness \cite[Appendix~A.1.6]{AGKRRV2}.
For $p:\Bun_G\to\operatorname{pt}$, the functor $p_!$ is left adjoint
to the continuous functor $p^!$ \cite[Appendix~A.1.7]{AGKRRV2}, so it
preserves compactness. Its value is therefore compact in
$\operatorname{Vect}_{\E}$, hence perfect.
The Hecke functor for a finite-dimensional representation has a
continuous adjoint supplied by its dual representation, and likewise
preserves compactness. Frobenius invariance of
\eqref{inp:duality-pairing} follows by change of variables and the
compatibility of Frobenius with star tensor product.
For compact Weil objects the numerical trace of
\eqref{inp:duality-pairing} is consequently
\begin{equation}\label{inp:function-pairing}
[u,g]=\sum_{b\in S}\frac{u(b)g(b)}{|\Aut(b)|}.
\end{equation}
This is the Grothendieck trace formula with groupoid counting measure;
see \cite[\S\S1.1.9 and~1.2.3]{GLR}.
The pairing is bilinear over $\E$, without complex conjugation.
The compact objects of the categorical dual are the opposites of the
compact objects of $\cC$, and their evaluation is enriched Hom
\cite[\S\S0.5.3 and~5.1.2]{AGKRRV2}.
Thus the second input may be transported to a compact Hom-test object.
Its Weil arrow is reversed on passing to the opposite category. For
arrows $a:\Phi M\to M$ and $b:N\to\Phi N$, the induced endomorphism of
$\Hom(M,N)$ is
\begin{equation}\label{inp:hom-action}
h\longmapsto\Phi^{-1}(b\circ h\circ a).
\end{equation}
This convention fixes the direction of every Frobenius action below.
The later spanning theorem will justify testing against every function
in \eqref{inp:function-pairing}.
\subsection{The geometry of a spectral component}
\begin{theorem}[Components and deformation theory]\label{inp:components}
Every connected component $Z_\sigma$ of the restricted local-system
stack has a unique semisimple isomorphism class $\sigma$. Its
automorphism group $J=\Aut(\sigma)$ is reductive, possibly disconnected.
There is a formal affine coarse space $S_\sigma$ with reduced space a
point, and $Z_\sigma\to S_\sigma$ is relatively algebraic.
After choosing a frame at a geometric point, the component is formal
affine. For an affine infinitesimal stage $S_n\to S_\sigma$, its
base change is affine; almost finite type stages and bounded Postnikov
truncations may be used. These affine stages have coherent cohomology
in each bounded range. The tangent complex at a local system $\tau$ is
\begin{equation}\label{inp:tangent}
T_\tau\LocSys^{\mathrm{restr}}_{\Ghat}(\overline X)
\simeq R\Gamma(\overline X,\ghat_\tau)[1].
\end{equation}
In particular the stack is quasi-smooth.
\end{theorem}
The component and finiteness statements assemble
\cite[Theorems~1.4.5 and~1.6.3, Remarks~1.4.6--1.4.7,
Corollary~3.7.4, and Theorem~5.4.2]{AGKRRV1}.
The affine fiber assertion is Corollary~5.4.6 there; the assertion for
general affine infinitesimal stages is made explicitly in
\cite[\S7.9.10]{AGKRRV1}.
The deformation formula is \cite[Proposition~2.2.2 and
\S\S2.4,~25.4.3]{AGKRRV1}.
Formal quasi-smoothness is established in
\cite[\S\S21.2.1--21.2.3]{AGKRRV1}; the deformation complex has the
required amplitude because the curve has cohomological dimension two.
Semisimplicity of $\sigma$ makes its geometric monodromy
reductive, and its centralizer $J$ is reductive in characteristic zero.
For $d$ frames, write $K=\Ghat^d$. Adding the remaining frames is an
affine torsor, so the same finite-stage assertions hold. The unique
closed $K$-orbit is $K/J$: closed orbits correspond to semisimple local
systems, and the stabilizer of a framed representative is $J$.
Every nonempty invariant closed subset of a finite stage meets this
orbit. These statements supply the finite-stage setting for the
reconstruction in the next section; they do not yet identify the
component with a quadratic moment-map model.
\begin{theorem}[Pure Weil lift]\label{inp:pure-lift}
Fix an abstract field isomorphism $\iota:\E\xrightarrow{\sim}\C$.
If $Z_\sigma$ is Frobenius invariant, its semisimple point $\sigma$
admits a Weil structure that is $\iota$-pure of weight zero. In
particular, writing $\ghat_\sigma$ for its adjoint local system, every
Frobenius eigenvalue on
$H^i(\overline X,\ghat_\sigma)$ has $\iota$-absolute value $q^{i/2}$,
for $0\leq i\leq2$.
\end{theorem}
The pure-lift application to a Frobenius-fixed semisimple point is
stated in \cite[Appendix~A.5.4--A.5.5]{GLR}, using
\cite[Corollary~3.7.4 and Proposition~25.4.6]{AGKRRV1}.
The literal statement of that proposition treats irreducible Weil
parameters; the cited appendix records the semisimple consequence used
here. Purity of $H^1$ is the smooth proper curve application of
Weil~II \cite[Corollary~3.3.6 and \S3.3.10]{Deligne}, also stated explicitly in
\cite[Appendix~A.5.5]{GLR}. For $H^0$ it follows by viewing horizontal
sections in a pure fiber, and for $H^2$ by curve duality.
No semisimplicity of the cohomological Frobenius operators is asserted
or required.
Put $\mathfrak j=\Lie(J)$ and
$V=H^1(\overline X,\ghat_\sigma)$.
An invariant form on $\ghat$ and curve duality give
\[
H^0(\overline X,\ghat_\sigma)=\mathfrak j,
\qquad H^2(\overline X,\ghat_\sigma)\simeq\mathfrak j^*(-1),
\]
and a $J$-invariant symplectic form
$V\otimes V\to\E(-1)$. Its moment map is the cup--bracket
obstruction. After forgetting the Tate factor it is the usual
quadratic map $\mu:V\to\mathfrak j^*$.
We will use purity to linearize Frobenius on the full formal component,
including this obstruction and the evaluation bundles.
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/02-inputs.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/03-components.tex ====
\section{A quadratic model with its Frobenius action}
\label{mod:section}
We now replace each Frobenius-fixed retained component by a derived
quadratic moment-map model, retaining its evaluation bundles and the map
defining nilpotent singular support. The main issue is to extend
coordinates constructed at its closed semisimple orbit over the formal
invariant base.
Fix an abstract field isomorphism $\E\simeq\C$. All absolute values in
this section refer to this isomorphism. An invertible linear operator has
\emph{absolute weight $w$} if all its eigenvalues have modulus $q^{w/2}$.
This terminology does not assert that the operator is semisimple. It is
distinct from both cohomological degree and the weights of an
$\mathfrak{sl}_2$-triple.
The spectral stack remains the prime union $Z$ from
Section~\ref{inp:section}. Its components form a categorical coproduct.
In the twisted bar construction, an arrow string closes only in a
component preserved by $\Phi$. Consequently
\begin{equation}
\label{mod:component-trace}
\Tr(\Phi,\IndCoh_Y(Z))
\simeq\bigoplus_{Z_\sigma\text{ preserved by }\Phi}
\Tr(\Phi,\IndCoh_Y(Z_\sigma)).
\end{equation}
The sum is a direct sum, including when there are infinitely many
components. There are no arrows between distinct components; this also
proves their orthogonality for the Hom pairings used below.
Choose such a component and its unique closed semisimple parameter
$\sigma$. Choose a pure Weil lift as in Theorem~\ref{inp:pure-lift},
and put
\[
J=\Aut(\sigma),\qquad \mathfrak j=\Lie(J),\qquad
V=H^1(X_{\overline{\mathbb F}_q},\ghat_\sigma).
\]
The group $J$ is reductive; it need not be connected. Curve duality and
an invariant form identify $H^2(\ghat_\sigma)$ with
$\mathfrak j^*(-1)$ and give a $J$-invariant symplectic form $\omega$ on
$V$, after forgetting its Tate factor. The moment map is
\begin{equation}
\label{mod:moment}
\mu:V\longrightarrow\mathfrak j^*,\qquad
\langle\mu(x),z\rangle=\tfrac12\omega(zx,x).
\end{equation}
Its zero fiber will always mean its \emph{derived} zero fiber. In
particular no regular-sequence assumption on the components of $\mu$
is being made.
\subsection{What is recovered from completion at the closed orbit}
We first explain why constructing a model near $\sigma$ suffices here.
The invariant base must be retained in this argument. Choose a closed
point $v$ of degree $d$ and frames at its $d$ geometric points. Write
$K=\Ghat^d$. The framed component supplied by
Theorem~\ref{inp:components} is an ind-affine derived scheme over a
formal invariant base with reduced scheme a point. Its unique closed
$K$-orbit is $K/J$. It has affine almost-finite-type stages over
nilpotent thickenings of that invariant point. At each such stage the
classical coordinate ring is noetherian, its invariant ring is
Artinian, and its cohomology sheaves in any bounded Postnikov range are
coherent. Adding the other frames to a one-frame presentation is an
affine operation. These are precisely the finite-stage properties we
use.
\begin{lemma}[Finite covariants and completion]
\label{mod:finite-covariants}
Let $K$ be a reductive complex algebraic group, possibly disconnected,
and let $A$ be a finitely generated $K$-algebra. Suppose $A^K$ is
Artinian and $\Spec A$ has a unique closed $K$-orbit $O$, with ideal
$I$. For an equivariant finite $A$-module $M$ and a finite-dimensional
$K$-representation $D$, the inverse system
\[
\Hom_K(D,M/I^nM)
\]
is eventually constant with value $\Hom_K(D,M)$. In particular the
rational, locally $K$-finite part of the $I$-adic completion recovers
$M$, and recovers $A$ together with its multiplication when $M=A$.
\end{lemma}
\begin{proof}
Choose a finite-dimensional $K$-stable generating space $U\subset M$.
The surjection $A\otimes U\to M$ induces a surjection on
$\Hom_K(D,-)$, by reductivity. Finite generation of the covariants
of $A$ over $A^K$ \cite[Lemma~2.3]{Brion} therefore makes
$M_D=\Hom_K(D,M)$ finite over $A^K$, hence finite-dimensional.
This applies to disconnected $K$: its identity component is reductive
and its component group is finite. Let $N=\bigcap_{n\geq1}I^nM$. It is a finite
equivariant submodule because $A$ is noetherian. At every point of $O$
its localization vanishes by Krull intersection \cite[Tag~00IP]{Stacks}. If $N$ were nonzero,
its nonempty invariant closed support would contain a closed orbit,
hence would meet $O$. Therefore $N=0$.
The descending subspaces $\Hom_K(D,I^nM)$ of $M_D$ have zero
intersection and thus are eventually zero. Reductivity in
characteristic zero makes $\Hom_K(D,-)$ exact, so
\[
\Hom_K(D,M/I^nM)
=M_D/\Hom_K(D,I^nM).
\]
This proves the assertion. Every rational representation is the union
of its finite-dimensional subrepresentations; applying the assertion
to all $D$ therefore recovers $M$. Applied also to finite tensor
products of representation spaces, it recovers multiplication and
every equivariant finite-presentation map.
\end{proof}
For example, let $\Gm$ act with weights $1,-1$ on the coordinate
functions $x,y$, and put
\[
A_N=\C[x,y]/((xy)^N),\qquad
A_N^{\Gm}=\C[t]/(t^N),\quad t=xy,\quad N\ge1.
\]
The origin is the unique closed orbit. Each weight space of $A_N$ is
finite-dimensional, so the locally $\Gm$-finite part of its
$(x,y)$-adic completion recovers $A_N$. The ordinary inverse limit
also contains arbitrary formal series such as $\sum_{m\ge0}x^m$,
which are not locally finite. Thus rational recovery retains the
nonzero points on both axes, not only the closed orbit. As $N$
varies, these stages give the formal neighborhood of $xy=0$,
including the formal invariant parameter $t$. The next lemma carries
out this recovery for derived algebras and their maps.
\begin{lemma}[Restoring the invariant base]
\label{mod:reconstruction}
For the framed component above, an equivariant isomorphism of its
completion along $K/J$ with another such completed model extends
uniquely over the formal invariant bases to an isomorphism of their
invariant-fiber completions. The same assertion holds for equivariant
maps and their full mapping spaces, and for finite equivariant
presentation data. Here the extension is unique up to a contractible
space of choices.
\end{lemma}
\begin{proof}
We give the derived argument, since recovering coordinate rings alone
would not recover homotopies. The proof has three stages: finite
covariants recover each affine stage and its mapping spaces; trivial
covariants recover the invariant base; derived Koszul thickenings then
give compatible extensions over that base.
Write $\mathcal R_K$ for the derived
category of rational $K$-representations. All limits in this paragraph
are taken in $\mathcal R_K$, not in the category of vector spaces with
an arbitrary, possibly nonrational, $K$-action.
\emph{Recovery at a fixed invariant-base stage.}
Let $A$ be its connective coordinate algebra, viewed as a commutative
algebra in $\mathcal R_K$. The precise hypotheses are that $H^0(A)$
is finitely generated, $H^0(A)^K$ is Artinian,
$\Spec H^0(A)$ has a unique closed orbit $O$, and every $H^i(A)$
is finite over $H^0(A)$. Choose a
finite-dimensional $K$-stable generating space for the ideal of $O$
in $H^0(A)$, lift its generators to degree-zero cycles, and extend
them to a finite-dimensional stable generating space for $H^0(A)$.
Reductivity allows these lifts to be equivariant. This gives a map
from a polynomial $K$-algebra $P$ to $A$, surjective on $H^0$.
Let $I_P$ be the ideal generated by the chosen orbit equations and put
\[
A_n=A\otimes_P^{\mathbf L}P/I_P^n.
\]
These derived orbit jets compute completion along $O$; the
module-level comparison is \cite[Tag~0BKH]{Stacks}.
The resulting formal neighborhood is independent of the generating
presentation \cite[Theorem~A.1.3(c) and \S A.1.5]{AGKRRV1}.
Here is the cohomological calculation at these jets. Every $H^i(A)$
is a finite $P$-module. Noetherian Artin--Rees makes the positive-Tor
systems
$\{\operatorname{Tor}^P_j(H^i(A),P/I_P^n)\}_n$, $j>0$,
pro-zero. Since $P$ has finite global dimension, the hyper-Tor
calculation has finite width in every cohomological degree, even if
$A$ is unbounded below. Consequently
\[
\{H^i(A_n)\}_n\simeq
\{H^i(A)/I_P^nH^i(A)\}_n
\]
as pro-modules. This is the coherent derived-completion calculation
of \cite[Tags~00MA, 0A06, and~0BKH]{Stacks}, applied here over the
ordinary noetherian polynomial ring $P$.
For every finite-dimensional $K$-representation $D$,
Lemma~\ref{mod:finite-covariants} and exactness of covariants therefore
give a natural pro-isomorphism
\begin{equation}
\label{mod:derived-covariants}
\{H^i\RHom_K(D,A_n)\}_n
\simeq H^i\RHom_K(D,A),
\end{equation}
with a constant system on the right. In particular these systems
have zero $\lim^1$. The functors $\RHom_K(D,-)$ commute with limits
and jointly detect equivalences in $\mathcal R_K$. Applying the
Milnor exact sequence to~\eqref{mod:derived-covariants} proves
\begin{equation}
\label{mod:rational-derived-completion}
A\xrightarrow{\sim}\operatorname*{holim}_n A_n
\quad\text{in }\operatorname{CAlg}(\mathcal R_K).
\end{equation}
Indeed the forgetful functor from commutative algebras creates limits
and detects equivalences. Thus this is a natural equivalence of
algebras, not merely an identification of their cohomology. It gives,
for every rational equivariant coordinate algebra $B$, an equivalence
of full mapping spaces
\begin{equation}
\label{mod:mapping-spaces}
\operatorname{Map}_{\operatorname{CAlg}(\mathcal R_K)}(B,A)
\simeq
\operatorname*{holim}_n
\operatorname{Map}_{\operatorname{CAlg}(\mathcal R_K)}(B,A_n),
\end{equation}
compatible with composition. No index $n$ uniform in $i$ or $D$ has
been asserted or used. The rational limit is essential: as in the preceding example, the
ordinary vector-space limit would retain arbitrary formal series.
The same calculation applies to an $A$-module with coherent
cohomology. Finite equivariant module presentations and their maps
are recovered by testing their generating representation spaces;
their relations and coherent homotopies are recovered by the
corresponding derived mapping spaces.
\emph{Recovery of the invariant base.}
The formal coarse base is constructed from the system of trivial
covariants of the framed affine stages
\cite[\S\S5.3.5--5.3.6]{AGKRRV1}. We do not identify an arbitrarily
chosen base stage with the spectrum of its pullback's invariant
algebra. The trivial case of~\eqref{mod:derived-covariants} instead
recovers this entire system. On classical invariant functions the
topologies are cofinal: if $\mathfrak m$ is the invariant ideal of
definition, then $\mathfrak m\subset I$, and, after passage to a
fixed nilpotent invariant-base stage,
Lemma~\ref{mod:finite-covariants} makes the trivial covariants of high
powers of $I$ vanish. Applying this in every bounded range gives the
corresponding comparison for the derived invariant systems.
