==== TABLE OF CONTENTS ====
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//main.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//preamble.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/01-introduction.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/02-categorical.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/03-amplitude.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/04-rotation.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/05-extraction.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/06-global.tex
Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//references.bib
===========================
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Ramanujan--Arthur Decompositions of Cuspidal Functions\\
at Full Finite Level}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove rational and $\overline{\mathbb Q}_\ell$ decompositions of cuspidal
automorphic functions for split semisimple groups over global function
fields into subspaces indexed by nilpotent orbits of the dual group.
The decompositions hold at every full finite level, with arbitrary divisor
multiplicities. Each summand is governed by the same orbit at every
unramified place and every complex embedding. At trivial level this
proves Conjectures~3.4.5 and~3.4.6 of Gaitsgory--Lafforgue--Raskin.
For split connected adjoint absolutely simple groups, we also prove that a
cuspidal automorphic representation with one generic unramified local
component is tempered at every unramified place. Thus globally generic
cuspidal representations in this scope satisfy the unramified part of the
generalized Ramanujan conjecture.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-categorical}
\input{sections/03-amplitude}
\input{sections/04-rotation}
\input{sections/05-extraction}
\input{sections/06-global}
\bibliographystyle{support/alpha-linked}
\bibliography{references}
\end{document}
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//main.tex ====
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//preamble.tex ====
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pdftitle={Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level},
pdfsubject={Finite-level nilpotent-orbit decomposition and unramified temperedness for generic cuspidal representations},
pdfauthor={OpenAI},
pdfkeywords={automorphic forms, generalized Ramanujan conjecture, geometric Satake, nilpotent orbits, shtukas, tempered representations, Whittaker models}
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==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//preamble.tex ====
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
The absolute values of unramified Hecke eigenvalues measure the departure
of an automorphic representation from temperedness. The Ramanujan--Arthur
prediction organizes the departures that occur in the discrete spectrum
by homomorphisms from \(\mathrm{SL}_2\) into the Langlands dual group;
the additional algebraic factor is central to Arthur's conjectural
description of non-tempered discrete automorphic representations
\cite[Sections~4 and~8]{Arthur1989}.
For cuspidal automorphic functions over a global function field, this leads to a decomposition
indexed by nilpotent orbits. We prove that decomposition with arbitrary
full finite level. Its common orbit also turns genericity at one
unramified place into temperedness at all unramified places in the
adjoint absolutely simple case.
\subsection{The statement}
Let \(X\) be a smooth projective geometrically connected curve over
\(\mathbb F_q\), let \(F=\mathbb F_q(X)\), and let \(G\) be a split
connected semisimple group over \(\mathbb F_q\). Write \(\mathbb A_F\)
for the adele ring and \(\mathcal O_{\mathbb A}\) for its integral
subring. Let \(D\) be any effective divisor on \(X\) defined over
\(\mathbb F_q\); its multiplicities are arbitrary. Set
\begin{equation}\label{intro:level}
K_D=\ker\bigl(G(\mathcal O_{\mathbb A})\longrightarrow
G(\mathcal O_D)\bigr),
\qquad
\mathcal X_D=G(F)\backslash G(\mathbb A_F)/K_D.
\end{equation}
For \(D=0\), this means \(K_D=G(\mathcal O_{\mathbb A})\).
Equivalently, \(\mathcal X_D\) is the set of isomorphism classes in
\(\Bun_{G,D}(\mathbb F_q)\), where the level is a trivialization on
the whole finite subscheme \(D\).
Let \(C_{D,\mathbb Q}\) be the space of compactly supported
\(\mathbb Q\)-valued functions on \(\mathcal X_D\) satisfying
\begin{equation}\label{intro:cuspidality}
\int_{U(F)\backslash U(\mathbb A_F)} f(ug)\,du=0
\end{equation}
for every proper \(F\)-parabolic subgroup \(P\subset G\), its
unipotent radical \(U\), and every \(g\in G(\mathbb A_F)\).
This condition is independent of the nonzero normalization of \(du\).
For a prime \(\ell\ne\operatorname{char}(\mathbb F_q)\), put
\[
C_{D,\ell}=\overline{\mathbb Q}_\ell\otimes_{\mathbb Q}C_{D,\mathbb Q}.
\]
All these cusp spaces at fixed level are finite dimensional.
Fix the pinned split rational form of the dual group \(\Gd\), with Lie
algebra \(\gd\), and the rational conventions of
\cite[Sections~3.1.1 and 3.4.1--3.4.3]{GLR}.
For a nilpotent \(\Gd\)-orbit \(\mathcal O\subset\gd\), choose a
Jacobson--Morozov homomorphism
\[
\phi_{\mathcal O}:\mathrm{SL}_2\longrightarrow\Gd,
\qquad
H_{\mathcal O}=Z_{\Gd}\bigl(\phi_{\mathcal O}(\mathrm{SL}_2)\bigr).
\]
The centralizer \(H_{\mathcal O}\) may be disconnected. Define
\(H_{\mathcal O}^{c}(\overline{\mathbb Q})\) to consist of its
semisimple algebraic points whose images under every embedding
\(\overline{\mathbb Q}\hookrightarrow\mathbb C\) are conjugate
into a maximal compact subgroup of \(H_{\mathcal O}(\mathbb C)\).
For a prime power \(r\), write
\begin{equation}\label{intro:orbit-spectrum}
\begin{split}
r_{\mathcal O}
&=\phi_{\mathcal O}
\begin{pmatrix}r^{1/2}&0\\0&r^{-1/2}\end{pmatrix},\\
R_{r,\mathcal O}
&=\operatorname{image}\left(
r_{\mathcal O}H_{\mathcal O}^{c}(\overline{\mathbb Q})
\longrightarrow
(\Gd/\!/\operatorname{Ad}(\Gd))(\overline{\mathbb Q})
\right).
\end{split}
\end{equation}
We fix compatible square roots and use the rational Satake normalization
of \cite[Equation~(3.2)]{GLR}. Changing a square root multiplies the displayed
factor by \(\phi_{\mathcal O}(-I)\), a finite-order element of
\(H_{\mathcal O}\), and does not change the set
in Equation~\eqref{intro:orbit-spectrum}.
For a closed point \(x\notin\operatorname{supp}(D)\), put
\(d_x=[k(x):\mathbb F_q]\) and \(q_x=q^{d_x}\). Its spherical
Hecke algebra is
\[
A_x=\operatorname{Funct}_c
\bigl(G(\mathcal O_x)\backslash G(F_x)/G(\mathcal O_x),\mathbb Q\bigr),
\]
with Haar measure normalized by
\(\operatorname{vol}(G(\mathcal O_x))=1\). We regard each character
of \(A_x\) over \(\overline{\mathbb Q}\) as a semisimple dual-group
conjugacy class by this fixed Satake convention.
Define \(C_{D,\mathcal O,\mathbb Q}\subset C_{D,\mathbb Q}\) by the
following spectral-support condition: for every
\(x\notin\operatorname{supp}(D)\), every character occurring in
\[
\overline{\mathbb Q}\otimes_{\mathbb Q} A_xf
\]
has Satake class in \(R_{q_x,\mathcal O}\). Set
\(C_{D,\mathcal O,\ell}
=\overline{\mathbb Q}_\ell\otimes_{\mathbb Q}
C_{D,\mathcal O,\mathbb Q}\).
The same orbit is required at all good points.
\begin{theorem}\label{main:decomposition}
For every \(X,G,D,\ell\) as above, the natural addition maps
\begin{equation}\label{intro:decomposition}
\bigoplus_{\mathcal O}C_{D,\mathcal O,\mathbb Q}
\xrightarrow{\ \sim\ }C_{D,\mathbb Q},
\qquad
\bigoplus_{\mathcal O}C_{D,\mathcal O,\ell}
\xrightarrow{\ \sim\ }C_{D,\ell}
\end{equation}
are isomorphisms. The sums range over the nilpotent orbits of \(\Gd\).
\end{theorem}
Taking \(D=0\) resolves Conjecture~3.4.6 of
Gaitsgory--Lafforgue--Raskin positively, as well as its equivalent rational
form, Conjecture~3.4.5. Theorem~\ref{main:decomposition} also establishes
the stated finite-level extension. The compactness in
Equation~\eqref{intro:orbit-spectrum} is the archimedean condition at
every embedding; no extra nonarchimedean condition is included.
\subsection{Generic cuspidal representations}
For a split group, fix a Borel subgroup with unipotent radical \(N\).
A local representation of \(G(F_x)\) is \emph{generic} if it admits a
nonzero Whittaker functional for a nondegenerate character of \(N(F_x)\).
A cuspidal automorphic representation is \emph{globally generic} if one
of its vectors has a nonzero global Whittaker coefficient for a
nondegenerate character of \(N(F)\backslash N(\mathbb A_F)\).
The generalized Ramanujan conjecture predicts that globally generic
cuspidal representations are tempered at every place
\cite[Section~1.1]{CiubotaruHarris}. We obtain its unramified assertion
in the following scope, with genericity required at only one unramified
place.
\begin{corollary}[Unramified generalized Ramanujan]
\label{main:generic-ramanujan}
Let \(F\) be the function field of a smooth projective geometrically
connected curve over a finite field, and let \(G/F\) be split connected
adjoint absolutely simple. Let
\(\pi=\bigotimes'_x\pi_x\) be a complex cuspidal automorphic
representation of \(G(\mathbb A_F)\). If \(\pi_v\) is unramified and
generic at one place \(v\), then \(\pi_x\) is tempered at every place
\(x\) at which it is unramified. In particular, every globally generic
cuspidal automorphic representation of \(G(\mathbb A_F)\) is tempered
at every unramified place.
\end{corollary}
Here unramified means spherical for a hyperspecial maximal compact
subgroup. The conclusion is only about these local components. The
proof, given at the end of Section~\ref{sec:global}, combines the common
orbit with the local geometric-parameter criterion of Ciubotaru and
Harris.
\subsection{Context}
The Langlands correspondence for general linear groups over function
fields, established by Drinfeld in rank two and by Laurent Lafforgue
in general, gives the purity theorem used at the end of this proof
\cite[Introduction and Theorem~VII.6]{LLafforgue}.
Vincent Lafforgue's construction of
global parameters extends the parameterization of cuspidal functions to
split reductive groups with arbitrary finite level
\cite[Theorem~1.1]{VLafforgue}. These results provide global parameters
and control the weights of their general-linear constituents. The main
issue here is to constrain the absolute-value part of an unramified
dual-group parameter to the neutral cocharacter of a nilpotent orbit.
Gaitsgory--Lafforgue--Raskin connect nilpotent-orbit filtrations on
automorphic functions with Hecke spectral support and formulate the
rational and \(\ell\)-adic cusp decompositions in
\cite[Section~3.4]{GLR}. Their Remark~3.4.7 identifies the latter as
the \(\ell\)-adic Ramanujan--Arthur conjecture. We address this
spectral-support assertion directly.
Using the present decomposition at \(D=0\), the companion manuscript
identifies each cuspidal canonical
Arthur-filtration step with the scalar extension of the corresponding sum
of the rational summands here, under its four characteristic hypotheses
\cite[Assumption~1.1 and Corollary~7.5]{ArthurRationality}. This is the
common-scope form of \cite[Conjectures~3.4.9 and~3.4.11]{GLR}.
Earlier function-field results constrain the unramified spectrum under
additional local hypotheses. In characteristic greater than two, Sawin and
Templier prove unramified temperedness under monomial geometric
supercuspidality and base-change assumptions
\cite[Theorem~1.1]{SawinTemplier}; their Theorem~11.7 also shows that a
cuspidal representation of a split semisimple group tempered at one
unramified place is tempered at all such places. Ciubotaru and Harris
obtain restrictions to complementary series indexed by nilpotent orbits,
and generalized Ramanujan under additional local assumptions, from
excursion parameters, Frobenius weights, and the spherical unitary dual
\cite[Theorems~1.11 and~5.4]{CiubotaruHarris}.
Theorem~\ref{main:decomposition} supplies the nilpotent-orbit restriction
for the whole cuspidal space at full finite level.
Corollary~\ref{main:generic-ramanujan} uses their local criterion
\cite[Corollary~6.32\textup{(1)}]{CiubotaruHarris}: the algebraic
\(\mathrm{SL}_2\) cocharacter in our decomposition makes the Satake
parameter geometric, and the
common orbit propagates the resulting temperedness without the additional
tempered-place hypothesis of their global Theorem~5.4.
The geometric ingredients have a separate ancestry. Geometric Satake
identifies spherical sheaves with representations of the dual group
\cite{MV}; its derived form and the regular-sheaf and Springer
calculations are due to Ginzburg, Bezrukavnikov--Finkelberg, and
Arkhipov--Bezrukavnikov--Ginzburg \cite{GinzburgLoop,BF,ABG}.