An equivariant map of orbit completions therefore gives the formal
base map as well.
\emph{Factorization and extension.}
The noetherian formal-affine presentation of that base uses finitely
many invariant generators $t_1,\ldots,t_r$ of an ideal of definition.
The noetherian and coherent presentation is supplied by
\cite[Theorem~A.1.3(b),(c) and Remark~A.1.4]{AGKRRV1}.
Its infinitesimal stages may be taken to be the derived Koszul
thickenings
\begin{equation}
\label{mod:koszul-base-stages}
R_N=R/\!/(t_1^N,\ldots,t_r^N)
:=R\otimes_{\C[u_1,\ldots,u_r]}^{\mathbf L}\C,
\qquad u_j\longmapsto t_j^N.
\end{equation}
These are precisely the formal-affine stages used in
\cite[\S7.1 and \S\S7.9.8--7.9.10]{AGKRRV1}. The relevant
framed base changes are affine and almost of finite type.
Restrict a map of orbit completions to the completed orbit of one
source stage $A$. Compose it with the target's map to its formal base
and then with the structural map to $\Spec R$.
Equation~\eqref{mod:mapping-spaces}, applied to the base algebra $R$
with trivial $K$-action, recovers a map $R\to A$ and its homotopies.
The images of the $t_j$ vanish at the reduced invariant point, so
they are nilpotent in the Artinian algebra $H^0(A)^K$. Choose a common power $N$ and invariant nullhomotopies of
these $t_j^N$. Such nullhomotopies exist because vanishing in $H^0$
means being a boundary; they may be chosen invariant by reductivity.
The universal property of~\eqref{mod:koszul-base-stages} now factors
this base map through a derived Koszul stage. Pulling back the target
framed component along that stage factors the completed-space map
through an affine target stage. Nilpotence
alone would not justify factorization through an ordinary quotient.
If $B_N$ is the coordinate algebra of the resulting affine target
stage, the completed-stage map is a compatible point of the
right-hand side of~\eqref{mod:mapping-spaces}, with $B=B_N$.
It therefore extends to a map on the source stage $A$, with all its
homotopies and composition data.
For clarity, the choices of powers and nullhomotopies do not add
extra maps. The Koszul stages are cofinal tests for a formal
neighborhood. One can check this directly on mapping spaces: above
a fixed map with nilpotent $t_j$, the choices of nullhomotopy are
torsors for the corresponding additive loop spaces, and increasing
$N$ to $M$ acts on their differences by multiplication by
$t_j^{M-N}$. Nilpotence kills every such homotopy group after passing
far enough in this filtered system. Its nonempty space of choices
is therefore contractible. This verifies cofinality for maps and
homotopies, without interchanging a filtered colimit with a
Postnikov limit. Equation~\eqref{mod:mapping-spaces} makes the
extended maps on different source stages compatible. Applying it
also to inverse maps and to their compositions completes the proof.
\end{proof}
Thus completion along the closed orbit is used to \emph{construct}
coordinates. It is not substituted for completion along the invariant
null fiber. For example, nonclosed points of that null fiber are
retained by Lemma~\ref{mod:reconstruction}.
\subsection{Simultaneous linearization}
Here is the elementary nonresonance calculation that removes the
higher relations in the local deformation problem.
The classical Frobenius-compatible quadratic hull in the smooth proper,
pure weight-zero setting is established in
\cite[Theorem~2.11 and Corollary~2.12]{Pridham}; its proof also uses
Jordan decomposition on finite jets. We retain the derived presentation
and equivariant data, then apply Lemma~\ref{mod:reconstruction} to
restore the formal invariant base.
\begin{lemma}[Nonresonance with Jordan blocks]
\label{mod:nonresonance}
Let a minimal complete free graded-commutative algebra have generators
$V^*$ in cohomological degree $0$ and $R^*$ in degree $-1$. Suppose a
continuous graded automorphism has linear parts of absolute weights
$-1$ and $-2$ on these two generator spaces. It is conjugate, by a
formal graded coordinate change tangent to the identity, to its
linear part. If it commutes with a minimal differential, that
differential, in the resulting coordinates, sends $R^*$ to quadratic
polynomials in $V^*$. These assertions hold equivariantly for a
reductive group normalized by the automorphism.
\end{lemma}
\begin{proof}
Filter coordinate changes by total generator order. At order $n\geq2$
the error in a degree-zero generator belongs to
$\Hom(V^*,\Sym^nV^*)$. All eigenvalues of the conjugacy operator
``linear target action times inverse linear source action'' on this
space have modulus $q^{-(n-1)/2}$, hence differ from $1$. An error in
a degree-minus-one generator contains exactly one degree-minus-one
factor and $n-1$ degree-zero factors. The corresponding operator on
\[
\Hom(R^*,R^*\otimes\Sym^{n-1}V^*)
\]
has the same modulus $q^{-(n-1)/2}$. In either case its difference
from the identity is invertible. This assertion follows from its
eigenvalues and holds on generalized eigenspaces; it requires no
diagonalization. Solve the conjugacy equation at this order and
continue. The corrections converge in the generator-adic topology.
In the equivariant case the errors and corrections lie in the
equivariant subspaces of these finite-dimensional jet spaces. The
conjugacy operator preserves these subspaces and its inverse does
also, so the same construction is equivariant. Transport the
differential along the coordinate changes. The differential of a
degree-zero generator is zero by degree. That of a degree-minus-one
generator has no constant or linear term by minimality. Since the
differential commutes with the now linear automorphism, an order-$n$
term can occur only if its weight $-n$ agrees with $-2$. Thus only
$n=2$ remains.
\end{proof}
\begin{theorem}[The framed quadratic model]
\label{mod:model}
\label{mod:quadratic-model}
The retained component $Z_\sigma$, with its Frobenius action,
evaluation bundles at the chosen cyclic tuple, and singular-support
map, admits the model
\begin{equation}
\label{mod:quadratic-equation}
Z_\sigma\simeq
\bigl[(\mu^{-1}(0))^{\wedge}_{\mathrm{unst}}/J\bigr].
\end{equation}
Here $\mathrm{unst}$ is the inverse image of the origin under
$V\to V/\!/J$, and completion is along this invariant null fiber.
In this model:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the singularity scheme has points $(x,z)$ satisfying
$\mu(x)=0$ and $zx=0$, with $z\in\mathfrak j$;
\item the support condition defined by $Y$ is the condition that the
image of $z$ under $\mathfrak j\hookrightarrow\ghat$ belongs
to $Y$;
\item every evaluation representation of $K=\Ghat^d$ is the bundle
associated to its restriction along $J\hookrightarrow K$;
Frobenius has constant cyclic transport on these bundles;
\item after removing the scalars $q^{1/2}$ on deformation points and
$q^{-1}$ on singular points, Frobenius is a linear
automorphism of the moment-map data, normalizing $J$ and the
framed representation labels. Its eigenvalues on these
normalized linear data have modulus one.
\end{enumerate}
The meaning of Frobenius on Hom complexes in \textup{(iv)} is fixed
explicitly in Lemma~\ref{mod:hom-action} below.
\end{theorem}
\begin{proof}
We first work in the completion of the framed component along $K/J$.
The smooth affine orbit admits an equivariant formal retraction:
lift the identity retraction through successive nilpotent
neighborhoods, using smoothness, and average the lifting torsors
using reductivity. The fiber is a formal slice with stabilizer $J$;
the completed framed space is its induction from $J$ to $K$.
Equivalently, this follows by splitting the orbit tangent directions
and comparing tangent complexes at every lifting step.
The transverse tangent and obstruction spaces are respectively
$V=H^1(\ghat_\sigma)$ and $H^2(\ghat_\sigma)$. Quasi-smoothness
therefore gives a minimal presentation by complete free coordinates
dual to these spaces in degrees $0$ and $-1$. One obtains this
presentation from a free resolution by removing the contractible
linear pairs; the splittings can be chosen $J$-equivariantly.
We recall the second-order obstruction calculation to fix its coefficient
and sign, working over $E$ before the chosen coefficient transport.
Use the simplicial descent model for local systems underlying
\eqref{inp:tangent}, computed by hypercover cochains with the
$\ell$-adic limit retained. Its adjoint cochains compute
$R\Gamma(\overline X,\ghat_\sigma)$, and its cup bracket combines the
cup product with the Lie bracket of the adjoint coefficients. For a
formal parameter $t$, write a perturbation of the descent data modulo
$t^3$ as $g_{ij}\exp(t x_{ij}+t^2 y_{ij})$, with the cochains twisted by
the original transition data $g_{ij}$. Expanding the cocycle equation
by the Baker--Campbell--Hausdorff formula gives
\[
dx=0,\qquad dy=-\tfrac12[x,x].
\]
Thus the quadratic obstruction is the class of $\tfrac12[x,x]$ in
$H^2(\overline X,\ghat_\sigma)$. This calculation uses a cochain model
computing the deformation problem, rather than an assumed finite
trivializing cover. Invariance of the coefficient pairing and graded
commutativity give, with the duality and sign convention of
\eqref{mod:moment},
\[
\bigl\langle\tfrac12[x,x],z\bigr\rangle
=\tfrac12\omega(zx,x)=\langle\mu(x),z\rangle.
\]
This identifies the quadratic obstruction with $\mu(x)$.
We next arrange that both the retraction and these coordinates respect
the chosen Frobenius transporter. Purity gives absolute point
weights $0,1,2$ on the orbit tangent, transverse tangent, and
obstruction spaces. A positive-transverse-order error in the
retraction has positive source weight and orbit target weight zero.
Its Frobenius conjugacy operator therefore has no eigenvalue $1$.
At every finite jet order it can be removed uniquely by solving the
same linear equation used in Lemma~\ref{mod:nonresonance}. The errors
are equivariant sections of the tangent bundle of the smooth orbit;
they are determined by their $J$-equivariant values at its base point.
The stabilizer transport and the cyclic permutation of frames have
weight zero. This constructs a Frobenius-compatible retraction and
an invariant slice.
Represent its single cyclic automorphism on the minimal free
presentation. Such a representative can be chosen equivariantly by
free lifting and reductivity. Its invertible linear part makes it an
automorphism of the complete graded algebra. There is no additional
relation to impose on the generator of an infinite cyclic action.
On formal coordinate functions use inverse transport; its generator
weights are $-1,-2$. Lemma~\ref{mod:nonresonance} makes this
automorphism linear and forces the transported differential to be
quadratic. A coordinate change tangent to the identity preserves
that quadratic part, so the differential is precisely $\mu$.
This proves the moment-map presentation on the orbit completion.
Lemma~\ref{mod:reconstruction} extends it over the invariant base,
giving~\eqref{mod:quadratic-equation}.
For its singularities one computes directly that
\[
\langle d\mu_x(v),z\rangle=\omega(zx,v).
\]
Nondegeneracy of $\omega$ identifies
$\ker(d\mu_x^*)$ with $\{z:zx=0\}$, proving \textup{(i)}.
The singular evaluation map is fiberwise linear in $z$. At $x=0$
curve duality identifies it with the centralizer inclusion
$\mathfrak j\hookrightarrow\ghat$, with its duality Tate factor.
The target singular points have absolute weight $-2$. A correction
of positive degree $n$ in $x$, linear in $z$, would have weight
$n-2$, and cannot intertwine this Frobenius action. Comparison on
each finite jet, again valid for Jordan blocks, excludes every such
term. This proves \textup{(ii)}. Changing the invariant form only
rescales the simple factors of $\ghat$, which preserves every
nilpotent orbit.
Finally, a $K$-frame is exactly the datum trivializing all the
evaluation bundles in question. Write $\mathscr S$ for the formal
slice. The equivariant retraction identifies the framed completion
with $K\times^J\mathscr S$, with its map to $K/J$ given by
$[k,s]\mapsto kJ$. For a representation $D$ of $K$, quotienting the
trivial framed bundle by $K$ therefore gives the bundle on
$[\mathscr S/J]$ associated to $D|_J$. The retraction is
Frobenius-compatible, so the transport on its $K/J$ coordinate is
the closed-orbit transport, independent of $s$. The same is therefore
true of the cyclic transport on every evaluation bundle.
Purity on the chosen Weil
parameter gives modulus-one eigenvalues on every normalized frame
representation. A cyclic action can be tested on its $d$th power.
Together with purity on $H^1$ and $H^2$, this proves
\textup{(iii)} and \textup{(iv)}.
\end{proof}
\subsection{Which action contracts Hom complexes}
The next convention is essential: the category still has the
endomorphism $\Phi$ specified in the problem.
\begin{lemma}[Inverse transport on arrows]
\label{mod:frobenius}
\label{mod:hom-action}
Choose structures $\alpha_M:\Phi M\xrightarrow{\sim}M$ on compact
objects. On an arrow $h:M\to N$ put
\begin{align}
T(h)&=\alpha_N\,\Phi(h)\,\alpha_M^{-1},\label{mod:T}\\
S(h)&=\Phi^{-1}(\alpha_N^{-1}h\alpha_M)=T^{-1}(h).
\label{mod:S}
\end{align}
The action in the bilinear Hom pairing is $S$. On the quadratic
model it gives absolute weight $-1$ to deformation functions and
absolute weight $+2$ to degree-two obstruction operators. Thus its
associated point weights on $(x,z)$ are $(1,-2)$, and
$W(x,z)=\langle z,\mu(x)\rangle$ has weight zero.
The graph identification in~\eqref{tr:graph-identification} turns
the original $\Phi$-graph bimodule into the graph of the transported
arrow action. In those coordinates the induced simultaneous bar
actions $T$ and $S$ are inverse. If the paracyclic homotopy is
$bH+Hb=1-T$, then
\begin{equation}
\label{mod:inverse-homotopy}
b(-T^{-1}H)+(-T^{-1}H)b=1-S.
\end{equation}
In particular using contracting $S$-weights does not replace
$\Tr(\Phi,-)$ by $\Tr(\Phi^{-1},-)$.
\end{lemma}
\begin{proof}
The Frobenius compatibility in Section~\ref{inp:section} identifies
$\Phi$ with $f^!$, where $f$ is pullback of parameters by geometric
Frobenius of the curve. Since $f$ is an automorphism, this is its
usual pullback functor. A covariant trace structure is
$b:N\to\Phi N$, whereas a structure on the opposite input is
$a:\Phi M\to M$. Their pairing takes an arrow to
\[
h\longmapsto\Phi^{-1}(b\,h\,a).
\]
With $b=\alpha_N^{-1}$ and $a=\alpha_M$ this is~\eqref{mod:S}.
It is the inverse of~\eqref{mod:T}, directly by composition.
For the signs, consider a coordinate and a relation generator with
$f^\#x=a_1x$ and $f^\#\epsilon=a_2\epsilon$, where
$|x|=0$, $|\epsilon|=-1$. Inverse transport on the endomorphisms of
the free module sends multiplication by $x$ to multiplication by
$a_1^{-1}x$. In a Koszul resolution the degree-two operator dual to
$\epsilon$ transforms by $a_2$ under the same transport. Thus
purity, $|a_1|=q^{1/2}$ and $|a_2|=q$, gives the stated signs.
The calculation is linear-algebraic and applies to generalized
eigenspaces. The polynomial operator dual to $\epsilon$ is a linear
function on the singular variable $z$; point weights are the
contragredient weights, giving $-2$ on $z$.
There is a distinction between ordinary arrows and raw graph
coefficients. In the left-graph convention for classes $M\to\Phi M$,
a raw coefficient is $m\in\Hom(M,\Phi N)$. It is identified with
an ordinary arrow by $m\mapsto\alpha_Nm$. The simultaneous operator on this
raw coefficient is
\[
m\longmapsto\Phi(\alpha_N)\Phi(m)\alpha_M^{-1},
\]
not the ordinary-arrow formula~\eqref{mod:T}. Under the stated graph
identification it becomes the transported ordinary-arrow action.
The full multiobject identification, including permuted labels and
the last twisted bar face, is proved in
Lemma~\ref{tr:bar-contraction}; see
\eqref{tr:graph-identification}--\eqref{tr:graph-transport}.
Thus the paracyclic homotopy is transported through an actual
isomorphism of graph bimodules. In these coordinates $T$ commutes
with $b$, and multiplying $bH+Hb=1-T$ by $-T^{-1}$ proves
\eqref{mod:inverse-homotopy}. For permuted labels a common positive
power is used only to read their weights. The graph throughout is
canonically identified with the original $\Phi$-graph; neither a
power of $\Phi$ nor $\Phi^{-1}$ is substituted for it.
\end{proof}
\subsection{Compact normalization, including the frame labels}
For the measure calculation, modulus-one eigenvalues before twisting
are not enough: they must remain of modulus one after every compact
stabilizer twist. We establish a simultaneous normalization of the
deformation and frame data. Choose a faithful representation $R$ of
$\Ghat$ and include in the linear data the direct sum of its fibers at
all $d$ framed points. The $d$th power of the cyclic transporter acts
on these summands by the pure weight-zero closed-point Weil
transports. Hence the cyclic transporter itself has modulus-one
eigenvalues. Together with $V$ and $\mathfrak j$, these data give a
faithful finite-dimensional realization of the normalized algebraic
data group, a subgroup of a product of linear groups and
$K\rtimes\langle\text{cyclic permutation}\rangle$.
\begin{lemma}[Simultaneous compact normalization]
\label{mod:compact-normalization}
Let $J.e$ be a nilpotent orbit preserved by the normalized transporter
in Theorem~\ref{mod:quadratic-model}. After changing that transporter
by an element of $J$, choose a triple $(e,h,f)$ in this orbit and
write the normalized transporter as
\[
u=t\,v_0.