We use the enhanced arithmetic Satake formulation of
Ben-Zvi--Sakellaridis--Venkatesh \cite{BZSV} together with the
tensor-product and trace formalism of Ben-Zvi--Francis--Nadler
\cite{BFN}.
Gaitsgory's nearby-cycle construction supplies central sheaves
\cite{GaitsgoryCentral}, and the Wakimoto filtration and its central
compatibilities are developed by Arkhipov--Bezrukavnikov and, in the
integral parahoric setting, Cass--van den Hove--Scholbach
\cite{AB,CvdHS}. Section~\ref{sec:categorical} states the precise
versions used here and proves the additional specialization,
root-datum, and cyclic-transfer compatibilities needed for the argument.
\subsection{Proof strategy}
The main intermediate result is local. An ordered Bernstein weight is
a joint character of the commuting Iwahori translation operators,
before passage to the Weyl-group quotient. If its absolute-value
exponent is dominant and it occurs in the discrete automorphic
spectrum, then it is a compact
factor times a Jacobson--Morozov value
(Theorem~\ref{ext:discrete}). We prove this by contradiction and
induction on semisimple rank. For a hypothetical violating weight \(t\)
at \(x\), choose a complex realization and write
\(\lambda=\log_{q_x}|t|\) in the dominant chamber; unitarity makes
its central component zero. The contradiction comes from two degree bounds on
the same spherical cohomological summand.
First, standard translations define bounded open shtukas that detect
the prescribed ordered weight. A polynomial filter removes irrelevant
continuous spectral families; induction gives a strict spectral gap
for those that remain. The rotation trace identity then forces the
weight to occur in compact cohomology. Deligne's weight bound puts its
degree at least
\(2d_x\langle\lambda,\mu\rangle-C'\), where \(\mu\) is the
translation weight and \(C'\) is independent of \(\mu\).
The occurrence itself, through integrality of Frobenius weights, also
forces \(\lambda\) to be rational
(Corollary~\ref{ext:rationality}). A fixed rational Weyl-invariant
metric therefore lets us choose integral highest weights
\(\mu=m\lambda\) for sufficiently divisible positive integers \(m\),
including when \(\lambda\) lies on a chamber wall.
The Wakimoto filtration compares this translation cohomology with
cohomology for the spherical Satake label of highest weight \(\mu\).
Since \(\mu=m\lambda\) is the unique weight maximizing pairing
with \(\lambda\), its extreme part cannot cancel against another
filtration term. The hyperspecial projections retain that part at a
spherical level. There the categorical trace realizes the projection
on complexes, and a finite specialization argument preserves its top
nonzero cohomology. This produces the spherical summand on which both
degree bounds can be compared.
The upper bound is prepared independently in
Section~\ref{sec:amplitude}. Derived Satake and the categorical trace
calculation express the spherical trace as an equivariant module.
Its group-valued trace variable \(s\in\Gd\) and Lie-algebra variable
\(e\in\gd\) satisfy the derived relation
\begin{equation}\label{intro:resonance}
\operatorname{Ad}(s)e=q_xe.
\end{equation}
At semisimple part \(t\), the resonant space is
\(E_t=\{e\in\gd:\operatorname{Ad}(t)e=q_xe\}\).
Uniform bounds from costandard translations and nef Springer line
bundles pass, by Levi restriction and localization on opposite flags,
to bounds on whole derived fibers, with degrees corrected by compatible
\(\mathfrak{sl}_2\)-triples. A finite-orbit induction then
gives the upper bound for equivariant modules satisfying these tests,
without a coherence hypothesis. Applied to the summand just constructed, the two
bounds on its top nonzero degree \(N\) are
\[
2d_x\langle\lambda,\mu\rangle-C'
\ \leq\ N\ \leq\
C+\max_{e\in E_t}d_x\langle\mu,h_e\rangle,
\]
where \(h_e\) is the neutral cocharacter of a compatible triple and
\(C,C'\) are independent of \(m\). On \(\mu=m\lambda\), the
difference of the leading terms is
\[
2d_xm\min_{e\in E_t}\|\lambda-h_e/2\|^2.
\]
It is positive for a violating parameter, contradicting the two
bounds as \(m\) grows. This proves the local theorem.
Two parts of the argument can be used separately. Propositions
\ref{amp:levi} and \ref{amp:fiber} extract corrected fiber bounds for
arbitrary equivariant modules from uniform flag tests.
Lemma~\ref{rot:picard} compares two descent conditions
by realizing a divisible rotation in a connected generalized Picard
group and applying Lang's theorem. The latter construction keeps the
entire level modulus and treats arithmetic components of a splitting
cover explicitly.
Finally, exact root equations for compatible triples show that the
compact factor can be chosen algebraically and is compact at every
complex embedding. Purity of the irreducible constituents of the global
parameter makes the dominant neutral cocharacter independent of the good place.
The finite rational Hecke algebra then gives the direct sum in
Equation~\eqref{intro:decomposition}.
\subsection{Organization and conventions}
Section~\ref{sec:categorical} constructs the cyclic trace and records
its Satake, central, and Springer descriptions.
Section~\ref{sec:amplitude} proves the amplitude estimate.
Section~\ref{sec:rotation} proves the rotation trace identity with
full finite level.
Section~\ref{sec:extraction} extracts the local constraint from the
discrete spectrum.
Section~\ref{sec:global} proves Theorem~\ref{main:decomposition} and
deduces Corollary~\ref{main:generic-ramanujan}.
The local arguments are formulated for split connected reductive groups
so that they apply inductively to Levi subgroups. Central degree is
bounded modulo a discrete lattice where necessary. On the sheaf side
the coefficients have characteristic zero, and all derived tensor
products and fibers are retained. A complex realization is chosen only
when absolute values are used. At a place of degree \(d\), the Hecke
parameter is \(q^d\), while the global cohomological Frobenius has base
\(q\). Standard translations are used for the trace identity;
costandard translations are used for upper amplitude estimates.
The established sheaf-theoretic and spectral inputs are stated with
their precise roles in the sections where they enter.
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/01-introduction.tex ====
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/02-categorical.tex ====
\section{Local kernels and their Frobenius traces}\label{sec:categorical}
We compare compact cohomology of Frobenius-fixed Hecke paths with
dual-group geometry. The comparison has three outputs. At spherical
level, the trace is a pairing with a dg module over the Koszul algebra
for $\operatorname{Ad}(s)e=q^d e$, where $s\in\Gd$ records transport
around the degree-$d$ Frobenius cycle and $e$ is the Lie-algebra
coordinate (Proposition~\ref{cat:loops}). Nef line bundles on the
Springer resolution give the costandard tests used to bound this
pairing (Proposition~\ref{cat:springer}). At Iwahori level, central
kernels and their Wakimoto filtration give Hecke actions and transfers
to spherical level (Proposition~\ref{cat:hecke}). The spherical
description retains dg morphisms; the Iwahori Hecke-algebra action
will be needed on cohomology.
All constructible
categories in this section have their stable, $\mathbf k$-linear enhancements.
Over a field of positive characteristic, $\mathbf k=\overline{\mathbf Q}_\ell$.
Comparisons with complex constructible sheaves are made with characteristic-zero
coefficients. On the dual side put
\[
S=\operatorname{Sym}(\gd^*[-2]).
\]
Thus a linear polynomial has cohomological degree $2$. Tensor products of
dg modules are derived unless explicitly stated otherwise.
\subsection{Conventions and change of characteristic}
Fix a split maximal torus, an alcove, and its standard parahorics. For
parahorics $P,Q$, write $\mathscr H_{P,Q}$ for the stable category of
equivariant constructible complexes on $P\backslash LG/Q$ generated by
constant orbit complexes and having finite Schubert support. We use the
usual convolution, with compact direct image, for which the point complex
is the unit. The standard and costandard complexes on an Iwahori orbit
of length $n$ are normalized as
\[
\Delta_w=(j_w)_!\mathbf k[n](n/2),\qquad
\nabla_w=(j_w)_*\mathbf k[n](n/2).
\]
Fix a square root $q^{1/2}$. Geometric Frobenius acts on
$\mathbf k(-1)$ by $q$. Satake uses the usual correction of the
commutativity constraint by the component parity, so that its target is
ordinary $\operatorname{Rep}(\Gd)$.
\begin{lemma}[Ordinary specialization]\label{cat:specialization}
For every prime $p\ne\ell$, the geometric characteristic-zero and
$\overline{\mathbf F}_p$ categories $\mathscr H_{P,Q}$ admit simultaneous
enhanced equivalences preserving standard, costandard, and intersection
complexes, convolution, and change of parahoric level. They also preserve
restriction of equivariance from $L^+G$ to the pro-unipotent radical $I^u$
of an Iwahori, including the induced maps of mapping complexes.
These are assertions about ordinary geometric categories; Weil structures
will be specified separately.
\end{lemma}
\begin{proof}
Use the split integral group with the given root datum and the strictly
henselian trait $R=\mathbf Z_{(p)}^{\mathrm{sh}}$. Its closed geometric
point is $\overline{\mathbf F}_p$; its geometric generic point has
characteristic zero. First forget the left parahoric equivariance.
On each relative partial affine flag variety $LG_R/Q_R$, use its
\emph{Iwahori-cell} stratification, whose strata are affine spaces over
$R$, and let $\mathscr C_R$ be the thick category generated by their
relative standard constant complexes, with finite Schubert support.
A parahoric orbit is a union of these cells, and its standard constant
complex belongs to $\mathscr C_R$ by localization. We do not regard
parahoric orbits themselves as affine spaces.
Here are the base-change hypotheses being used. Start the Tate lifts
over $S_0=\operatorname{Spec}\mathbf Z[1/\ell]$, which is an admissible
base in \cite[Notation~2.1]{RicharzScholbach}. In that paper's
Section~2.4 the target $T$ of a base change $T\to S_0$ need only be
Noetherian of finite Krull dimension. Thus its Lemma~2.12 applies to
$T=\operatorname{Spec}R$ and to both geometric fibers. It says that
$*$-extension and $!$-restriction of a stratum commute with these
pullbacks on the Tate lifts. Its proof first treats Iwahori cells and
then obtains the parahoric version by localization. Realize these
comparison maps in ordinary etale complexes. Transitivity gives the
maps for restriction from $R$ to either fiber; closing their
equivalence loci under cones and retracts gives the assertion on
$\mathscr C_R$. The same argument shows that restrictions and
corestrictions to a fine cell lie in the thick category generated by
its constant complex. No motivic $t$-structure or new
Beilinson--Soul\'e assertion over $R$ is needed.
On an affine stratum we have
\[
R\Gamma(\mathbf A_R^n,\mathbf k)=\mathbf k
=R\Gamma(\mathbf A_{\bar\eta}^n,\mathbf k)
=R\Gamma(\mathbf A_{\bar s}^n,\mathbf k).
\]
Consequently fiber restriction identifies mapping complexes between
objects generated by its constant complex. Induct on a finite closed
union of strata containing the supports of $A,B\in\mathscr C_R$.
For a closed stratum $i:Z\hookrightarrow Y$ with complementary open
$j:U\hookrightarrow Y$, the fiber sequence
\[
R\operatorname{Hom}_Z(i^*A,i^!B)
\longrightarrow R\operatorname{Hom}_Y(A,B)
\longrightarrow R\operatorname{Hom}_U(j^*A,j^*B)
\]
and the boundary comparison prove full faithfulness on mapping
complexes. Fiber restriction is essentially surjective on the specified
categories because it contains their standard generators and preserves
cones and retracts. This also proves compatibility with composition:
the comparisons are induced by one enhanced restriction functor.
Here is the equivariant version of this argument. On a bounded support
choose a finite-type smooth quotient $K_R$ of the acting parahoric;
its discarded kernel is split pro-unipotent and has trivial positive
equivariant cohomology. Compatible quotients can be used for inclusions
of parahorics and for $I^u\subset L^+G$
\cite[Lemma~2.18]{RicharzScholbach}. Each $K_R$ is an extension of a
split reductive group by split unipotent groups. The cohomology of
$K_R^a$ commutes with restriction to either geometric fiber: this
follows from the Bruhat decomposition of its reductive quotient,
homotopy invariance for the unipotent factors, and localization and
duality for products of affine spaces and split tori. Compute
equivariant mapping complexes by the bar presentation
\[
Y_R,\quad K_R\times Y_R,\quad K_R^2\times Y_R,\quad\ldots.
\]
At each simplicial degree use the strata
$K_R^a\times\mathbf A_R^n$ coming from the fine Iwahori cells of
$Y_R$. The underlying coefficients in the bar calculation are pulled
back from $Y_R$. The preceding recollement argument therefore applies,
with the single-stratum calculation replaced by that for $K_R^a$.
It gives an equivalence of the mapping complexes at that degree.
Totalizing these equivalences gives the equivariant comparison. No
exchange of a scalar extension with an infinite totalization is used.