\]
There are simultaneous compact forms of the group and frame data
such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $t$ belongs to a compact group, preserves the compact form
$J_{\mathrm{cpt}}$, and centralizes the triple;
\item $v_0$ is unipotent, centralizes $J$, and commutes with $t$;
\item for $L_e=Z_J(\mathrm{SL}_{2,e})$,
$L_{e,\mathrm{cpt}}=L_e\cap J_{\mathrm{cpt}}$ is a maximal
compact including its components, and every
$k t v_0$, $k\in L_{e,\mathrm{cpt}}$, has modulus-one
eigenvalues on all the normalized linear data.
\end{enumerate}
These assertions also hold on their tensor, symmetric, exterior,
subquotient, and triple-weight constructions. After the
orbit-fixing translation by $\lambda_e(\sqrt q)$, the semisimple
value of the frame transport around the $d$-cycle belongs to
\begin{equation}
\label{mod:cyclic-compact-value}
\bigl\{[\lambda_e(q^{d/2})s]:
s\in Z_{\Ghat}(\mathrm{SL}_{2,e})_{\mathrm{cpt}}\bigr\}
=B_{\Ghat.e}.
\end{equation}
This holds for every $k\in L_{e,\mathrm{cpt}}$.
\end{lemma}
\begin{proof}
Write $u_1=s_1v_1$ for the Jordan decomposition of the original
normalized transporter in its algebraic normalizer. Both factors
preserve the data and normalize $J$. The semisimple part has compact
closure of its powers in the faithful realization just chosen.
We first remove the inner action of the unipotent part; this step
requires some care when $J$ is disconnected. For a complex reductive
group,
\begin{equation}
\label{mod:connected-automorphisms}
\Aut(J)^\circ=\im\bigl(J^\circ\longrightarrow\Aut(J)\bigr).
\end{equation}
Indeed an infinitesimal automorphism is inner on the semisimple part
of $J^\circ$ and trivial on its central torus, whose character
lattice has discrete automorphism group. After subtracting this
inner action, its values on the other components give a cocycle of
the finite group $\pi_0(J)$ with values in
$\Lie Z(J^\circ)$. Averaging makes that cocycle a coboundary,
which is an inner infinitesimal action by the central torus. This
proves surjectivity of the differential of the inner-automorphism
map and hence~\eqref{mod:connected-automorphisms}.
The kernel of $J^\circ\to\Aut(J)$ is $Z(J)\cap J^\circ$, a diagonalizable central
group. Therefore the automorphism induced by $v_1$ has a unique
unipotent lift $a\in J^\circ$: take the unipotent part of any
lift; uniqueness follows because two such lifts differ by a central
semisimple element. Since $s_1$ commutes with $v_1$, uniqueness
gives $s_1as_1^{-1}=a$. The element $v_0=a^{-1}v_1$ is unipotent,
centralizes $J$, and commutes with $s_1$: here $v_1av_1^{-1}=a$
because the action of $v_1$ on $J$ is conjugation by $a$, so the
two unipotent factors defining $v_0$ commute. Changing the
transporter by $a^{-1}\in J$ thus leaves $s_1v_0$. Since $v_1$
acts on $\mathfrak j$ by an inner automorphism, it preserves every
$J$-orbit; the assumed stable orbit is therefore also preserved by
$s_1$.
Let $A_1$ be the Zariski closure of the group generated by $s_1$.
It is diagonalizable, normalizes $J$, and $M=JA_1$ is reductive:
its unipotent radical maps trivially to $M/J$ and would be a
connected normal unipotent subgroup of $J^\circ$. The
maximal-compact theorem for complex reductive groups, including
disconnected groups \cite[Definition~3.3 and
Theorems~3.6--3.7]{AdamsTaibi}, gives a
maximal compact $M_{\mathrm{cpt}}$ containing the compact closure
of $\langle s_1\rangle$. Its intersection with $J$ is a maximal
compact $J_{\mathrm{cpt}}$: conjugate a maximal compact of $J$
into $M_{\mathrm{cpt}}$ and use normality of $J$ and maximality
inside the compact intersection. The element $s_1$ preserves this
intersection. Averaging one
Hermitian form makes this whole compact group unitary on the
faithful simultaneous realization.
Choose the triple in $J.e$ so that its $\mathrm{SU}(2)$ lies in
$J_{\mathrm{cpt}}$. The two homomorphisms $\mathrm{SL}_2\to J$
given by this triple and its $s_1$-translate are $J$-conjugate,
because their nilpotents lie in the same orbit. Their restrictions
to $\mathrm{SU}(2)$ land in $J_{\mathrm{cpt}}$, so they are
already conjugate by an element $k_0\in J_{\mathrm{cpt}}$.
For completeness, if a complex conjugator is $b=kp$ in the polar
decomposition relative to $J_{\mathrm{cpt}}$
\cite[Lemma~3.5]{AdamsTaibi}, unitarity of the two
homomorphisms implies that $b^*b$ commutes with the first one. Its
positive square root $p$ therefore commutes with it, and $k$ is
the required compact conjugator. This works in every component.
Set $t=k_0s_1$, choosing the conjugator's direction so that $t$
centralizes the triple. Since $v_0$ centralizes $J$, it commutes
with $k_0$, with $t$, and with every compact twist from $L_e$.
The product $kt$ belongs to $M_{\mathrm{cpt}}$ and is unitary for
the same Hermitian form. Multiplication by the commuting unipotent
$v_0$ changes neither its eigenvalues nor its semisimple
character. The centralizer of $\mathrm{SU}(2)$ is stable under
the compact conjugation, so its compact intersection is a compact
real form of the full complex triple centralizer
\cite[Lemma~3.10]{AdamsTaibi}, including every component. This
proves \textup{(i)}--\textup{(iii)}. The same proof
applies to all the indicated functorial linear constructions.
It remains to check the cyclic frame value, rather than merely the
action on $V$. Embed the compact frame image of
$M_{\mathrm{cpt}}$ in a maximal compact of
$K\rtimes\langle\text{cyclic permutation}\rangle$. Its
intersection with $K$ is a product of compact forms of the $d$
factors. For $k\in L_{e,\mathrm{cpt}}$, the $i$th factor of
$(kt)^d$ is the product transport around the cycle starting at the
$i$th frame. It is compact and centralizes the triple in that
factor. Since $v_0$ commutes with $kt$, the corresponding factor
of $(ktv_0)^d$ has that compact element as its semisimple part.
The translation by $\lambda_e(\sqrt q)$ commutes with both
$kt$ and $v_0$, and $v_0$ commutes with $kt$. Thus, writing $w=kt$,
\[
\bigl(\lambda_e(\sqrt q)wv_0\bigr)^d
=\lambda_e(q^{d/2})w^dv_0^d.
\]
No commutation between $k$ and $t$ is needed. This proves
\eqref{mod:cyclic-compact-value}. Compact forms of the triple
centralizer are conjugate inside that centralizer, so the set of
ambient conjugacy values is independent of the compact form
chosen in this proof.
\end{proof}
Translation of a transporter by $J$ does not change an equivariant
calculation: it changes the chosen identification of the same
Frobenius functor. After the additional translation by
$\lambda_e(\sqrt q)$ the point $e$ is fixed, since singular
points originally scale by $q^{-1}$ and $e$ has triple weight $2$.
The resulting weights and the distinction between the original and
lifted gradings are recorded in Section~\ref{orb:section}.
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/03-components.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/04-orbits.tex ====
\section{The category on a nilpotent orbit}\label{orb:section}
Fix an isomorphism $\E\simeq\C$ and a Frobenius-fixed retained
component. Use the quadratic model of Theorem~\ref{mod:quadratic-model},
with $J=\Aut(\sigma)$, $\mathfrak j=\Lie(J)$ and
$V=H^1(\overline X,\ghat_\sigma)$. All representations in this section
are algebraic; rational representations of an algebraic group may be
unions of finite-dimensional representations. The group $J$ is reductive
but need not be connected. We first omit the completion along the
invariant null fiber. Section~\ref{tr:section} restores that condition,
on both traces and their actual compact representatives.
On each nilpotent orbit we construct compact generators, extend them
with Weil structures to its closure, and compute their morphisms. With
pure coefficient normalizations, the generators and their weight-zero
morphism cohomology form a semisimple category, and the remaining weights for the
inverse action $S$ are strictly negative. Section~\ref{tr:section} uses
these facts to contract the nonconstant part of each orbit trace.
\subsection{The curved model and supported localization}
Put $R=\C[V]$. The derived zero fiber of the moment map is represented
by the Koszul differential graded algebra
\begin{equation}\label{orb:derived-koszul}
A=\bigl(R\otimes\Lambda(\mathfrak j[1]),d_\mu\bigr),
\qquad |\eta_a|=-1,\qquad d_\mu\eta_a=\mu_a.
\end{equation}
Here a basis of $\mathfrak j$ indexes the components $\mu_a$ of
$\mu:V\to\mathfrak j^*$. In particular, no regularity of this section
is asserted or needed. Its graded Koszul dual is the curved algebra
\begin{equation}\label{orb:potential}
B=\C[V\times\mathfrak j],\qquad
|x^*|=0,\quad |z^*|=2,\qquad
W(x,z)=\langle z,\mu(x)\rangle.
\end{equation}
A curved module has a differential $\delta$ of degree one satisfying
$\delta^2=W$. A simultaneous reversal of the curvature sign gives the
same calculations after the corresponding convention change.
\begin{lemma}[Koszul model and support]\label{orb:koszul}
The compact ind-coherent category on the uncompleted derived quotient
$[\mu^{-1}(0)/J]$ is the category of finite coherent graded
$J$-equivariant factorizations of \eqref{orb:potential}.
Coherent singular support in a closed cone $\Omega\subset\mathfrak j$
corresponds to local vanishing of the factorization off
$V\times\Omega$. The completion in
Theorem~\ref{mod:quadratic-model} additionally imposes ordinary support
on the inverse image of $0\in\Spec R^J$.
\end{lemma}
\begin{proof}
The uncompleted derived quotient is a QCA stack: it is
quasi-compact, with affine stabilizers, and its classical inertia is of
finite presentation, as for every finite-type quotient here. Its compact
ind-coherent objects are therefore its coherent objects by
\cite[Theorem~3.3.5]{DGFiniteness}. We recall the Koszul construction
to specify the derived and grading conventions.
A coherent $A$-module is perfect as an $R$-complex because
$R$ is regular. Choose a bounded finite projective $R$-resolution and
transfer the actions of the degree-minus-one generators to it, allowing
higher homotopies. The Koszul twisting cochain, on the symmetric
\emph{divided-power coalgebra} of the dual generators, turns those
homotopies into a differential on its extension to $B$. Its square is
$\sum_a z_a\mu_a$. The linear coefficients encode nullhomotopies of
$\mu_a$, and the higher coefficients encode all compatibility
homotopies for the exterior-generator actions. Conversely, the
coefficients of such a curved differential, specialized at $z=0$,
recover that homotopy-coherent $A$-action.
The usual free Koszul resolution proves that these constructions are
inverse on derived modules and morphisms: after filtering by exterior
length, it is the ordinary exterior/symmetric Koszul equivalence, and
the equation differential contributes precisely the curvature term.
There is no completed polynomial ring here. Between bounded finite
projective models only finitely many $z$-monomials can occur in any
fixed total degree. Derived morphisms on the curved side are sections
of differential Hom; its two curvature terms cancel.
One may equivalently define coherent curved modules modulo
totalizations of finite exact sequences. On the smooth ambient
space, finite locally free replacements compute this category.
They are obtained by curved free covers, with the partner part of the
differential correcting the square to $W$; the underlying coherent
resolutions terminate by regularity. Reductivity permits equivariant
choices for $J^\circ$, followed by equivariance for the finite component
group. This is also the equivariant derived-zero-locus Koszul
equivalence of \cite[Theorem~2.3.3]{Toda}, with its differential graded
formulation in \cite[Theorem~2.1]{PT-koszul}.
The variables $z_a$ are the degree-two complete-intersection operators
on the $A$-side. Their localization is the localization defining
coherent singular support. Under the equivalence, the category with
that singular support is exactly the kernel of restriction to the
complementary open, with the corresponding quotient description;
this is \cite[Proposition~2.3.9]{Toda}. Support here is support of the
object in the curved category: a locally free underlying module can
represent an object vanishing on an open set.
The remaining completion imposes a different, ordinary support
condition. For a DG scheme almost of finite type,
\cite[Proposition~7.4.5]{DGIndschemes} identifies ind-coherent sheaves
on its formal prestack completion with the kernel of restriction to
the complementary open; its compact objects are the coherent objects
with that ordinary support by \cite[Proposition~4.1.7(b)]{IndCoh}.
Smooth descent, \cite[Corollary~10.4.5]{IndCoh}, identifies the
presentable supported categories on the quotient by $J$. Compactness
on the quotient also uses the QCA theorem just invoked and the continuous
support right adjoint of \cite[Corollary~4.1.5]{IndCoh}. If $i$ denotes
the supported inclusion and $\Gamma$ its continuous right adjoint, then
$\Hom(iM,-)=\Hom(M,\Gamma(-))$ shows that $i$ preserves compactness.
Conversely, since $i$ preserves colimits, an ambient compact whose
restriction to the complementary open vanishes is compact in the
supported category. Its compact objects are therefore precisely the
supported coherent objects. This identifies compacts after presentable
descent, without commuting compact objects with a descent limit.
Applied to the invariant null fiber in our derived affine model, it
proves the last condition in the lemma.
The completion is the formal prestack, not the spectrum of a completed
coordinate ring. This ordinary support condition leaves the
singular-direction operators unchanged and can be intersected with
the prescribed $z$-support.
\end{proof}
Write $\mathscr B_\Omega$ for the resulting compact category with
$z$-support in $\Omega$, and $\mathscr B_{\Omega,0}$ when the invariant
null-fiber condition is also imposed. We use the same notation with
$\operatorname{Ind}$ when a presentable category is required for a trace. All
compact quotients below are idempotent completed.
\begin{lemma}[Supported localization]\label{orb:localization}
Let $\Omega'\subset\Omega$ be closed invariant cones. Restriction to
the complementary open gives the compactly generated localization
with compact quotient $\mathscr B_\Omega/\mathscr B_{\Omega'}$.
If $\mathcal O$ is an orbit closed in that open, its supported compact
category is generated by coherent curved pushforwards from
$V\times\mathcal O$. Morphisms are computed in the open before taking
derived equivariant sections. The statements also hold with invariant
null-fiber support.
\end{lemma}
\begin{proof}
Choose finite homogeneous equivariant generators of the relevant
ideal sheaves on a finite affine cover. Koszul complexes on their
powers, and their duals, give the ordinary local-cohomology projector
and complementary localization. Tensoring a curved module by these
ordinary complexes is defined because their differentials have square
zero. A compact object's identity factors through a finite stage of
the projector if the object is supported on its center. Thus it is a
summand of a finite supported stage.
Let $i:Z_0\hookrightarrow U$ be the smooth closed locus in the
smooth ambient open, with ideal sheaf $I$, and let $K_n$ be the finite
ordinary perfect projector stage just chosen. Its locally free terms
need not be supported on $Z_0$. Its bounded ordinary cohomology
sheaves are coherent, supported on $Z_0$, and annihilated by powers
of $I$. Since $U$ is smooth, those coherent sheaves are perfect.
Ordinary truncation triangles therefore express $K_n$ in their thick
span. Each such sheaf has a finite $I$-adic filtration whose
successive quotients have the form $i_*F$, for coherent sheaves $F$
on $Z_0$. The truncations, ideal filtrations, and their maps retain
equivariance and grading.
Ordinary perfect complexes act exactly by tensor product on coherent
curved modules. The extra ordinary differential has square zero,
so this action takes the preceding finite triangles to curved
triangles without changing the potential. For a locally free curved
representative $M$, the projection formula gives
\[
(i_*F)\otimes M\simeq i_*\bigl(F\otimes\mathbf L i^*M\bigr).
\]
The right side is a coherent curved pushforward with the restricted
potential. Consequently $K_n\otimes M$ lies in the thick span of
such pushforwards. The factorization of the identity through this
stage makes $M$ a summand, proving generation. The ordinary
truncations have been applied only to $K_n$, before its exact action
on the curved factor.
The same local-cohomology construction computes restriction and the
quotient on morphisms; the support/quotient compatibility is also the
one in \cite[Proposition~2.3.9]{Toda}. On the ambient open, finite
covers and finite \v{C}ech descent compute the ordinary perfect-complex
statements. On a homogeneous orbit, equivariant descent takes the
finite Lie-cochain complex for its unipotent stabilizer, followed by
exact invariants for the reductive quotient. These bounded operations
commute with the finite triangles and direct-sum folding of the
grading. The argument also applies on homogeneous vector bundles.