The common relative parahoric standard complexes are equivariant
objects; their fiber restrictions generate the specified equivariant
categories. Full faithfulness on the bar mapping complexes and
closure under cones and retracts therefore give essential
surjectivity on those categories as well.
The maps of bar presentations for subgroup inclusions give the claimed
compatibility with restriction of equivariance.
Convolution is twisted exterior product followed by proper direct
image on bounded supports. Smooth descent for the torsor defining the
twisted product and proper base change therefore commute with fiber
restriction. The comparison for all iterated products uses the same
iterated convolution diagram and respects all associativity
compatibilities. This is also the realized convolution comparison of
\cite[Proposition~3.14 and Lemma~3.15]{RicharzScholbach}.
The argument applies to the smooth proper projections between the
partial affine flags. The stratumwise perverse conditions agree on
the two fibers: the restriction and corestriction degrees agree,
and the relative cell dimensions are the same. The equivalence is
therefore perverse $t$-exact and preserves intermediate extensions,
hence intersection complexes. Wakimoto complexes
are finite convolutions of standard and costandard complexes, so they
are preserved as well.
An algebraic closure of the generic field embeds in $\mathbf C$.
On each finite support and each finite bar term, algebraically closed
extension in characteristic zero is fully faithful on bounded
constructible $\overline{\mathbb Q}_\ell$-complexes
\cite[Lemma~A.1]{JKLMBaseChange}.
The etale--Betti comparison is likewise fully faithful
\cite[Corollary~3.4.3, Proposition~3.5.9, and
Remark~3.5.12]{SunComparison}. These comparisons use the derived
pullback and site functors; their isomorphisms on Hom into every shift
give the mapping-complex equivalences. Full faithfulness suffices here:
the specified categories are generated from the matched standard
complexes by finite cones and retracts. Comparing the same bar
presentations at fixed coefficients before totalization gives the
asserted characteristic-zero comparison.
All constructions above use only $p\ne\ell$.
\end{proof}
\begin{theorem}[Derived Satake with Frobenius]\label{cat:satake}
The stable idempotent-complete tensor category generated by the
normalized spherical Satake complexes is monoidally equivalent to
\[
\operatorname{Perf}^{\Gd}(S).
\]
The normalized Satake object $\operatorname{Sat}(V)$ corresponds to
$S\otimes V$. With the split Weil structures, Frobenius pullback
corresponds to extension of scalars by
\[
S\longrightarrow S,\qquad \alpha\longmapsto q\alpha
\quad(\alpha\in\gd^*[-2]),
\]
and acts trivially on the representation labels. These assertions
include the enhanced monoidal structure and all morphisms between
the indicated objects.
\end{theorem}
\begin{proof}
The geometric input for reductive groups is the monoidal dg equivalence
of \cite[Theorem~6.6.1]{BZSV}, which strengthens the monoidal
triangulated calculation of \cite[Theorem~5]{BF}. We use the
corrected version of the latter paper; its purity and multiplicative
formality construction is in \cite[Sections~6.5--6.6]{BF}.
The Galois-compatible formulation in
\cite[Theorem~6.7.5]{BZSV} uses an arithmetic shearing. We verify below
the split Weil normalization and dilation required here.
The regular Satake Ext algebra is the graded equivariant algebra
$\operatorname{Sym}(\gd^*)$, with its linear generators in degree $2$;
the Ext spaces between the other Satake objects are
\[
\operatorname{Ext}^{\bullet}
(\operatorname{Sat}(V),\operatorname{Sat}(W))
=(S\otimes V^*\otimes W)^{\Gd}.
\]
These identifications respect tensor products and composition.
Lemma~\ref{cat:specialization} carries the complete ordinary diagram
of these calculations to every residue characteristic.
For the Weil calculation we need pointwise purity. Over every finite
field, the intersection complex of a split parahoric Schubert variety
is very pure and Tate with semisimple Frobenius
\cite[Section~1.2 and Theorem~2.2.1]{DCHL}. That theorem is stated for
arbitrary split connected reductive groups and all $p\ne\ell$.
Writing $\operatorname{IC}$ here for that paper's unshifted complex,
our normalization is $\operatorname{IC}[d](d/2)$. It says that a
stalk cohomology group in degree $a$ has Frobenius scalar $q^{a/2}$.
Normalized Verdier self-duality gives the same assertion for point
costalks. On a smooth orbit of dimension $d_O$, the identity
$x^!j^!B=(j^!B)_x[-2d_O](-d_O)$ then gives that scalar also on the
degree-$b$ cohomology of the ordinary fiber of $j^!B$.
The equivariant orbit filtration of a morphism complex has local
terms computed from these restrictions and the cohomology of the
split orbit stabilizer $H$. Its reductive quotient is a split Levi,
and the remaining unipotent factors are cohomologically trivial;
$H^c(BH)$ is Tate with scalar $q^{c/2}$. A local term
\[
\operatorname{Hom}(H^a(A_x),H^b((j^!B)_x))\otimes H^c(BH)
\]
therefore has degree $b-a+c$ and scalar $q^{(b-a+c)/2}$.
All spectral-sequence differentials vanish by their distinct
weights. This proves purity and the semisimplified scalar on the
Ext groups; it alone does not exclude extensions of equal-weight
Frobenius modules. On a degree-two adjoint generator the scalar is
nevertheless exactly $q$: its semisimplification is $q$, and it
commutes with $\Gd$; each simple adjoint summand occurs once. On the
central summand the generators are the first Chern classes of the
split torus, on which Frobenius is also exactly $q$. This determines
Frobenius on the generated algebra. The corresponding goodness
statement for convolution \cite[Corollary~2.2.3]{DCHL}, together with
the perversity of spherical convolution, makes its normalized
tensor multiplicity spaces Tate of weight zero with trivial
Frobenius. Thus the canonical top-cell Weil structures give the
split Satake tensor identifications used above.
For clarity, formality can be applied to this entire diagram at once.
For a mixed morphism complex $C=\bigoplus_w C_w$, use the
multiplicative intermediate complex
\[
\bigoplus_w\tau_{\leq w}C_w.
\]
Its inclusion into $C$ and its projection to
$\bigoplus_w H^w(C_w)$ are quasi-isomorphisms. Here $C_w$ denotes
the generalized weight summand; no semisimplicity of an arbitrary
pure Weil complex is being inferred. Composition and tensor product
add weights and degrees, so these quasi-isomorphisms are compatible
with every composition and tensor operation. They are
Frobenius-equivariant. Closing this enhanced diagram under finite
cones and retracts gives the assertion. Only its associative
monoidal structure is required below.
\end{proof}
\subsection{Central kernels and Springer tests}
Let $\Lambda=X_*(T)=X^*(\widehat T)$. We call the chamber in which
$J_a$ is standard the positive translation chamber. The opposite
chamber is the costandard chamber. A line bundle on the dual flag
variety is denoted $\mathcal O(\eta)$ with the convention that this
costandard chamber is its nef chamber.
\begin{theorem}[Central and Wakimoto kernels]\label{cat:central}
There is a tensor functor to the Drinfeld center of the homotopy
category
\[
Z:\operatorname{Rep}(\Gd)\longrightarrow
\mathcal Z(\operatorname{Ho}(\mathscr H_{I,I}))
\]
with its split Weil structure. Its interchanges are compatible with
projection to every standard parahoric. At a hyperspecial parahoric
the projected functor is Satake.
There are invertible kernels $J_a$, $a\in\Lambda$, with coherent
multiplication $J_a\star J_b\simeq J_{a+b}$. They are standard on
the positive chamber and costandard on its opposite. The central
object $Z(V)$ has a finite functorial Wakimoto filtration, tensor
compatible in $V$, with
\[
\operatorname{gr}_a Z(V)=J_a\otimes V_a.
\]
Its induced central interchange with $J_b$ is the ordinary interchange
of the two translation factors and the identity on the multiplicity
space, with the fixed Satake parity convention. The multiplicity
spaces have trivial semisimplified Frobenius. These statements hold
over every $\mathbf F_q$ under consideration.
For a fixed central label, its interchange in the other kernel
variable is a chosen natural equivalence of exact enhanced functors.
We use the central tensor identities in the homotopy category;
we do not assert an enhancement of $Z$ to an infinity-categorical
Drinfeld center.
\end{theorem}
\begin{proof}
The central construction and the Wakimoto filtration for every split
reductive group over the base in question follow from
\cite[Theorem~1.4 and Theorem~5.24]{CvdHS}, by realization.
The scope of its centrality assertion is exactly the homotopy
category. More precisely, its Theorem~4.38 constructs the two
natural maps in its Equation~(4.7) from the enhanced nearby-cycles
and convolution functors, using Lemma~4.20 and Theorem~4.36(3).
For a fixed central label these give the stated natural equivalence
of exact functors in the other kernel variable, so it applies to
an actual cofiber diagram. Theorem~4.38 supplies the homotopy
hexagon, and Theorem~4.41 the homotopy tensor compatibility.
These finite identities are the ones needed for products of
character operators on cohomology below. The maps are defined
over $\mathbf F_q$ and carry their actual Weil structures, after
fixing the nearby-cycles Frobenius lift.
We record explicitly the graded-interchange argument, to include all
root data. The Wakimoto graded category is the category of
$\Lambda$-graded vector spaces, and the graded central functor has
the weights of restriction from $\Gd$ to $\widehat T$. For every
dominant character $\lambda$ of the actual torus there is its
highest-weight quotient to the line labelled by $J_\lambda$.
On this quotient the central interchange is the ordinary
interchange: restrict the fusion diagram to the open Cartan-position
cell, where the sheaves are normalized constants and the quotient
is the highest-line restriction. The two tensor generators are
interchanged there, with exactly the fixed Satake parity correction.
Naturality and the central tensor identity now imply the same
interchange for an arbitrary central label with $J_\lambda$.
Indeed the two opposite central interchanges have product one,
and passage to this quotient identifies one of them with the
ordinary interchange, so identifies the other with its inverse.
This is the argument of \cite[Lemma~18 and Section~3.6.6,
Equation~(21)]{AB}. Dominant characters generate $\Lambda$ as a
group. Tensor compatibility and invertibility of the $J_\lambda$
extend the assertion to every $\lambda$. The argument uses the
actual character lattice, and applies to products and central torus
factors as stated. Its top-cell maps are defined over $\mathbf F_q$,
so it preserves the specified normalization and Weil maps.
Projection compatibility follows from proper base change for the
nearby-cycles construction: the projection of flag levels is a
smooth proper flag fibration, and the commutation before taking
nearby cycles is the commutation of modifications at disjoint points.
Using the same multiple-point diagram makes these compatibilities
simultaneous for all the parahorics and all products.
The split Weil multiplicities can also be checked without any
choice of splitting of the filtration. The trace of the central
object is the Bernstein central character, over each finite
extension. The Wakimoto trace functions are linearly independent.
Consequently all power traces on $V_a$ equal $\dim V_a$;
their semisimplified eigenvalues are one. This statement permits
unipotent extensions, which do not affect the eigenvalue
separations used later.
\end{proof}
\begin{proposition}[The spherical Springer test]\label{cat:springer}
Let $\operatorname{For}$ forget $L^+G$-equivariance to
$I^u$-equivariance on $\operatorname{Gr}_G$, without changing the
underlying complex, and let $\operatorname{Av}_*$ be its right
adjoint. Put $W_\eta=J_\eta\star\delta_0$. For every integral
weight $\eta$ in the closed costandard chamber, derived Satake
identifies
\[
\operatorname{Av}_*W_\eta
\quad\text{with}\quad
\bigl(R\pi_*\mathcal O_{\widetilde{\mathcal N}}(\eta)\bigr)_{\rm sh},
\qquad
\pi:\widetilde{\mathcal N}\longrightarrow\gd.
\]
The subscript means that coherent degree $a$ and polynomial degree
$b$ are assigned total degree $a+2b$. This is an equivalence of
ordinary equivariant dg $S$-modules. The $S$-action is polynomial
restriction along $\pi$, and the assertion is compatible with
tensoring by representations and with all morphisms between free
Satake labels. The displayed convention introduces no extra shift.
Any smooth-pullback normalization of $\operatorname{For}$ changes
the formula only by a fixed shift and Tate twist, independent of
$\eta$.
\end{proposition}
\begin{proof}
We give the algebra-to-module calculation, including its root-datum
and equivariance conventions. Let $\mathcal R$ be the regular Satake
ind-object. Its second regular action gives the $\Gd$-action on the
following algebras and modules. The regular calculations are
\[
\operatorname{Ext}^{\bullet}_{L^+G}(\delta_0,\mathcal R)=S,
\qquad
\operatorname{Ext}^{\bullet}_{I^u}(\delta_0,\mathcal R)
=\mathbf k[\mathcal N]_{\rm sh},
\]
and the Wakimoto calculation for nef weights, including its products, is
\begin{equation}\label{cat:section-algebra}
\bigoplus_{\eta\ \text{nef}}
\operatorname{Ext}^{\bullet}_{I^u}
(\delta_0,J_\eta\star\mathcal R)
\simeq
\left(\bigoplus_{\eta\ \text{nef}}
\Gamma(\widetilde{\mathcal N},\mathcal O(\eta))\right)_{\rm sh}.