Ordinary support in the invariant base is preserved at every step.
\end{proof}
\subsection{Gradings and Clifford generators}
The intersection of $\mathfrak j$ with $\Nhat$ is its nilpotent cone:
for a reductive subgroup, the semisimple and nilpotent parts of an
element agree with those in the ambient faithful representation.
There are finitely many nilpotent $J$-orbits. Fix one, $\mathcal O=J.e$,
and choose an $\mathfrak{sl}_2$-triple $(e,h,f)$ in $\mathfrak j$.
Set
\begin{equation}\label{orb:notation}
\lambda=\lambda_e,\qquad
C_e=J^e=L_e\ltimes U_e,\qquad
L_e=Z_J(\mathrm{SL}_{2,e}),\qquad
N_e=\mathfrak j/[\mathfrak j,e].
\end{equation}
The zero orbit is included, with trivial triple and $U_e=1$.
Subscripts $i$ denote $h$-weights, independently of cohomological
degree or Frobenius weight. The Lie algebra of $U_e$ has strictly
positive $h$-weights. The quotient $N_e$ consists of lowest-weight
vectors and has $h$-weights at most zero.
Suppose first that the orbit is Frobenius-stable. By
Lemma~\ref{mod:compact-normalization}, the weight-zero normalized
transporter can be chosen to fix the triple, preserve a compatible
compact form, and remain of absolute weight zero after compact
$L_e$-twists. Multiplication by $\lambda(\sqrt q)$ corrects the scalar
motion of $e$ and gives an action fixing $e$. Write $S_e$ for this
translated inverse action on the Hom calculation. For an orbit or a
finite list of labels stable only under a power, the same convention
is made for that power. Absolute weights are measured in units
$q^{1/2}$ (or $q^{m/2}$ for the $m$th power).
There is a separate translation of the grading. The original grading
acts on $z$-points by $t^{-2}$; lifting it by $\lambda(t)$ makes $e$
fixed. We call the resulting degree on a fiber at $e$ the
\emph{lifted degree}. The data needed below are
\begin{equation}\label{orb:weight-table}
\begin{array}{c|c|c}
\text{factor}&\text{lifted degree before an Ext shift}
&\text{absolute }S_e\text{-weight}\\ \hline
(V_i)^*\text{, as an }x\text{-function}&-i&-1-i\\
(N_e)_i\text{, as a normal tangent}&i-2&-2+i\\
\Lie(U_e)_i^*&-i&-i.
\end{array}
\end{equation}
Translating by a group element changes none of the final equivariant
invariants. In particular these are the actual weights there, rather
than weights for a newly chosen categorical Frobenius.
The original super sign must be retained during this change of
grading. It differs from the lifted-degree sign by the action of
$\lambda(-1)$. Consequently the lifted total-degree sign computes
true alternating traces \emph{after taking invariants}. In
particular, a polynomial coordinate of odd lifted degree remains a
commuting coordinate; it is not replaced by an exterior generator.
Put $Q_e(x)=W(x,e)$. With the symplectic pairing on $V$, this is, up
to the fixed moment-map sign, $\frac12\langle ex,x\rangle$.
Its restriction to $V_{-1}$ is nondegenerate: $e:V_{-1}\to V_1$ is an
isomorphism and the symplectic pairing identifies $V_1$ with
$V_{-1}^*$. Let
\[
D_e=\operatorname{Cl}(V_{-1},Q_e)
\]
be the ordinary Clifford algebra, with $c(v)^2=Q_e(v)$, and let
$\mathsf E_e$ be the category of finite-dimensional $D_e$-modules with
compatible algebraic $L_e$-action.
The related spectral Whittaker construction uses Clifford modules for a
quadratic cohomology complex \cite[Sections~4.5.2--4.5.5]{Raskin};
its orbit trace comparison is Theorem~J in Section~4.6.2 there.
Here we keep explicit generators and their Hom weights, as well as
compact Weil extensions, for the later bilinear tests.
For $P\in\mathsf E_e$, give its coefficient space lifted degree zero.
Its original coefficient parity is the involution
\begin{equation}\label{orb:parity}
\gamma_P=\rho_P(\lambda(-1)),\qquad
\gamma_Pc(v)\gamma_P^{-1}=-c(v)\quad(v\in V_{-1}).
\end{equation}
Define $M_P$ over $e$ as the curved pushforward from
$V_{\ge-1}\subset V$ of
\begin{equation}\label{orb:generator}
\bigl(\mathcal O_{V_{\ge-1}}\otimes P;\quad
\delta=c(x_{-1})\bigr).
\end{equation}
The unipotent radical acts on $P$ through its trivial quotient, and
acts trivially on the projection of $V_{\ge-1}$ to $V_{-1}$.
Only quadratic terms with both inputs of weight minus one contribute
to $Q_e$ on this subspace. Thus \eqref{orb:generator} has square $Q_e$, is
$C_e$-equivariant, and has lifted differential degree one.
The lifted grading and $C_e$-equivariance descend to the required
original graded object over $J.e$. Denote its curved pushforward into
a surrounding open by the same symbol $M_P$.
\begin{proposition}[Orbit generation]\label{orb:generation}
The objects $M_P$, for simple $P\in\mathsf E_e$, generate the compact
category supported on $J.e$ in an open where it is closed. Here
``generate'' allows finite cones, shifts and summands.
The category $\mathsf E_e$ is semisimple.
\end{proposition}
\begin{proof}
For a locally free factorization, differentiation of $\delta^2=Q_e$
gives
\[
\delta(\partial_v\delta)+(\partial_v\delta)\delta
=\partial_vQ_e.
\]
Thus the derivatives of $Q_e$ annihilate the identity up to homotopy.
Their common zero locus is $\ker(e:V\to V)$, which consists of
highest-weight vectors and lies in $V_{\ge0}$. Lemma~\ref{orb:localization}
therefore reduces generation to curved pushforwards from $V_{\ge0}$,
where the potential is zero.
On this affine space, finite graded free modules with finite
stabilizer-representation labels and their degree shifts generate the
zero-potential category. To justify this with the nonreductive
stabilizer, note first that all finite rational $C_e$-modules have a
finite filtration with $U_e$-trivial subquotients: take unipotent
invariants and iterate, retaining $L_e$-stability. Rational
$U_e$-cohomology has cohomological dimension $\dim U_e$ and is computed
by its finite Lie-cochain complex. The usual equivariant free tests
therefore detect underlying graded cohomology: tensor by all finite
representation labels, pass to their union in the regular
representation, and then take invariants. These operations preserve
colimits. The tests are compact and hence generate.
There is no additional curved object at zero potential invisible to
this test. After a finite projective replacement, an acyclic folded
differential module over the regular polynomial ring is absolutely
acyclic: resolve its cycles and boundaries by finite projective
resolutions and fold the resulting bounded exact complexes.
Equivariant descent adds only the bounded Lie-cochain complex above.
Equivalently, the lifted coordinate degrees on $V_{\ge0}$ are
nonpositive; filtering finite free representatives by generator degree
reduces to finite complexes over the degree-zero polynomial ring.
Stabilizing the zero section of the nondegenerate quadratic space
$V_{-1}$ replaces a generator on $V_{\ge0}$ by
\eqref{orb:generator} with the regular Clifford module. In coordinates
this is the Koszul differential for $x_{-1}=0$, together with its
Clifford partner; their square is $Q_e(x_{-1})$. The normal coordinates
have lifted degree one, so the Koszul shifts put the coefficient
module in lifted degree zero. Tensoring by $L_e$-representations and
allowing shifts gives all the preceding generators.
Finally $D_e$ is a finite-dimensional semisimple complex algebra,
including when $\dim V_{-1}$ is odd. A $D_e$-linear splitting of any
short exact sequence in $\mathsf E_e$ can be averaged over a compact
form of the reductive group $L_e$. Conjugation preserves $D_e$-linearity,
so the averaged splitting is also $L_e$-equivariant. This proves
semisimplicity, including the finite component group, and reduces the
generating list to its simple objects.
\end{proof}
\subsection{Actual extensions with Weil structures}
\begin{proposition}[Compact orbit-closure lifts]\label{orb:weil-lifts}
Every $M_P$ is the restriction of a compact object with $z$-support in
$\overline{J.e}$. If $P$ has a compatible locally finite cyclic
structure, the extension can be given that structure. For a simple
label fixed by Frobenius, a structure may be chosen whose
triple-translated coefficient eigenvalues have modulus one.
The same assertions hold for a power of Frobenius, for a finite cyclic
list of labels, and after tensoring $P$ by any finite-dimensional
$L_e$-representation with compatible cyclic structure, placed in
lifted degree zero. Inverse structures are available for opposite
category classes.
\end{proposition}
\begin{proof}
Let $P_e=J_{\ge0}$ be the subgroup of elements for which
$\lim_{t\to0}\lambda(t)g\lambda(t)^{-1}$ exists, and put
$J_0=Z_J(\lambda)$. The Jacobson--Morozov incidence map is
\begin{equation}\label{orb:incidence}
\pi:J\times^{P_e}\mathfrak j_{\ge2}
\longrightarrow\overline{J.e},
\qquad [g,z]\longmapsto\operatorname{Ad}(g)z.
\end{equation}
Its connected version is the resolution in
\cite[Section~2.3]{Namikawa}; the following argument records the
properties, including disconnected groups, that we use.
The $\mathfrak{sl}_2$ decomposition gives
$[\mathfrak j_{\ge0},e]=\mathfrak j_{\ge2}$. Hence
$P_e^\circ.e$ is open dense in $\mathfrak j_{\ge2}$. Every component
of $P_e$ preserves this unique open $P_e^\circ$-orbit, so
$P_e.e=P_e^\circ.e$. The centralizer decomposition in
\eqref{orb:notation} gives $J^e\subset P_e$.
Consequently the open part of the source of \eqref{orb:incidence} is
$J\times^{P_e}(P_e.e)=J/J^e$, mapped isomorphically to $J.e$.
The entire source is equidimensional of dimension $\dim(J/J^e)$.
Its complementary closed subset has smaller dimension and has
$J$-invariant image, so that image cannot meet $J.e$.
Thus this is the whole inverse image of the orbit.
Finally $J/P_e$ is a finite disjoint union of flag varieties; the
incidence space is closed in $(J/P_e)\times\mathfrak j$.
The map is therefore proper and its image is $\overline{J.e}$.
The degree-two component of this construction uses the open orbit
$J_0.e\subset\mathfrak j_2$, whose stabilizer is $L_e$.
Over that open orbit the module $P$ gives an equivariant coherent module
for the family of Clifford algebras of
\[
Q_z|_{V_{-1}},\qquad z\in\mathfrak j_2.
\]
This Clifford algebra is finite as a module over
$\mathcal O_{\mathfrak j_2}$, also where the quadratic form degenerates.
Extend the module coherently over $\mathfrak j_2$ as follows. Its
quasi-coherent direct image from the open orbit is a rational
$J_0$-module. Choose finitely many module generators over that open,
enlarge them to a finite-dimensional equivariant space, and take their
span under the finite Clifford algebra and the polynomial ring.
This is a coherent equivariant submodule whose restriction is the
original family. More explicitly, let $t$ act on this coefficient
family through the central element $\lambda(t^{-1})\in J_0$. It sends
$z\mapsto t^{-2}z$ and
$c_z(w)\mapsto c_{t^{-2}z}(tw)$. With the original $x$-points fixed,
$c_z(x_{-1})$ consequently has degree one. At $t=-1$ its coefficient
parity is precisely $\lambda(-1)$. Thus equivariant extension also
preserves the required original grading.
Inflate this extension along
$\mathfrak j_{\ge2}\to\mathfrak j_2$ and tensor it with
$\mathcal O_{V_{\ge-1}}$. The differential $c(x_{-1})$ has square
$W(x,z)$ on this locus: a term pairing $z_k$ with $x_i,x_j$ requires
$k+i+j=0$, with $k\ge2$ and $i,j\ge-1$, and hence
$k=2$, $i=j=-1$. The construction is $P_e$-equivariant because its
unipotent radical acts trivially on the two indicated associated
graded quotients. Induction to $J$ and proper pushforward from
\[
J\times^{P_e}(V_{\ge-1}\times\mathfrak j_{\ge2})
\longrightarrow V\times\mathfrak j
\]
give the required coherent factorization. This map is proper by the
same closed-incidence argument. Over $J.e$ its restriction is exactly
$M_P$, so this is an actual lift, with the asserted support.
For completeness, the equivariance in this extension can include a
chosen cyclic operator, rather than merely its action on isomorphism
classes. Take the Zariski closure of the group generated by the
algebraic equivariance data, the grading, and that operator; if
necessary take a common algebraic cover acting on the coefficient
module. Rational direct image is the union of finite-dimensional
representations of this group. Thus the finite generating space in
the preceding paragraph can be chosen stable under the operator, and
the entire extension and proper pushforward retain its structure.
Here is also why the pure coefficient choices exist. Use the
finite-type normalizer of the linear and framed data in
Lemma~\ref{mod:compact-normalization}; the action is an algebraic action
of this finite-type group, not of an abstract automorphism group of a
torus. For a fixed simple equivariant Clifford module, pairs
consisting of a data automorphism and an implementing linear map form
an algebraic group over the stabilizer of its isomorphism class. Its
kernel is the scalars, by Schur's Lemma. Given a lift $(u,A)$, take its
semisimple Jordan part and the diagonalizable algebraic group it
generates. Characterwise polar decomposition in this group separates
its positive-real and unit-modulus parts. The positive-real part
projects to the identity: the eigenvalues of $u$ on the faithful
normalized data realization have modulus one. The scalar-kernel
assertion therefore forces this positive-real part to be $(1,cI)$
for some $c>0$. All eigenvalues of $A$ have the same modulus $c$,
and $c^{-1}A$ is the required pure implementing map. The Jordan
unipotent part supplies the compatible commuting unipotent lift.
The label orbits are finite as well. Encode compatible Clifford and
$L_e$ actions as representations of the reductive group
$D_e^\times\rtimes L_e$. A simple compatible module is irreducible
for this group, because the units linearly span $D_e$. A connected
algebraic family of automorphisms preserves each irreducible class:
on a fixed maximal compact group, its multiplicities against
irreducible characters are continuous integer-valued functions of the
family, hence constant. The compact group is Zariski dense, so this
identifies the algebraic representations. Only the finite component
group of the data normalizer can therefore permute labels. The
preceding choices apply to a power, or around a finite orbit; a label
fixed by Frobenius needs no power. Undoing the triple translation
gives the original locally finite Weil structure. Tensor products
and inversion of the implementing maps preserve the extension
construction, proving the remaining assertions.
\end{proof}
\subsection{Morphisms, parity, and contracting weights}
\begin{proposition}[The orbit Hom calculation]\label{orb:hom-calculation}
For $P,P'\in\mathsf E_e$, the Hom complex between the orbit generators
has a finite equivariant calculation with Euler terms
\begin{equation}\label{orb:hom-euler}
\left[
\C[\ker(e:V\to V)]\otimes
\Hom_{D_e}(P,P')\otimes
\Lambda^\bullet(N_e[-1])\otimes
\Lambda^\bullet(\Lie(U_e)^*[-1])
\right]^{L_e}.
\end{equation}
Here the coefficient factor has lifted degree zero; the shifts in the
normal and Lie factors add one to the degrees in
\eqref{orb:weight-table}. The formula is an equality of alternating
equivariant characters, and a finite-filtration calculation bounding
weights, rather than an assertion that the Hom complex splits as the
displayed tensor product.
For pure coefficient normalizations, all nonconstant terms have
strictly negative absolute $S_e$-weight. The weight-zero cohomology is
\begin{equation}\label{orb:weight-zero-hom}
H^\bullet\Hom(M_P,M_{P'})_{\mathrm{wt}=0}
=\Hom_{D_e,L_e}(P,P')[0],
\end{equation}
with ordinary composition. Before the finite invariant calculation,
weight pieces are finite-dimensional and their dimensions grow at most
polynomially over bounded-length weight intervals. In particular,
the character sums in \eqref{orb:hom-euler} are absolutely convergent
on every normalized compact $L_e$-coset, uniformly on that coset.
\end{proposition}
\begin{proof}
Take an $L_e$-stable lowest-weight slice transverse to $J.e$ in the
$z$-space. Its tangent is $N_e$; moving this slice by $J$ is smooth
near $e$, so it computes the equivariant transverse fiber of Hom.
Then take $U_e$-cochains and finally exact $L_e$-invariants.
The second input is pushed forward from the equations $n=0$ in the
slice and $x_{<-1}=0$. A finite locally free stabilization of the
first input is obtained by tensoring its Clifford differential with
the Koszul factorizations of these equations and their potential
partners. Indeed the difference between the full potential and
$Q_e(x_{-1})$ is a sum of an equation times its partner. The terms in
$n$ have quadratic partners in $x$; the terms $x_i$ for $i<-1$ have
partners of grade $-i-2\ge0$. This explicit correction makes the
square of the differential equal to the full potential.
Hom into the second pushforward sets the normal equations to zero.
The $z$-normal Koszul generators remain as exterior \emph{normal
tangent} factors $\Lambda(N_e[-1])$.
Filter finitely by those exterior generators. The remaining
$x$-Koszul differential pairs every grade $i<-1$ with its grade
$-i-2$ partner. On an irreducible $\mathfrak{sl}_2$ summand these
pairings are nondegenerate except for its highest-weight vector.
Thus their cohomology is precisely $\C[\ker e]$. The terms involving
$n$ may give further differentials in this finite filtration, but do
not change its Euler terms.
It remains to calculate the nondegenerate quadratic part. On
$\C[V_{-1}]\otimes\Hom(P,P')$ its differential is the sum of the
coordinate functions times the Clifford supercommutators. Choose an
orthonormal Clifford basis, and on a homogeneous coefficient map $a$
put
\[
L_i(a)=c_i'a,\qquad R_i(a)=(-1)^{|a|}ac_i,\qquad
T_i=L_i-R_i,\qquad K_i=\tfrac14(L_i+R_i).
\]
Clifford anticommutation gives
\[
\{T_i,T_j\}=\{K_i,K_j\}=0,\qquad
\{T_i,K_j\}=\delta_{ij}.
\]
Thus the $K_i$ are contracting partners for the exterior operators
$T_i$. The differential $\sum_i x_iT_i$ is the polynomial Koszul
calculation: its cohomology is the constant common kernel of the
$T_i$, namely the Clifford super-intertwiners, in lifted degree zero.
All positive polynomial degrees are acyclic. This calculation works
in even and odd Clifford dimensions alike.
We explain its parity before taking invariants. A homogeneous map
$a:P\to P'$ of original parity $p$ is a super-intertwiner precisely
when
\[
a c(v)=(-1)^p c'(v)a.
\]
The map $a\mapsto a\gamma_P^p$ identifies that vector space with
ordinary Clifford intertwiners. It respects the $L_e$-action and the
triple-preserving cyclic action, because $\lambda(-1)$ is central in
$L_e$ and is fixed by that action. Only a representation-space
identification is claimed before taking invariants. An $L_e$-invariant
map commutes with $\gamma_P,\gamma_{P'}$, so an invariant
super-intertwiner has even parity. After invariants the identification
therefore has its ordinary composition and is exactly
$\Hom_{D_e,L_e}(P,P')$. This proves that no odd Clifford morphism is
being silently retained in degree zero.
Finally rational $U_e$-cohomology is computed by its finite
Chevalley--Eilenberg complex, adding the terms
$\Lambda(\Lie(U_e)^*[-1])$, with their Lie differential and action on
the other terms. This proves \eqref{orb:hom-euler} as an alternating
character identity with a finite cohomological filtration.
The weight assertions now follow factor by factor. Functions on
$\ker e$ have weights $-1-i\le-1$ because its $h$-weights satisfy
$i\ge0$. The normal factors have weights $-2+i\le-2$ because their
$h$-weights satisfy $i\le0$. Nontrivial Lie-cochain factors have
weights $-i<0$. The pure constant coefficient Hom has weight zero.
Thus nothing of any other degree has weight zero, and no differential
can enter or leave its invariant subspace. This proves
\eqref{orb:weight-zero-hom}.
There are finitely many polynomial generators, all with strictly
negative weights, and finitely many exterior and coefficient factors.
Their monomial counts in bounded-length weight intervals have
polynomial growth. Multiplication by the geometric decay of the
absolute eigenvalues makes their traces summable. The same estimates
hold uniformly with any compact $L_e$-twist by
Lemma~\ref{mod:compact-normalization}. Unipotent parts may be retained:
trace depends on eigenvalues with multiplicity, not on an assertion
that those operators are unitary.
\end{proof}
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/04-orbits.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/05-traces.tex ====
\section{Frobenius traces and compact Weil classes}\label{tr:section}
This section proves the concentration and spanning assertions needed
for the image definition of $\cF_Y$. They will also permit every
function in the second variable of the automorphic pairing to be
represented by a finite combination of compact spectral tests.
The argument has three parts: remove invariant-base support on the
trace, contract the nonconstant part of an orbit trace, and use finite
supported localization to assemble the result.
\subsection{The invariant-base projector and actual supported lifts}
\begin{lemma}[Invariant support on traces]\label{tr:invariant-support}
For every Frobenius-stable conical $z$-support bound $\Omega$, inclusion
induces an isomorphism
\begin{equation}\label{tr:remove-completion}
\Tr(\Phi,\operatorname{Ind}\mathscr B_{\Omega,0})
\xrightarrow{\;\sim\;}
\Tr(\Phi,\operatorname{Ind}\mathscr B_\Omega).
\end{equation}
A compact Weil object on the right has a compact Weil lift on the left
whose trace class maps to a nonzero scalar multiple of its class.
Both statements hold for the compact opposite categories.
Moreover, for Hom pairings of two such lifts, the finite Koszul
construction multiplies each orbit contribution by a constant nonzero
factor, independent of the compact stabilizer variable.
\end{lemma}
\begin{proof}
The ideal $I\subset R^J$ of the invariant origin is generated by a
finite Frobenius-stable homogeneous space $U$ of positive-degree
invariant polynomials. To obtain $U$, choose homogeneous algebra
generators and take their Frobenius spans inside the finitely many
finite-dimensional polynomial-degree spaces involved. The inverse
Hom action has strictly negative weights on $U$ by
Lemma~\ref{mod:hom-action}; its forward inverse has strictly positive
weights. In either convention the appropriate transport operator
$F_U$ has no eigenvalue one. Thus $1-F_U$ is invertible, even when
$F_U$ has Jordan blocks.
For a central scalar $a$, its two actions on a twisted Hochschild
complex, one transported by Frobenius, are chain homotopic. This is
the elementary homotopy moving $a$ once around the bar. Applying it
simultaneously to a basis of $U$ and multiplying by $(1-F_U)^{-1}$
shows that every generator of $I$ acts trivially on trace cohomology.
Equivalently, regard the complex as a module over the polynomial
algebra $\Sym U$ mapping to $R^J$; its cohomology is supported at the
origin. In particular it is $I$-torsion, a statement about the ordinary
complex rather than about its completion.
Let $i$ be inclusion of the support category and let $\Gamma_I$ be
its continuous colocalization. On the ambient category,
$i\Gamma_I$ is tensor product with the scalar local-cohomology complex
$\RGamma_I(R)$. The inclusion preserves compact objects. Cyclicity
of categorical trace, followed by continuity in its coefficient
bimodule, therefore identifies the map in
\eqref{tr:remove-completion} with
\[
\RGamma_I\Tr(\Phi,\operatorname{Ind}\mathscr B_\Omega)
\longrightarrow
\Tr(\Phi,\operatorname{Ind}\mathscr B_\Omega).
\]
One can compute this equality directly with the finite Koszul
complexes on powers of the chosen equations and their filtered union.
It is an isomorphism because the target cohomology is $I$-torsion.
This proves the trace assertion without replacing Hochschild chains
by a completed bar complex.
To lift an actual class, let $M$ be a compact Weil object and let
$K(U)$ be the finite equivariant Koszul complex on the map
$U\otimes R\to R$. Then $M\otimes K(U)$ is compact, has ordinary
support over $I=0$, retains its $z$-support, and has the induced Weil
structure. The finite exterior filtration and additivity of trace
give
\begin{equation}\label{tr:koszul-factor}
[M\otimes K(U)]
=\left(\sum_j(-1)^j\Tr(F_U\mid\Lambda^jU)\right)[M]
=\det(1-F_U\mid U)\,[M].
\end{equation}
The determinant is nonzero. The notation $F_U$ in this formula means
the coefficient transport on the chosen exterior terms; passing to
inverse Weil structures replaces it by its inverse, which still has
no eigenvalue one. Dividing the class by this nonzero complex scalar
gives any desired lift in the trace vector space, with the representative
itself still an actual compact Weil object.
The same finite filtration proves the pairing statement. In a Hom
from one Koszul lift to another, its associated terms have coefficient
factor $\Hom(\Lambda^aU,\Lambda^bU)$, with the dual transport on the
contravariant coefficient and the original transport on the covariant
one. Its alternating character is the product of the corresponding
two nonzero determinants. Since the equations are $J$-invariant,
these are constant functions of every compact $L_e$-variable.
Taking sections with support in an orbit commutes with this finite
filtration. Thus the same factor multiplies each orbit contribution,
not merely the sum of all contributions. This also proves the
assertions for opposite categories and their inverse structures.
\end{proof}
\subsection{An algebraic contraction of the twisted bar complex}
We first identify the original $\Phi$-graph with the transported model,
then use the contracting inverse action $S$ on that same graph.
On a finite stable list of models let $\pi$
be the permutation induced by Frobenius and choose
$\alpha_i:\Phi M_i\xrightarrow{\sim}M_{\pi i}$.
These identifications give a functor $\Psi$ on the chosen models.
For arrows $a:M_i\to M_j$ and $b:M_{\pi i}\to M_{\pi j}$,
its forward transport and inverse are
\begin{equation}\label{tr:forward-inverse}
\begin{split}
T(a)=\Psi(a)&=\alpha_j\Phi(a)\alpha_i^{-1},\\
S(b)=T^{-1}(b)&=\Phi^{-1}(\alpha_j^{-1}b\alpha_i).
\end{split}
\end{equation}
The action $S$ is the contracting Hom action of
Lemma~\ref{mod:hom-action}.
To match trace classes $M\to\Phi M$, use the left-twisted graph
$G_\Phi(i,j)=\Hom(M_i,\Phi M_j)$, whose left action is
composition through $\Phi$. The maps
\begin{equation}\label{tr:graph-identification}
\theta_{ij}:G_\Phi(i,j)\longrightarrow G_\Psi(i,j),
\qquad m\longmapsto\alpha_j m,
\qquad G_\Psi(i,j)=\Hom(M_i,M_{\pi j})
\end{equation}
identify the graph bimodules: for composable arrows,
$\theta(\Phi(a)mb)=\Psi(a)\theta(m)b$, with the appropriate
source and target indices. Transport the entire twisted bar
construction through this identification. In the $G_\Psi$
coordinates simultaneous forward transport is $\Psi$ on every
factor. On the original graph coefficient it is instead
\begin{equation}\label{tr:graph-transport}
T_G(m)=\Phi(\alpha_j)\Phi(m)\alpha_i^{-1}
\in G_\Phi(\pi i,\pi j),
\qquad
\theta_{\pi i,\pi j}(T_G(m))=\Psi(\theta_{ij}(m)).
\end{equation}
For example, for one algebra with automorphism $\phi$ and
$\psi=\operatorname{Ad}_\alpha\phi$, these formulas are
$\theta(m)=\alpha m$ and
$T_G(m)=\phi(\alpha)\phi(m)\alpha^{-1}$.
Thus the original $\Phi$-trace is retained, with its graph coefficient
canonically identified. Ordinary-arrow transport cannot be applied
to the raw graph factor without this identification.
\begin{lemma}[Locally finite contraction]\label{tr:bar-contraction}
Let a small generating differential graded category with an
autoequivalence be modeled in complexes of locally finite cyclic
representations. Suppose, after finite cyclic permutations of its
labels and pure choices of their structures, its morphism cohomology
has nonpositive absolute $S$-weights; composition adds weights.
Suppose its weight-zero morphism cohomology is concentrated in degree
zero and its additive idempotent completion is semisimple, with
endomorphism algebra $\C$ for every simple object. Then its twisted
trace is the twisted trace of that split semisimple category. It is concentrated in
degree zero, with one basis line for each fixed simple label.
Nontrivially permuted simple labels contribute zero.
\end{lemma}
\begin{proof}
Take the sum of the nonpositive-weight subcomplexes in every morphism
complex. This is a differential graded subcategory: identities have
weight zero, the differential preserves weights, and composition adds
them. It is quasi-equivalent to the original category because the
excluded weight pieces have zero cohomology. Locally finite
semisimple-weight decomposition is exact, so this assertion also holds
for unbounded morphism complexes.
Use the direct-sum realization of its twisted cyclic bar complex.
Every chain belongs to a finite sum of finite-dimensional cyclic
subrepresentations. On a finite permutation-stable list of labels,
take a common power fixing the labels; divided absolute weights for
that power define the same nonpositive grading on the bar terms.
Only the weights are read using this power: the functor and graph
whose trace is computed remain $\Phi$ and $G_\Phi$.
In the identified left-graph convention, cyclic rotation has the
one-algebra formula
$t_n(a_0\mid\cdots\mid a_n)
=(\Psi(a_n)\mid a_0\mid\cdots\mid a_{n-1})$,
with the usual Koszul signs for graded factors.
Its $(n+1)$-fold iterate applies $\Psi$ to every factor.
The usual paracyclic homotopy therefore gives
\begin{equation}\label{tr:paracyclic}
dH+Hd=1-T.
\end{equation}
Here $d$ is the total Hochschild and internal differential, and the
operator $T$ is simultaneous forward transport in the identified
$G_\Psi$ coordinates, or equivalently the ordinary-arrow transport
and \eqref{tr:graph-transport} on the original graph. The identity can be obtained
by inserting the unit and summing the cyclic rotations: consecutive
faces cancel in pairs, and the two remaining faces are the identity
and a full rotation. Transporting this identity through
\eqref{tr:graph-identification} gives the asserted identity on the
original $\Phi$-twisted bar. In degree zero the one-algebra convention
makes it explicit: with $d(m\mid a)=ma-\phi(a)m$, the term
$H_0(m)=\alpha^{-1}\mid\alpha m$ satisfies
$dH_0(m)=m-T_G(m)$.
Since $T$ commutes with $d$, the operator
\begin{equation}\label{tr:inverse-homotopy}
H_{\mathrm{inv}}=-T^{-1}H
\qquad\text{satisfies}\qquad
dH_{\mathrm{inv}}+H_{\mathrm{inv}}d=1-S.
\end{equation}
On every strictly negative-weight block, $1-S$ is invertible.
This inverse is algebraic: on each finite-dimensional cyclic
subrepresentation it is a polynomial in $S$, by its minimal
polynomial, and the resulting inverses agree on inclusions.
There is no infinite geometric series of chains.
The homotopy $(1-S)^{-1}H_{\mathrm{inv}}$ contracts that block.
The homotopy preserves total weight, and the same conclusion holds
when a power is used to read the weights, since every eigenvalue of
$S$ then has modulus strictly less than one.
Only weight zero remains. As every arrow has nonpositive weight, a
weight-zero bar term has only weight-zero arrows. Hence this is
exactly the bar complex of the weight-zero category. Its morphism
cohomology is in degree zero; the canonical differential graded
truncation gives a zigzag of quasi-equivalences to its degree-zero
cohomology category, compatibly with the cyclic action and composition.
A split semisimple category has the same trace as its full subcategory
of simple objects. By the scalar-endomorphism hypothesis, its only
nonzero morphisms are scalar endomorphisms. The twisted bar therefore gives $\C$ in degree
zero for a fixed simple and zero for a permuted block.
Finally, any chain uses only finitely many labels. Enlarge them to a
finite permutation-stable list and apply the preceding argument.
Filtered union gives the assertion for all labels. All realizations
are direct sums; neither an infinite product nor convergence of an
infinite Hochschild spectral sequence has been used.
\end{proof}
\begin{proposition}[Trace of an orbit]\label{tr:orbit-trace}
The trace on the supported category of a Frobenius-stable nilpotent
orbit is concentrated in degree zero. A basis is given by the classes
of $M_P$ for Frobenius-fixed simple $P\in\mathsf E_e$, with chosen
pure normalized structures. Every basis class lifts to an actual
compact Weil object supported on the orbit closure, and also to one
with invariant null-fiber support up to a nonzero scalar.
The same conclusions hold for the compact opposite category with
simultaneous Frobenius.
\end{proposition}
\begin{proof}
By Proposition~\ref{orb:generation}, the full subcategory on the simple
labels generates the supported category; Morita invariance permits
computing its trace on this generating subcategory. We first check
the local finiteness required by Lemma~\ref{tr:bar-contraction}, since
numerical weight bounds on cohomology alone would not suffice.
All models used in Section~\ref{orb:section} are algebraic equivariant
models with finite-dimensional starting representation data. Keep
the grading internal until direct-sum totalization. For a finite
cyclic list of labels, take a common finite-type algebraic cover $H$
of the equivariance and cyclic chain-map data, acting on the chosen
objects and the invariant quasi-compact
separated ambient open $U$. Polynomial functions, finite Koszul
resolutions, equivariant restriction, and proper pushforward are
defined in rational representations of this group.
One can compute derived sections without choosing a
transporter-stable affine cover. Equivariant derived pushforward
along $[U/H]\to BH$ produces complexes of rational
$H$-representations. Flat base change along $\operatorname{pt}\to BH$
recovers ordinary derived sections on $U$, and the lax monoidal
pushforward supplies their composition maps
\cite[\S1.3.10, Corollary~1.4.5, and \S\S6.4.3--6.4.6]{DGFiniteness}.
Here differential Hom is an ordinary square-zero complex because
the two curvature terms cancel. Functorial bar--cobar rectification
uses direct sums of finite tensor words, so it stays in rational
representations, preserves the cyclic action, and gives a compatible
differential graded model with strict composition
\cite[Proposition~2.1 and \S\S4.3, 4.8, 5.2]{KellerAInfinity}.