\end{equation}
The algebra calculation is
\cite[Theorems~7.3.1 and 7.6.1]{ABG}; the section algebra and its
product maps are \cite[Theorem~8.5.2 and Equations~(8.7.2)--(8.7.8)]{ABG}.
We explain why using the actual lattice $\Lambda$ entails no
isogeny restriction here. Wakimoto inversion identifies the left
side for $\eta$ with the appropriate fixed-point costalk of
$\mathcal R$. Its cohomological filtration is the principal-nilpotent
filtration on the $-\eta$ weight space of $\mathbf k[\Gd]$.
The fixed-point and filtration assertions of
\cite[Propositions~5.5.2 and 5.6.2]{GinzburgLoop} are stated for the
full cocharacter lattice of an arbitrary semisimple group.
The section-filtration identification used in
\cite[Equations~(8.6.2)--(8.6.3)]{ABG} applies to every dominant
character of the actual dual torus. Multiplication of the regular representation,
followed by the fixed-point-to-convolution base-change maps in
\cite[Equations~(8.7.6)--(8.7.8)]{ABG}, identifies the products in
Equation~\eqref{cat:section-algebra}. This proves the required
calculation for every semisimple isogeny type, without an assertion
about all objects in a generalized ABG equivalence.
For a torus the calculation is immediate: its Grassmannian is
$\Lambda$, its pro-unipotent group is cohomologically trivial, and
the algebra map is $\operatorname{Sym}(\operatorname{Lie}(\widehat
T)^*[-2])\to\mathbf k$. For a general reductive group, perform
the characteristic-zero calculation using its central isogeny to
the product of its adjoint group and its abelianization. Each
Grassmannian component identifies with the corresponding component
in that product; finite central kernels have trivial positive
cohomology with these coefficients. On the dual side this restricts
the regular representation and its products to the prescribed
central-character lattice. It therefore restricts precisely the
preceding section-algebra calculation. Central translations have
length zero, and central linear variables map to zero. Finally
Lemma~\ref{cat:specialization} transports this ordinary enhanced
diagram, including the forgetful map of algebras and the module
action, to every $p\ne\ell$. The regular ind-object causes no
problem: mapping out of the point in these Ind-completions commutes
with its filtered union.
We now identify the polynomial map. The equivariant-cohomology
spectral sequence for the regular internal Hom has second page
\[
H^{\bullet}(BG)\otimes
\operatorname{Ext}^{\bullet}_{I^u}(\delta_0,\mathcal R).
\]
Both factors are even, so its edge map induces
\[
\mathbf k\otimes_{H^{\bullet}(BG)}
\operatorname{Ext}^{\bullet}_{L^+G}(\delta_0,\mathcal R)
\simeq
\operatorname{Ext}^{\bullet}_{I^u}(\delta_0,\mathcal R).
\]
Freeness over $H^{\bullet}(BG)$ follows by lifting a homogeneous
basis on this second factor and comparing its associated graded.
Taking $\Gd$-invariants identifies $H^{\bullet}(BG)$ with
$S^{\Gd}$, since the invariant part of $\mathcal R$ is the unit.
The map thus has kernel $S(S^{\Gd})_{>0}$ and is restriction to
the nilpotent cone. On degree-two simple adjoint factors its
identification differs from the usual one at most by nonzero
scalars, by equivariance and surjectivity. Absorb these constant
scalars in the choice of coordinates; they commute with the
Frobenius dilation. This also identifies the action on each module
in Equation~\eqref{cat:section-algebra}.
The comparisons just used are comparisons of dg modules. In
characteristic zero the simultaneous purity construction of
\cite[Proposition~9.5.2 and Lemma~9.5.3]{ABG}, or the multiplicative
weight truncation in Theorem~\ref{cat:satake}, formalizes the
algebra, its map, and its positive modules together. The ordinary
enhanced specialization carries that diagram to the desired
characteristic. No identification of an arbitrary Weil structure
on a Springer test is needed for the present assertion.
For every finite-dimensional $V$, adjunction now gives
\begin{align*}
R\operatorname{Hom}(\operatorname{Sat}(V),\operatorname{Av}_*W_\eta)
&\simeq R\operatorname{Hom}(\operatorname{For}\operatorname{Sat}(V),W_\eta)\\
&\simeq
\bigl(R\Gamma(\widetilde{\mathcal N},
V^*\otimes\mathcal O(\eta))\bigr)^{\Gd}_{\rm sh}\\
&\simeq R\operatorname{Hom}_{S,\Gd}
(S\otimes V,(R\pi_*\mathcal O(\eta))_{\rm sh}).
\end{align*}
For every nef $\eta$, including walls, the higher line-bundle
cohomology vanishes by \cite[Theorem~2.4 and Remark~2.7(2)]{Broer}.
Thus the section-module computation indeed computes the derived
pushforward appearing here. This use of vanishing is entirely on
the characteristic-zero dual side. The polynomial computation
proves naturality in all morphisms between the free labels, which
compactly generate. Yoneda proves the asserted equivalence.
The pushforward is coherent on the smooth affine space $\gd$;
an equivariant graded finite free resolution, followed by shearing,
shows that it is a perfect $S$-module.
Finally our forgetful functor preserves the underlying sheaf. If
one instead uses $\operatorname{For}[c](c/2)$, its right adjoint
is $\operatorname{Av}_*[-c](-c/2)$. The integer $c$ is the
dimension fixed by that change of level, independently of $\eta$.
\end{proof}
\subsection{The geometric cyclic functor}
Bundle stacks below may have fixed full congruence level away from
the selected legs and fixed parahoric level at them. They may also
be divided by a discrete lattice of central modifications, as in
Section~\ref{sec:rotation}. At a closed place every geometric leg
is retained, with Frobenius cyclically permuting them. A kernel
may change these levels. It always has bounded relative position.
For a composite kernel $L$ on a bundle stack $\mathcal B$, define
\[
\mathsf K(L)=R\Gamma_c(\mathcal B,
\operatorname{Graph}(\mathrm{Fr})^*L).