Restriction to the algebraic closure of the cyclic operator makes
the action on this model locally finite. Stabilizer invariants are
computed by the finite Lie-cochain complex followed by exact
reductive invariants. These bounded constructions commute with the
direct-sum folding of the grading. Thus the locally finite cyclic
actions are defined on morphism complexes with their composition,
and decomposition by the characters of the semisimple part gives
the weight subcomplexes used in the contraction lemma.
Proposition~\ref{orb:weil-lifts} provides finite cyclic orbits of the
labels and consistently pure implementing maps, including when a
power is required to fix a label. On taking invariants, the
triple translation is by a group element and does not change the
actual arrow action $S$. Proposition~\ref{orb:hom-calculation}
therefore says that this model has no positive-weight cohomology and
that its zero-weight category is exactly the semisimple category
$\mathsf E_e$. Its simple objects have scalar endomorphisms: each is
a finite-dimensional complex module, so Schur's Lemma applies over
the algebraically closed field $\C$.
Lemma~\ref{tr:bar-contraction} proves concentration
and the stated basis. The incidence construction of
Proposition~\ref{orb:weil-lifts} supplies its actual compact
orbit-closure representatives; Lemma~\ref{tr:invariant-support}
supplies the completed representatives.
For the opposite category use the same object models, with reversed
arrows and inverse object structures. Its simultaneous forward
action on a reversed arrow is the forward action on the underlying
original arrow; its inverse is consequently the same contracting
$S$ on that underlying complex. The weight calculation and
paracyclic identity are unchanged. In particular, this argument
uses the autoequivalence $\Phi^{\mathrm{op}}$ of the opposite
category, not the inverse autoequivalence of the original category.
It proves the same spanning assertion with actual inverse Weil
structures.
\end{proof}
\subsection{Finite support filtrations and the full function space}
\begin{theorem}[Concentration, injection, and Weil spanning]\label{tr:spanning}
On a fixed retained spectral component, the trace of every closed
invariant nilpotent support category is concentrated in degree zero.
If $\Omega'\subset\Omega$ are such support bounds, then
\begin{equation}\label{tr:support-injection}
H^0\Tr(\Phi,\operatorname{Ind}\mathscr B_{\Omega',0})
\longrightarrow
H^0\Tr(\Phi,\operatorname{Ind}\mathscr B_{\Omega,0})
\end{equation}
is injective. A finite filtration by invariant unions of orbits has
the orbit traces as its actual successive quotients. Every trace
class is a finite linear combination of classes of compact objects
with Weil structures; those objects can be chosen with support in the
prescribed bound. All assertions hold on the compact opposite side.
After assembling retained components, these statements apply to
$\cC_Y$ and $\cC$. In particular, the image defining $\cF_Y\cA_{\E}$
is the injective image of the supported trace, and every function in
$\cA_{\E}$ and every vector in $\cF_Y\cA_{\E}$ is a finite linear
combination of the corresponding actual compact Weil trace functions.
The analogous statement supplies all tests through the
nonstandard self-duality of Theorem~\ref{inp:duality}.
\end{theorem}
\begin{proof}
Choose a finite filtration of the nilpotent support by closed invariant
unions of $J$-orbits, grouping Frobenius permutations into stable strata.
It exists because there are finitely many nilpotent orbits and
Frobenius preserves their closure order. At each step,
Lemma~\ref{orb:localization} gives a compactly generated localization.
Twisted Hochschild homology is localizing, so it gives a cofiber
sequence of traces; this is the trace-localization formalism of
\cite[Theorem~3.4 and Proposition~5.4]{HSS}, applied to the actual
Frobenius-compatible localization just constructed.
Disjoint orbits closed in the corresponding open
have orthogonal supported categories. If Frobenius permutes such
summands without a fixed summand, their twisted trace is zero: in the
bar model, nonzero chains would have to begin and end in the same
summand, while the graph coefficient ends in its Frobenius translate.
Every fixed orbit has the degree-zero trace of
Proposition~\ref{tr:orbit-trace}.
Induction up this finite filtration proves concentration in degree
zero. Each cofiber sequence is therefore a short exact sequence on
$H^0$. More generally, apply the same argument to the finite orbit
filtration of the complement of $\Omega'$ in $\Omega$. Its trace is
also concentrated in degree zero, so the localization sequence starts
\[
0\longrightarrow H^0\Tr(\Phi,\mathscr B_{\Omega',0})
\longrightarrow H^0\Tr(\Phi,\mathscr B_{\Omega,0})
\longrightarrow H^0\Tr(\Phi,\mathscr B_{\Omega,0}/
\mathscr B_{\Omega',0})
\longrightarrow0,
\]
where taking ind-completions in the trace notation is understood.
This proves \eqref{tr:support-injection} and identifies the successive
quotients with the computed orbit traces. It does not infer
injectivity from concentration of the full category alone.
At an induction step, each quotient basis class lifts by the compact
incidence construction and the finite invariant Koszul construction.
Subtract a finite linear combination of these lifts from a given
class; the remainder lies in the preceding supported trace.
The induction terminates after finitely many strata. Thus every
class is spanned by actual compact Weil classes with the required
closed support. This argument also proves spanning on the opposite
side; its localization sequences and orbit lifts have already been
checked in Proposition~\ref{tr:orbit-trace}.
For the retained union of components, compact objects involve only
finitely many components and there are no morphisms between distinct
components. The same bar argument shows
\begin{equation}\label{tr:component-sum}
\Tr(\Phi,\cC)
=\bigoplus_{\substack{Z_a\subset Z\text{ retained}\\
\Phi Z_a=Z_a}}
\Tr(\Phi,\cC|_{Z_a}),
\end{equation}
and likewise for supported categories and their compact opposites.
A component permuted nontrivially contributes zero even if a power
fixes it. The direct sum in \eqref{tr:component-sum} is essential:
no vector acquires infinitely many component contributions.
The Frobenius-compatible restricted equivalence of
Theorem~\ref{inp:langlands} transports these results to the categories
in the problem, retaining exactly the prescribed prime union.
The trace/functions isomorphism and ordinary compact local-term
compatibility of Theorem~\ref{inp:trace} take each compact Weil class
to its actual finitely supported trace function. This identifies the
supported trace injection with the image used to define
$\cF_Y\cA_{\E}$. Finally the simultaneous-Frobenius nonstandard duality
identifies the trace on the compact opposite category with the full
space of tests. The opposite spanning statement therefore supplies
every such test by a finite linear combination of the constructed
classes. All calculations were performed after an arbitrary
coefficient isomorphism with $\C$; concentration, injections, and finite
spanning transport back along that isomorphism.
\end{proof}
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/05-traces.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/06-measures.tex ====
\section{Hecke matrix coefficients and support detection}
\label{meas:section}
Fix an abstract coefficient isomorphism $\E\simeq\C$ and a closed point
$v\in X$ of degree $d$. Put $r=q^d$. We use the compact sets $B_o$ and
the bilinear pairing $[-,-]$ of \eqref{intro:compactsets} and
\eqref{intro:pairing}. Write
\[
B=\bigcup_{o\subset\Nhat}B_o,
\qquad B_Y=\bigcup_{o\subset Y}B_o.
\]
There are finitely many ambient nilpotent orbits. Lemma
\ref{desc:compactsets}, whose proof is independent of this section, says
that the $B_o$ are disjoint compact sets and that representation characters
are uniformly dense in $C(B)$. We will use this elementary fact for
uniqueness and for separating different orbit contributions.
\begin{theorem}[Bilinear support detector]\label{meas:detector}
For every $f,g\in\cA_{\C}$ there is a unique finite complex Borel measure
$\nu_{f,g}$ on $B$ such that, for every finite-dimensional representation
$D$ of $\Ghat$,
\begin{equation}\label{meas:moments}
[T_Df,g]=\int_B\chi_D\,d\nu_{f,g}.
\end{equation}
For every closed invariant subset $Y\subset\Nhat$,
\begin{equation}\label{meas:criterion}
f\in\cF_Y\cA_{\C}
\quad\Longleftrightarrow\quad
\operatorname{supp}(\nu_{f,g})\subset B_Y
\quad\text{for every }g\in\cA_{\C}.
\end{equation}
\end{theorem}
The theorem concerns ordinary finitely supported functions. In its proof,
infinite sums occur only in auxiliary character calculations on compact
groups. Every automorphic input and test remains a finite linear
combination of compact Weil classes.
\subsection{The bilinear Hom formula}
We first identify exactly which scalar trace is being calculated. Use the
nonstandard self-duality and compactness of Theorem~\ref{inp:duality}
\cite[Theorem~3.2.2 and Appendix~A.1]{AGKRRV2}.
On the automorphic side its evaluation is
\[
(\mathcal F,\mathcal G)\longmapsto
\Gamma_c\bigl(\Bun_G,\mathcal F\mathbin{\overset{*}{\otimes}}
\mathcal G\bigr).
\]
On compact objects the categorical dual is the opposite category.
Simultaneous invariance under $\Phi$ identifies its action there with
$\Phi^{\mathrm{op}}$. Thus, on the spectral side, take a covariant class
$b:N\longrightarrow\Phi N$ and an opposite-input class
$a:\Phi M\longrightarrow M$. Evaluation acts on an arrow $h:M\to N$ by
\begin{equation}\label{meas:hom-action}
S_{M,N}(h)=\Phi^{-1}(b\circ h\circ a).
\end{equation}
The action with a Hecke insertion includes its specified cyclic Weil
transport. Formula~\eqref{meas:hom-action} is the inverse transported action
$S$ of Section~\ref{mod:section}; it uses the original $\Phi$-trace.
\begin{lemma}\label{meas:hom-functions}
Let $N$ and $M$ be compact spectral objects with the structures just
specified, and let $f_N$ and $g_M$ be their automorphic trace functions,
where $M$ is transported through the categorical duality. Then
\begin{equation}\label{meas:hom-function-formula}
[T_Df_N,g_M]
=\operatorname{str}\!\left(
S_{M,N,D};\operatorname{RHom}(M,\mathcal E_{D,v}\otimes N)\right).
\end{equation}
Here $\mathcal E_{D,v}$ denotes the evaluation insertion at the cyclic tuple
above $v$. The scalar evaluation complex is perfect. Finite linear
combinations of these classes give all inputs and all tests in
$\cA_{\C}$. Inputs in $\cF_Y\cA_{\C}$ may be represented by such $N$ with
the prescribed $Y$-support.
\end{lemma}
\begin{proof}
The compact embedding into ambient sheaves, preservation of compactness
by the star tensor product in two compact variables, and preservation by
the finite Hecke insertion imply that the displayed compact-cohomology
evaluation is perfect. Compact pushforward to a point preserves
compactness; its target is the category of complexes of vector spaces,
whose compact objects are precisely the perfect complexes. These are the
compactness assertions recorded with the duality input in
Section~\ref{inp:section}. They are needed in addition to abstract
self-duality.
The ordinary local-term comparison of Theorem~\ref{inp:trace} and the
sheaf--function dictionary
identify the Frobenius trace of this evaluation with
\[
\sum_{b\in\Bun_G(\mathbf F_q)/\cong}
\frac{(T_Df_N)(b)g_M(b)}{|\Aut(b)|}.
\]
For each fixed Hecke label the correspondences are bounded, and the compact
inputs have quasi-compact $!$-support. Thus the compact local-term theorem
applies to precisely this evaluation. Transport through the duality gives
\eqref{meas:hom-action}, proving
\eqref{meas:hom-function-formula}. Finally, Theorem~\ref{tr:spanning}
gives actual compact Weil-class spanning on both the covariant and opposite
sides, and on every closed-support trace image. Combined with the
trace--functions isomorphism, it proves the last assertions.
\end{proof}
The duality need not preserve individual spectral components or their
support labels. We choose $M$ on the Hom side and then obtain its function
test by duality. Distinct components are orthogonal because their spectral
Hom complexes vanish.
\subsection{Supported traces on an orbit}
Fix a retained Frobenius-stable component and the model of
Theorem~\ref{mod:model}. In the curved description, write
$W(x,z)=\langle z,\mu(x)\rangle$. For two compact lifts let
$\mathcal H=\mathcal H\!om(M,N)$ be their differential Hom, computed with
locally free representatives. Its differential squares to zero.
Its derived $J$-invariant sections, with the evaluation insertion, compute
the right side of \eqref{meas:hom-function-formula}. The support of
$\mathcal H$ lies in the intersection of the two factorization supports:
where either object vanishes in the local factorization category, the
differential Hom is acyclic.
We filter these sections by \emph{local cohomology with supports} in the
nilpotent $z$-orbits. Ordinary restriction to an orbit would give the wrong
answer on boundary strata. The following calculation fixes both the
normal-direction sign and the convergence issue.
For the motivating use of local cohomology, normal symmetric powers and
distributional convergence, see Kazhdan--Okounkov
\cite[Sections~2.6.1--2.6.4, equations~(107)--(111)]{KazhdanOkounkov}.
The following calculation supplies the supported convergence needed for
the present bilinear detector.
\begin{lemma}[Supported convergence]\label{meas:supported-convergence}
Let $O=J.e$ be a stable nilpotent orbit, closed in an invariant open subset
of $\mathfrak j$. The contribution with support on $O$ to the Hom
calculation has an absolutely convergent alternating character on the
compact transporter coset of $L_e$. Convergence is uniform on that coset,
also after any fixed finite-dimensional evaluation insertion. These
characters are additive through the finite orbit filtration and compute
the ordinary supertrace of the final perfect evaluation complex.
\end{lemma}
\begin{proof}
Let $i:O\hookrightarrow U\subset\mathfrak j$ be the closed immersion and
$c=\operatorname{codim}_{U}(O)$. It is a regular immersion, with normal
bundle whose fiber at $e$ is $N_e=\mathfrak j/[\mathfrak j,e]$. If $I$ is
its ideal, local cohomology is the filtered colimit
\[
R\Gamma_O(\mathcal H)
=\underset{n}{\operatorname{colim}}\,
R\mathcal H\!om(\mathcal O_U/I^{n+1},\mathcal H).
\]
The successive derived cofibers of this filtration are
\begin{equation}\label{meas:normal-filtration}
i_*\left(Li^*\mathcal H\otimes\Sym^n N
\otimes\det N[-c]\right),\qquad n\geq0.
\end{equation}
Indeed, $I^n/I^{n+1}=\Sym^n N^*$, while regular-immersion duality gives
$i^!\mathcal H=Li^*\mathcal H\otimes\det N[-c]$. One may verify the same
formula directly by the regular Koszul resolution. Each such resolution
is bounded before folding the grading, so it also applies to the locally
free differential Hom used here. In particular the determinant in
\eqref{meas:normal-filtration} is the \emph{normal} determinant.
For example, a degree-two coordinate $z$ with $S(z)=qz$ gives
\begin{equation}\label{meas:one-normal-coordinate}
H^1_{(z)}\C[z]=\bigoplus_{m\geq1}\C z^{-m},
\qquad S(z^{-m})=q^{-m}z^{-m}.
\end{equation}
The total degree of $z^{-m}$ after including local-cohomology degree is
$1-2m$. This checks the normal sign and the shift simultaneously.
Take the fiber at $e$ and then derived $C_e=L_e\ltimes U_e$ invariants.
The cohomology of the restricted differential Hom is a coherent graded
module over $\C[V]$. Its support is contained in $\ker e$. To see the
last assertion directly, differentiate the factorization differential:
if $d_N^2=W$, then
\[
d_N(\partial d_N)+(\partial d_N)d_N=\partial W.
\]
Left composition with $\partial d_N$ is a homotopy for multiplication by
$\partial W$ on differential Hom. At $z=e$ the derivatives in the
$V$-directions give the linear equations $ex=0$. This also proves the
assertion using any finite locally free replacement.
Coherence implies that some power of the ideal of $\ker e$ annihilates
this cohomology. Consequently only \emph{finitely many} polynomial orders
transverse to $\ker e$ in the $x$-space occur. The orders $n$ in
\eqref{meas:normal-filtration}, in contrast, remain unbounded.
After the triple translation, the weights from
Proposition~\ref{orb:hom-calculation} give
\[
\begin{array}{c|c|c}
\text{space}&\text{absolute }S\text{-weight}&\text{bound}\\ \hline
(\ker e\cap V_i)^*&-1-i\quad(i\geq0)&\leq-1\\
(N_e)_i&-2+i\quad(i\leq0)&\leq-2\\
\Lie(U_e)_i^*&-i\quad(i>0)&<0.
\end{array}
\]
Absolute weight $w$ means eigenvalue modulus $q^{w/2}$. Thus the
unrestricted $x$-variables and the normal symmetric powers both contract.
The finitely many transverse $x$-orders, the normal determinant, and the
finite complex of $U_e$-cochains affect only a finite coefficient factor.
Here is a precise summability bound. Choose a finite-dimensional invariant
generating space for the restricted cohomology, including its finite
transverse thickening. The locally finite models constructed in the proof of
Proposition~\ref{tr:orbit-trace} permit this choice for
the algebraic group generated by the normalized transporter and
equivariance. There are constants $C,a,b$ and $0<\rho_x,\rho_z<1$ such
that the sum of eigenvalue moduli, counted with multiplicities, in the
terms of $x$-degree $m$ and normal order $n$ is at most
\begin{equation}\label{meas:geometric-bound}
C(m+1)^a(n+1)^b\rho_x^m\rho_z^n.
\end{equation}
For example one can take $\rho_x=q^{-1/2}$ and $\rho_z=q^{-1}$ after
absorbing the finite coefficient spaces into $C$. Polynomial dimension
growth follows from the finite number of polynomial variables. Subquotients
can only decrease the multiplicity bound on each generalized weight.
By Lemma~\ref{mod:compact-normalization} these estimates are uniform when a
compact element of $L_e$ is included in the transporter. The semisimple
normalized parts preserve a compact form; the residual unipotent commutes
with it. Hence compact twists do not change the contracting moduli of the
linear data. Finite coefficient representations supply only a uniform
constant. This reasoning concerns traces and eigenvalue multiplicities;
it does not require a unipotent operator to be unitary.
For completeness, the bound also justifies passage from the graded pieces
to their colimit. For every $\epsilon>0$, only finitely many pairs $(m,n)$
can have eigenvalues of modulus at least $\epsilon$, since the coefficient
weights lie in a fixed finite set. In the category of locally finite
representations, taking a sum of generalized eigenspaces in such a range
is exact. On that range the filtered calculation therefore stabilizes
after finitely many normal orders, and the ordinary long exact sequences
give trace additivity. Filtered colimits of vector spaces are exact, so
this identifies the same part of the cohomology of the colimit. Finally
\eqref{meas:geometric-bound} is summable in $(m,n)$, allowing
$\epsilon$ to decrease to zero. The argument applies also to the finite
orbit filtration. It uses direct-sum folding of the graded complexes;
bounded Koszul and group-cochain calculations commute with this folding.
If Frobenius permutes a finite set of strata without fixing any, use a
power $S^t$ to make the preceding estimates on each summand. The same
geometric bound is summable after raising each eigenvalue modulus to
any positive power, in particular $1/t$. It therefore gives summability
also for $S$, whose one-step action has zero trace on the resulting
permuted sum. On fixed strata the
preceding calculation applies directly. The full scalar evaluation is
perfect by Lemma~\ref{meas:hom-functions}, so its convergent alternating
character is its ordinary finite supertrace.
\end{proof}
\subsection{Constructing the measures}
Let $L_{e,c}$ be the maximal compact subgroup of $L_e$ chosen in
Lemma~\ref{mod:compact-normalization}, including all components. Denote by
$\tau_e$ the triple-adjusted $S$-action, using that normalized transporter
and retaining the absolute-weight scalars in the displayed table above.
On coefficient modules in lifted degree zero it is the normalized
transporter itself. Expressions on the corresponding compact coset are
written as functions of $k\in L_{e,c}$, with operator $k\tau_e$ on the
space in question. Haar measure $dk$ is probability measure on the entire
compact group. This notation allows $\tau_e$ to act nontrivially on $L_e$.
Projection to reductive invariants is integration over $L_{e,c}$:
\begin{equation}\label{meas:compact-projection}
\operatorname{str}(\tau_e;H^{L_e})
=\int_{L_{e,c}}\operatorname{str}(k\tau_e;H)\,dk.
\end{equation}
For a finite-dimensional representation this is the Reynolds projection,
which commutes with $\tau_e$ because it normalizes the compact group.
Lemma~\ref{meas:supported-convergence} proves the formula for all the
convergent characters here by uniform absolute convergence. It also
applies when $L_e$ is disconnected: averaging over its finitely many
components is part of the same Reynolds projection.
The Hecke insertion contributes a finite-dimensional cyclic trace. For
linear maps around $d$ copies of $D$ one has
\begin{equation}\label{meas:cyclic-trace}
\operatorname{tr}\bigl(\operatorname{cyc}\circ
(A_1\otimes\cdots\otimes A_d);D^{\otimes d}\bigr)
=\operatorname{tr}(A_d\cdots A_1;D).
\end{equation}
Expanding in a basis proves the identity: the cyclic identification of
indices leaves precisely the matrix entries of the indicated product.
The framed evaluation description in Theorem~\ref{mod:model} applies
this identity to the geometric points above $v$. The triple translations
contribute $\lambda_e(q^{d/2})$, and the residual semisimple product is
compact and centralizes the triple. Unipotent factors may be discarded
in this trace. We obtain a continuous map
\begin{equation}\label{meas:parameter-map}
\beta_e:L_{e,c}\longrightarrow B_{\Ghat.e}.
\end{equation}
Any lifted-degree sign on the insertion contributes
$\lambda_e(-1)$ to the compact part. This is already included in
$B_{\Ghat.e}$. Contravariant character conventions give the inverse
parameter, which lies in the same compact set by
Lemma~\ref{desc:compactsets}.
Remove the insertion and write $a_{M,N,e}(k)$ for the alternating character
density of the supported contribution. The lifted-degree sign may be
used inside \eqref{meas:compact-projection}: after taking invariants it is
the true cohomological sign, as established in
Proposition~\ref{orb:hom-calculation}. The density is continuous and
absolutely integrable by Lemma~\ref{meas:supported-convergence}.
The contribution of $O$ is therefore the finite complex measure
\begin{equation}\label{meas:orbit-measure}
(\beta_e)_*\bigl(a_{M,N,e}(k)\,dk\bigr).
\end{equation}
Its total variation is bounded by $\int|a_{M,N,e}|\,dk$. Formula
\eqref{meas:cyclic-trace} shows that its $\chi_D$-moment is exactly the
orbit contribution with the Hecke insertion.
\begin{proposition}\label{meas:common-support}
Suppose $M,N$ belong to one retained component and have closed
$z$-support bounds $Z_M,Z_N\subset\mathfrak j$. Their measure satisfies
\begin{equation}\label{meas:common-support-bound}
\operatorname{supp}(\nu_{f_N,g_M})
\subset
\bigcup_{e\in Z_M\cap Z_N}B_{\Ghat.e}.
\end{equation}
\end{proposition}
\begin{proof}
Sum \eqref{meas:orbit-measure} over the finitely many stable nilpotent
orbits in the common support. Lemma~\ref{meas:supported-convergence}
gives additivity, and Lemma~\ref{meas:hom-functions} identifies all its
moments. Orbits outside the common support give zero differential Hom;
permuted orbits give zero trace. The support bound follows from
\eqref{meas:parameter-map}. Uniqueness follows from the density of
characters in $C(B)$ in Lemma~\ref{desc:compactsets}.
\end{proof}
By Theorem~\ref{tr:spanning}, every pair of functions is a finite linear
combination of the pairs just considered. Each vector uses only finitely
many retained fixed components, because the categorical trace is the
direct sum on such components. Cross-component Homs vanish. Adding the
finite measures above therefore proves existence in
Theorem~\ref{meas:detector} for every $f,g$, and character density proves
uniqueness independent of the chosen expansion. The construction is
bilinear. Proposition~\ref{meas:common-support} already proves the forward
implication in \eqref{meas:criterion}.
\subsection{Nondegeneracy of the top-orbit contribution}
The decisive test concerns total mass. Canceling a common nonvanishing
density will isolate a simple coefficient, so injectivity of the
parameter map $\beta_e$ is unnecessary.
We must also show that a nonzero quotient class cannot be hidden by
cancellation of its measure. Fix a stable orbit $O=J.e$. Let $P,Q$ be
finite-dimensional equivariant $D_e$-modules, with compatible cyclic
structures, and take the compact orbit-closure lifts of
Proposition~\ref{orb:weil-lifts}. We put the test coefficient $Q$ first
and the input coefficient $P$ second in a Hom.
On an open subset where $O$ is closed and the proper boundary of its
closure has been removed, both lifts are already supported on $O$.
Consequently local cohomology with that support is the actual Hom
complex. There is no additional series of inverse normal powers to put
into its Euler character. Proposition~\ref{orb:hom-calculation} gives
\begin{equation}\label{meas:top-density}
a_{Q,P,e}(k)=p_e(k)\,
\operatorname{tr}\bigl(k\tau_e;\Hom_{D_e}(Q,P)\bigr),
\qquad k\in L_{e,c}.
\end{equation}
The action on the Hom uses the inverse structure on the test as in
\eqref{meas:hom-action}. The factor $p_e$ is the graded Euler character of
\begin{equation}\label{meas:universal-density-factors}
\C[\ker e]\otimes
\Lambda^\bullet(N_e[-1])\otimes
\Lambda^\bullet(\Lie(U_e)^*[-1]).
\end{equation}
The identification of constant Clifford super-intertwiners with ordinary
intertwiners, and its compatibility with lifted parity and tensor twists,
are part of Proposition~\ref{orb:hom-calculation}.
The invariant-base Koszul complexes used to obtain actual compact lifts
do not alter \eqref{meas:top-density} except by constant nonzero factors.
Indeed, expand their finite exterior terms by additivity in each input.
Their equation spaces are $J$-invariant, so these factors are independent
of $k$; their Frobenius eigenvalues and inverse eigenvalues are never one.
The exterior traces are therefore nonzero, as in
Lemma~\ref{tr:invariant-support}. We rescale the trace classes by these
constants. This permits computation first on the uncompleted orbit model
while retaining actual compact tests throughout.
\begin{lemma}[Cancellation of the universal density]
\label{meas:cancel-density}
The function $p_e$ is continuous and nowhere zero on $L_{e,c}$. Its
reciprocal is a uniform limit of finite complex linear combinations of
twisted characters of finite-dimensional $L_e$-representations with
compatible transporter structures. Such a twisted character can be
inserted in \eqref{meas:top-density} by tensoring the test coefficient with
the dual representation, in lifted degree zero.
\end{lemma}
\begin{proof}
Decompose each of the finite-dimensional spaces in
\eqref{meas:universal-density-factors} by lifted degree. The symmetric
algebra contributes inverse determinants, and each exterior algebra
contributes determinants. Thus $p_e$ is a finite product of terms
\begin{equation}\label{meas:determinant-factor}
\det(1-\varepsilon kA;W)^{\eta},
\qquad \varepsilon\in\{1,-1\},\quad \eta\in\{1,-1\},
\end{equation}
where $W$ is a finite-dimensional $L_e$-representation with transporter
$A$, and all eigenvalues of $kA$ have modulus strictly less than one,
uniformly in $k$. The sign $\varepsilon$ incorporates the specified
lifted-degree shift. The strict inequality follows from the weight
bounds: all factors outside the constant coefficient Hom have negative
absolute weight. Each determinant in \eqref{meas:determinant-factor} is
therefore nonzero, proving the first assertion.
The identities
\[
\det(1-A;W)=\sum_{j=0}^{\dim W}(-1)^j
\operatorname{tr}(A;\Lambda^jW),
\qquad
\det(1-A;W)^{-1}=\sum_{n\geq0}
\operatorname{tr}(A;\Sym^nW)
\]
apply also with the sign $\varepsilon$. The second series converges
uniformly here: its $n$th term is bounded by
$\dim(\Sym^nW)\rho^n$ for a fixed $\rho<1$. Finite products of these
identities express $1/p_e$ as the required uniform limit. Every finite
term is the twisted character of a finite tensor product of symmetric and
exterior powers of the allowed representation data; scalar factors may
be put into its coefficient or cyclic structure.
Finally,
\[
\Hom_{D_e}(Q\otimes R^*,P)
\simeq R\otimes\Hom_{D_e}(Q,P)
\]
with the induced cyclic actions. Under the parity identification already
used for \eqref{meas:top-density}, its character multiplies that density
by the twisted character of $R$. The extension construction in
Proposition~\ref{orb:weil-lifts} applies to $Q\otimes R^*$ as well. Each
finite approximant is consequently realized by a finite linear
combination of permitted compact test classes.
\end{proof}
\begin{lemma}[Top quotient detection]\label{meas:top-detection}
Every nonzero finite linear combination of fixed-simple classes in the
$O$-quotient is detected by a compact test supported on $\overline O$:
its top-orbit measure has nonzero total mass on $B_{\Ghat.e}$.
\end{lemma}
\begin{proof}
Write the quotient class as $u=\sum_P c_P[P]$, with finitely many nonzero
coefficients. Suppose its top mass were zero against every permitted test
lift. Fix a test coefficient $Q$. By tensoring $Q$ with the duals of the
representations in Lemma~\ref{meas:cancel-density}, the assumption gives
\[
\int_{L_{e,c}}p_e(k)h(k)
\sum_P c_P\operatorname{tr}
\bigl(k\tau_e;\Hom_{D_e}(Q,P)\bigr)\,dk=0
\]
for every finite character combination $h$ used there. Letting these
$h$ converge uniformly to $1/p_e$ gives
\begin{equation}\label{meas:unweighted-coefficient-pairing}
0=\sum_P c_P\operatorname{tr}
\bigl(\tau_e;\Hom_{D_e,L_e}(Q,P)\bigr).
\end{equation}
This is legitimate integration against a fixed continuous function, not
a limit of automorphic functions.
The equivariant Clifford coefficient category is semisimple. Choose
$Q$ to be any fixed simple appearing in $u$, with the matching structure.
For $P\not\simeq Q$ its Hom is zero; for $P\simeq Q$ it is the scalar
endomorphism line and the action is the identity. Thus
\eqref{meas:unweighted-coefficient-pairing} says $c_Q=0$. Repeating this
contradicts $u\ne0$. Some actual finite test must therefore have nonzero
top mass. The argument requires no injectivity of
$\beta_e:L_{e,c}\to B_{\Ghat.e}$: the total mass alone suffices.
\end{proof}
\subsection{Excluding boundary cancellation}
\begin{lemma}\label{meas:boundary-rank}
If $e'\in\overline{J.e}\setminus J.e$, then
$\dim(\Ghat.e')<\dim(\Ghat.e)$. In particular the two ambient nilpotent
orbits, and their compact sets, are distinct.
\end{lemma}
\begin{proof}
Since $J$ is reductive in characteristic zero, choose a $J$-stable
complement $W$ in $\ghat=\mathfrak j\oplus W$. The adjoint rank on
$\mathfrak j$ is the dimension of the $J$-orbit and strictly drops on a
proper boundary orbit. The rank on $W$ cannot increase under
specialization, because the condition that a matrix have rank at most a
given integer is closed. Hence
\[
\rk(\operatorname{ad}(e')|\ghat)
<\rk(\operatorname{ad}(e)|\ghat).
\]
These ranks are the ambient orbit dimensions. The same argument works
for a disconnected $J$, whose orbits are finite unions of orbits of its
identity component. Disjointness of the compact sets follows from
Lemma~\ref{desc:compactsets}.
\end{proof}
We finish the converse in Theorem~\ref{meas:detector}. Suppose
$f\notin\cF_Y\cA_{\C}$. By Theorem~\ref{tr:spanning}, choose a
retained fixed component on which its trace class does not lie in the
supported trace for $Y\cap\mathfrak j$. Starting with this closed subset,
filter the nilpotent cone in $\mathfrak j$ by closed invariant subsets,
adding one Frobenius orbit of $J$-orbits at a time, compatibly with
closure. Write $T_0\subset T_1\subset\cdots\subset T_m$ for the resulting
supported trace subspaces, with $T_0$ the trace for $Y\cap\mathfrak j$.
Let $i$ be the least index for which the
component class lies in $T_i$. Then $i>0$, and its image
$u\in T_i/T_{i-1}$ is nonzero. A nontrivially permuted stratum has zero
trace, so this step adds a single Frobenius-stable orbit $O=J.e$ outside
$Y\cap\mathfrak j$.
Express $u$ as a finite combination of the fixed-simple classes in the
$O$-quotient, and choose their compact orbit-closure lifts as in the proof
of Theorem~\ref{tr:spanning}. Subtract their sum from the component
class. The remainder belongs to $T_{i-1}$ and, by the same theorem,
is represented by a finite combination of compact Weil classes with
that earlier support. Apply
Lemma~\ref{meas:top-detection} to $u$ and these lifts to obtain a compact
test supported on $\overline O$ with nonzero top mass. The earlier
support bound defining $T_{i-1}$ meets $\overline O$ only in
$\overline O\setminus O$. Proposition~\ref{meas:common-support} and
Lemma~\ref{meas:boundary-rank} therefore show that the remainder,
and the proper-boundary contributions of the chosen lifts, have zero
mass on $B_{\Ghat.e}$. This also excludes contributions from incomparable
$J$-orbits in the same ambient nilpotent orbit. Other components give
zero Hom. Therefore
\[
\nu_{f,g}(B_{\Ghat.e})\ne0
\]
for the resulting genuine finitely supported function test $g$.
Since $e\notin Y$, the invariant set $Y$ does not contain its ambient
orbit, and $B_{\Ghat.e}$ is disjoint from $B_Y$. The asserted support
condition fails. This proves the converse and completes the proof of
Theorem~\ref{meas:detector}.
No step used a rational point of $X$, a cuspidality condition, or
Hecke-finiteness. Empty and full $Y$ are included, and the closed point
$v$ may have any positive degree.
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/06-measures.tex ====
==== BEGIN Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/07-descent.tex ====
\section{Compact spectral sets and rational descent}\label{desc:section}
We first verify the elementary properties of the sets in
\eqref{intro:compactsets}. These properties supply the uniqueness used in
Theorem~\ref{meas:detector}; their proof is independent of that theorem.
\begin{lemma}\label{desc:compactsets}
There are finitely many sets $B_o$. Each is compact and independent of
the triple and compact form used to define it, and distinct orbits give
disjoint sets. The algebra of representation characters restricted to
$B=\bigcup_oB_o$ contains the constants, separates points and is closed
under complex conjugation. It is uniformly dense in $C(B,\C)$.
Each $B_o$ is preserved by inversion and by multiplication by a
finite-order central element of $\Ghat_\C$.
\end{lemma}
\begin{proof}
Nilpotent orbits in a complex semisimple Lie algebra are finite in number.
Triples for a fixed orbit are conjugate, and maximal compact forms of a
complex reductive group, including its components, are conjugate.
Consequently their images in the affine conjugacy quotient give the same
$B_o$. The image of a compact group under the displayed continuous map is
compact. The finite union $B$, as a compact subspace of a complex affine
variety, is metrizable.
Choose a maximal torus $T\subset\Ghat_\C$. Polar decomposition in
$T(\C)$ gives
\[
T(\C)=T(\mathbb R_{>0})\times T_{\cpt},
\]
where the positive factor is intrinsically described by the absolute
values of all characters. This decomposition is Weyl-equivariant.
The centralizer of the torus $\lambda_e(\Gm)$ in $\Ghat_\C$ is a
connected reductive Levi subgroup. It contains $s$, and a maximal
torus of this Levi containing the semisimple element $s$ also contains
its central torus $\lambda_e(\Gm)$. Thus $\lambda_e$ and $s$ can be
placed in a common maximal torus. The positive part of
$\lambda_e(\sqrt r)s$ is exactly $\lambda_e(\sqrt r)$. Its Weyl orbit
determines the dominant cocharacter $\lambda_e$: evaluation of
cocharacters at $\sqrt r>1$ is injective.
For completeness, that cocharacter determines the nilpotent orbit
without forgetting any distinction in type~D. Fix $\lambda$ and write
$\ghat_i$ for its weight spaces. Whenever a triple has diagonal
cocharacter $\lambda$, the $\mathfrak{sl}_2$ decomposition gives
\[
[\ghat_0,e]=\ghat_2.
\]
Indeed, the raising map from weight zero to weight two is surjective
on each irreducible $\mathfrak{sl}_2$ summand. Thus the centralizer of
$\lambda$ has an open orbit through $e$ in the irreducible vector space
$\ghat_2$. Two such open orbits must coincide. It follows that two
parameters with the same positive part have the same ambient nilpotent
orbit. This proves disjointness; it is also the description underlying
\cite[Lemma~3.1.3]{GLR}.
Representation characters separate semisimple conjugacy values in a
connected complex reductive group: restriction to $T$ identifies their complex
span with $\C[T]^W$ \cite[Theorem~22.38]{MilneAG}.
Hence their restrictions separate points of $B$.
To check complex conjugation, write $a=\lambda_e(\sqrt r)$.
For a representation $D$, $D(a)$ is diagonalizable with positive real
eigenvalues and commutes with $D(s)$; on each of its eigenspaces $D(s)$
is unitary after choosing an invariant Hermitian form for the compact
centralizer. Therefore
\[
\overline{\chi_D(as)}=\chi_D(as^{-1}).
\]
A Weyl element in the triple's $\mathrm{SL}_2$ conjugates $a$ to
$a^{-1}$ and commutes with $s$. Thus
\[
\chi_D(as^{-1})=\chi_D(a^{-1}s^{-1})=\chi_{D^\vee}(as).
\]
The same representation $D^\vee$ works simultaneously on every $B_o$.
The character algebra is consequently self-adjoint, and the complex
Stone--Weierstrass Theorem proves its density on the finite compact union.
The triple Weyl element also shows that $(as)^{-1}$ is conjugate to
$as^{-1}$, proving invariance under inversion. A finite-order central
element lies in the chosen compact triple centralizer: adjoining it
gives a compact group, and maximality leaves that group unchanged.
Multiplication by it therefore preserves $B_o$.
\end{proof}
\begin{corollary}\label{desc:unique-measure}
A finite complex Borel measure on $B$ is determined by its integrals
against all representation characters.
\end{corollary}
\begin{proof}
Integration against a finite complex measure is a continuous functional
on $C(B,\C)$ for the uniform norm. Lemma~\ref{desc:compactsets} makes the
character algebra dense. Equality on the characters therefore gives
equality of the continuous functionals and hence of the measures.
\end{proof}
\subsection{Independence of the coefficient isomorphism}
All algebraically closed characteristic-zero coefficient fields below
are considered as abstract fields. Both $E$ and $\C$ have transcendence
degree $2^{\aleph_0}$ over $\Q$, so there are field isomorphisms
$E\simeq\C$. No continuity or compatibility with a selected embedding of
the algebraic numbers is used.
\begin{proposition}\label{desc:coefficient-independence}
For a fixed closed invariant support $Y$, the subspace
$\iota_*(\cF_Y\cA_E)\subset\cA_\C$ is independent of the abstract
isomorphism $\iota:E\xrightarrow{\sim}\C$, after matching nilpotent
orbits by the split root datum. Consequently $\cF_Y\cA_E$ is invariant
under every automorphism of $E$ fixing $\Q$.
\end{proposition}
\begin{proof}
The point functions identify every transported ambient space with the
same $\C^{(S)}$. The pinned split form identifies representation
characters by highest weight across coefficient fields. The pairing \eqref{intro:pairing} has rational
coefficients. Spherical Hecke correspondences count the same finite-field
modifications under each transport. Their normalization may be chosen
on the complex side using the fixed positive $\sqrt q$, so their
operators $T_D$ are the same for each $\iota$.
If a transported choice of $\sqrt q$ differs from this one, set
\[
z=(2\rho_G)(-1)\in Z(\Ghat)[2],
\]
where $2\rho_G$ is the sum of the positive roots of $G$, viewed as a
cocharacter of its dual group. The Satake parity convention at the
degree-$d$ point introduces $z^d$: a highest weight $\mu$ acquires the
sign $(-1)^{d\langle 2\rho_G,\mu\rangle}$, which is exactly its central
character at $z^d$. This convention is also encoded by the modified
dual group in \cite[Remark~6.7 and Section~6.4]{RicharzScholbach}.
On characters the change is therefore translation by $z^d$, and on
moment measures it is the corresponding pushforward. By
Lemma~\ref{desc:compactsets}, this translation preserves each $B_o$.
The contragredient convention similarly preserves each $B_o$ by
inversion. Thus these conventions do not change the support test.
Nilpotent orbits of the split group are identified across these fields
using the fixed pinned $\Q$-form and their triple cocharacters, or
equivalently their weighted Dynkin diagrams. No outer automorphism is
introduced in this identification. In particular the orbit labels in
$Y$ and the compact set
$B_Y$ are fixed independently of $\iota$. Theorem~\ref{meas:detector}
and Corollary~\ref{desc:unique-measure} now characterize the transported
subspace by the same moment equations and the same support condition
for every test $g\in\cA_\C$. The subspaces are equal.
Fix one $\iota$ and let $\tau\in\Aut(E/\Q)$. Comparing $\iota$ with
$\iota\circ\tau$ gives
$\iota_*(\cF_Y\cA_E)=\iota_*\tau_*(\cF_Y\cA_E)$.
Applying $\iota^{-1}$ proves invariance.
\end{proof}
\subsection{Descent on finite coordinate sets}
\begin{lemma}\label{desc:descent}
Let $E$ be an algebraically closed extension of $\Q$, and let
$V_\Q$ have a specified basis, possibly infinite. If
$W\subset E\otimes_\Q V_\Q$ is invariant under every
$\tau\in\Aut(E/\Q)$, then
\[
E\otimes_\Q(W\cap V_\Q)\longrightarrow W
\]
is an isomorphism.
\end{lemma}
\begin{proof}
The fixed field of $\Aut(E/\Q)$ is $\Q$. Indeed, an algebraic element
outside $\Q$ has a different conjugate, and the corresponding embedding
of its number field extends to an automorphism of $E$. A transcendental
element can be included in a transcendence basis; the substitution
$t\mapsto t+1$ on that basis element extends from the rational function
field to its algebraic closure $E$. Thus it too is moved by an
automorphism over $\Q$.
Let $I$ be a finite subset of the specified basis, choose an order on
$I$, and put
\[
W_I=W\cap E^I.
\]
This is an invariant finite-dimensional subspace. Its unique reduced
row-echelon basis matrix is unchanged by every automorphism over $\Q$:
field automorphisms preserve the row space, the pivot positions and the
normalization conditions. Every matrix entry therefore belongs to the
fixed field $\Q$. Its nonzero rows give a rational basis, and hence
\[
W_I=E\otimes_\Q(W_I\cap\Q^I).
\]
Every vector of $E\otimes_\Q V_\Q$ has finite support in the specified
basis. Taking the directed union over $I$ gives the claimed
surjectivity. Injectivity follows by tensoring the inclusion
$W\cap V_\Q\hookrightarrow V_\Q$ with the flat $\Q$-module $E$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{intro:main}]
Proposition~\ref{desc:coefficient-independence} makes
$W=\cF_Y\cA_E$ invariant under all coefficient automorphisms over $\Q$.
Apply Lemma~\ref{desc:descent} to $V_\Q=\cA_\Q$ with its point-function
basis. This is exactly the map \eqref{intro:main-map}.
The argument applies to every closed invariant $Y$, including the empty
set and the full nilpotent cone. The trace calculation uses all retained
fixed components and actual finite linear combinations of Weil trace
classes. The detector allows every finitely supported test function.
Consequently the result holds for the entire specified space, without
any additional finiteness or cuspidality assumption.
\end{proof}
The argument also compares coefficient primes on the full function space
under the same four conditions of Assumption~\ref{intro:characteristic}.
Put $p=\operatorname{char}(\mathbb F_q)$ and
$E_\ell=\overline{\Q}_\ell$ for $\ell\ne p$. Let
$Y_\ell\subset\Nhat_{E_\ell}$ be closed invariant supports with the same
nilpotent-orbit labels under the pinned split dual group. Then, as
subspaces of $\cA_\Q$,
\[
\cA_\Q\cap\cF_{Y_\ell}\cA_{E_\ell}
=\cA_\Q\cap\cF_{Y_{\ell'}}\cA_{E_{\ell'}}
\qquad(\ell,\ell'\ne p).
\]
Indeed, choose the same closed point $v$ and abstract isomorphisms
$E_\ell\simeq\C$. With the positive square-root Satake normalization,
the moment equations for every $f,g\in\cA_\C$ and the compact support
sets in Theorem~\ref{meas:detector} are the same for every prime, as in
the proof of Proposition~\ref{desc:coefficient-independence}. Uniqueness
of the measures gives the same transported filtration in $\cA_\C$.
Intersecting with $\cA_\Q$, which every coefficient isomorphism fixes,
proves the equality. As the matched closed orbit subset varies, these
common rational subspaces form a filtration of $\cA_\Q$ whose
$E_\ell$ scalar extension is the original Arthur filtration, by
Theorem~\ref{intro:main}. This compares rational function subspaces;
it does not identify spectral support categories or individual
eigenobjects.
\subsection{Unramified cuspidal comparison}
\begin{corollary}[Cuspidal comparison across coefficient primes]
\label{desc:cuspidal-comparison}
Keep the curve $X/\mathbb F_q$, the split connected semisimple group
$G/\mathbb F_q$, and all four conditions of
Assumption~\ref{intro:characteristic}. Put
$p=\operatorname{char}(\mathbb F_q)$ and
$E_\ell=\overline{\Q}_\ell$ for $\ell\ne p$. At unramified level $D=0$,
let $C_{0,\Q}\subset\cA_\Q$ be the cuspidal function space and
$C_{0,\mathcal O,\Q}$ its rational orbit summands from
\cite[Theorem~1.1]{CuspidalArthur}, and set
$C_{0,\ell}=E_\ell\otimes_\Q C_{0,\Q}$.
For each $\ell\ne p$, let $Y_\ell\subset\Nhat_{E_\ell}$ be closed and
invariant, with the same nilpotent-orbit subset $\mathscr Y$ under the
fixed pinned split rational dual group. Write $Y=(Y_\ell)_{\ell\ne p}$ and
define
\[
F^{\Q}_{Y,\ell}:=\cA_\Q\cap\cF_{Y_\ell}\cA_{E_\ell},
\qquad
V_{Y,\Q}:=\bigoplus_{\mathcal O\in\mathscr Y}C_{0,\mathcal O,\Q}.
\]
Then
\begin{equation}\label{desc:cuspidal-equalities}
\begin{split}
C_{0,\Q}\cap F^{\Q}_{Y,\ell}&=V_{Y,\Q},\\
C_{0,\ell}\cap\cF_{Y_\ell}\cA_{E_\ell}
&=E_\ell\otimes_\Q V_{Y,\Q}.
\end{split}
\end{equation}
The intersections are taken in the corresponding finitely supported
function spaces. The equalities include the zero cusp space and empty or
full $Y_\ell$.
\end{corollary}
This is precisely the unramified cuspidal common-scope form of
\cite[Conjectures~3.4.9 and~3.4.11]{GLR}. The additional identification
with the companion's orbit summands concerns the $D=0$ cuspidal subspace.
\begin{proof}
For split $G$, the companion's unramified double quotient is the set $S$
of bundle classes \cite[Section~6.1]{CuspidalArthur}, giving the stated
point-function embeddings. Fix $\ell$, an abstract isomorphism
$\iota:E_\ell\xrightarrow{\sim}\C$, and a closed point $v$ with
$r=q^{\deg v}$. The finite reduced global Hecke image on the cusp space
gives finitely many joint eigenspaces \cite[Lemma~6.1]{CuspidalArthur}.
Restrict to the spherical algebra at $v$ and group all joint characters
with the same one-place Satake class. The ordinary Hecke compatibility of
Theorem~\ref{inp:trace} and the companion's volume-one normalization give
\[
C_{0,\C}:=\C\otimes_\Q C_{0,\Q}
=\bigoplus_{t\in\Sigma_v}C_t,
\qquad T_V|_{C_t}=\chi_V(t),
\]
where the $t$ are distinct classes in the positive $\sqrt r$ convention
and $V$ is any finite-dimensional representation of $\Ghat_\C$.
The possible square-root correction is a finite central translate
\cite[Remark~6.3]{CuspidalArthur}, and a contragredient convention gives
inversion; both preserve every $B_{\mathcal O}$ by
Lemma~\ref{desc:compactsets}.
The companion's orbit descent assigns each joint character one orbit
label, independent of the good place and complex embedding, and the complex
scalar extension of each rational orbit summand is the sum of the joint
eigenspaces with that label
\cite[Lemma~6.2 and Propositions~6.5--6.6]{CuspidalArthur}. Every occurring
embedded class of label $\mathcal O$ lies in $B_{\mathcal O}$ by the
all-embeddings compactness defining $R_{r,\mathcal O}$. Since the
$B_{\mathcal O}$ are disjoint, all joint eigenspaces grouped into one
$C_t$ have the same label. Thus $\Sigma_v\subset B$ and
\[
\C\otimes_\Q C_{0,\mathcal O,\Q}
=\bigoplus_{t\in\Sigma_v\cap B_{\mathcal O}}C_t.
\]
For the common support put
$B_Y=\bigcup_{\mathcal O\in\mathscr Y}B_{\mathcal O}$.
Write $f=\sum_t f_t\in C_{0,\C}$. For every $g\in\cA_\C$, the finite
atomic measure $\mu_{f,g}:=\sum_t[f_t,g]\delta_t$ on $B$ satisfies
\[
[T_Vf,g]=\sum_t\chi_V(t)[f_t,g]
=\int_B\chi_V\,d\mu_{f,g}.
\]
Corollary~\ref{desc:unique-measure} identifies $\mu_{f,g}$ with the
detector measure $\nu_{f,g}$. If $f_t\ne0$, choose $b\in S$ with
$f_t(b)\ne0$ and take the point function $g=\delta_b$. Then
\[
\mu_{f,\delta_b}(\{t\})=[f_t,\delta_b]
=\frac{f_t(b)}{|\Aut(b)|}\ne0.
\]
The atoms are distinct, so no other local component cancels this mass.
The test need not be cuspidal. Consequently all detector measures are
supported in $B_Y$ exactly when $f_t=0$ for every $t\notin B_Y$.
Theorem~\ref{meas:detector} and the preceding orbit decomposition give
\[
C_{0,\C}\cap\iota_*(\cF_{Y_\ell}\cA_{E_\ell})
=\C\otimes_\Q V_{Y,\Q}.
\]
Pulling back by $\iota$, which fixes $\Q$, proves the second equality in
\eqref{desc:cuspidal-equalities}. Intersect it with $C_{0,\Q}$ and use
$C_{0,\Q}\cap(E_\ell\otimes_\Q V_{Y,\Q})=V_{Y,\Q}$ to obtain the first.
\end{proof}
==== END Rationality-of-the-Canonical-Unramified-Arthur-Filtration//sections/07-descent.tex ====