\]
Equivalently this is compact cohomology of the stack of paths from
$P$ to ${}^{\mathrm{Fr}} P$, with all intermediate bundles and all Hecke
coefficients retained. Fixed kernels at other places may be
included in every path. Compact cohomology of a locally finite
type stack is the colimit over finite-type open substacks, using
extension by zero.
\begin{proposition}[Coherent path rotation]\label{cat:cyclic}
The preceding construction is an exact enhanced Frobenius-twisted
cyclic functor. Put $\Phi=\mathrm{Fr}^*$ on local kernels. With convolution
ordered in the direction of the path, its cyclic maps are
\begin{equation}\label{cat:last-first}
\rho_{U,V}:\mathsf K(U\star V)
\xrightarrow{\sim}\mathsf K(\Phi(V)\star U).
\end{equation}
They are natural in all enhanced morphisms and coherently
compatible with units, iterated products, and change of level.
Here $\rho_{U,\mathbf1}$ is the identity. The full slide
$\rho_{\mathbf1,V}:\mathsf K(V)\to\mathsf K(\Phi V)$,
followed by the Weil structure of $V$, is cohomological
Frobenius. Slides at disjoint sets of legs commute; their product
is the full slide.
\end{proposition}
\begin{proof}
For two kernels, the path description is
\[
(P,Q;\ U(P,Q),\ V(Q,{}^{\mathrm{Fr}}P)).
\]
Rotating the first step sends $(P,Q)$ to $(Q,{}^{\mathrm{Fr}}P)$.
To obtain pullback in that direction use its inverse, which sends
$(P,Q)$ to $({}^{\mathrm{Fr}^{-1}}Q,P)$. The two coefficients on this
rotated path are $\Phi(V)({}^{\mathrm{Fr}^{-1}}Q,P)$ and $U(P,Q)$.
This is precisely Equation~\eqref{cat:last-first}. In this
description $\mathrm{Fr}^{-1}$ may be interpreted on perfections. Etale
constructible sheaves and their compact direct images are
invariant under the resulting universal homeomorphisms, so the
construction also defines the same map before perfection.
For any finite list of kernels use the stack of all its
intermediate bundles. Multiplication forgets an intermediate
bundle, a unit repeats one, and rotation moves an end bundle
through Frobenius. The identities between these operations are
identities of the corresponding path diagrams. Applying compact
direct image, base change, and the projection formula in the
enhanced six-operation formalism preserves these identities
coherently. The required precise finite-type foundations are
the enhanced adic operations and their base-change and projection
formula compatibilities in
\cite[Section~7.1 and Theorem~7.2.7]{LiuZheng}.
One can express the coherence as a balanced bar construction.
In simplicial degree $n$ retain all intermediate kernels in the
path; inner faces compose adjacent kernels, degeneracies insert
units, and the end face is the rotation just described.
Their common refinements are again the same path stack, which
supplies every simplicial identity and its higher compatibility.
This verifies coherence in the enhancement, rather than only
pairwise isomorphisms in its homotopy category.
The same reasoning applies to the locally finite type stack.
A fixed finite diagram uses finitely many bounded kernels.
The image and inverse image of a finite-type bounded open under
the relevant bounded-position maps are contained in some larger
finite-type bounded open. Thus that diagram is computed on a
finite-type stage, after enlarging stages when necessary.
The finite-type maps are compatible under extension by zero,
so their filtered colimit preserves the same identities.
Central-lattice quotients are treated by the corresponding
locally finite sums of these diagrams.
When $U=\mathbf1$, inverse rotation of the path is $\mathrm{Fr}^{-1}$.
Its compact direct image is pullback by $\mathrm{Fr}$. Composing with the
Weil structure is therefore exactly cohomological Frobenius.
Moving disjoint groups of labels gives commuting rotations of
the common path diagram, with product the complete rotation.
\end{proof}
\subsection{Hecke characters and finite flag transfers}
If $U$ is a rigid Weil kernel with left dual $U^\vee$ and with
an interchange $U\star L\simeq L\star U$, define its character
operator on $\mathsf K(L)$ by
\begin{equation}\label{cat:character}
\begin{split}
\mathsf K(L)&\longrightarrow
\mathsf K(L\star U^\vee\star U)
\xrightarrow{\rho}
\mathsf K(\Phi U\star L\star U^\vee)\\
&\longrightarrow\mathsf K(U\star L\star U^\vee)
\longrightarrow\mathsf K(L\star U\star U^\vee)
\longrightarrow\mathsf K(L).
\end{split}
\end{equation}
The arrows use coevaluation, rotation, the Weil structure,
interchange, and evaluation, respectively. We denote this
operator by $\chi_U$. The same formula for a rigid kernel
between two different levels is called transfer.
The individual formula is a morphism of complexes. The Iwahori
Hecke-algebra action asserted next is its action on
$H^\bullet\mathsf K(L)$; no enhanced action of that algebra is
deduced from the homotopy central functor.
\begin{proposition}[Hecke action and transfer]\label{cat:hecke}
Suppose that $L$ has a central label at a closed place $x$.
The operators in Equation~\eqref{cat:character} give the usual
affine Hecke algebra at $x$, with parameter $q_x$, on
$H^\bullet\mathsf K(L)$. They give ordinary Hecke operators on
$H^\bullet\mathsf K(\mathbf1)$.
More explicitly, let $H_w$ denote the operator with characteristic
function $1_{IwI}$ and Iwahori volume one. Then
\[
H_s^2=(q_x-1)H_s+q_x,
\qquad H_wH_{w'}=H_{ww'}\quad
\text{if }\ell(ww')=\ell(w)+\ell(w').
\]
The positive normalized Bernstein translations are
\begin{equation}\label{cat:translation-sign}
T_a=(-1)^{\ell(t^a)}\chi_{J_a},
\qquad
T_a=q_x^{-\ell(t^a)/2}H_{t^a}\quad(a\text{ positive}),
\end{equation}
and extend multiplicatively to the entire translation lattice.
For a standard parahoric $P\supset I$, its two level transfers
identify the hyperspecial or parahoric trace cohomology with the image of
\[
e_P=\left(\sum_{w\in W_P}q_x^{\ell(w)}\right)^{-1}
\sum_{w\in W_P}H_w.
\]
At hyperspecial level the spherical character action agrees
with the Bernstein center. The individual-leg slides commute
with these actions and transfers. On the Wakimoto graded terms
the Bernstein action is the translation action tensored with
the weight multiplicities in Theorem~\ref{cat:central}.
The Wakimoto filtration gives long exact sequences equivariant for
these Bernstein translations and for the selected-leg slides.
Its graded terms carry the stated translation action. Thus flat
completion, ordered-weight projections, and spectral projectors
may be applied to these cohomology sequences.
There is also the following geometric form of the rank-one
assertion. If an exact translation position is parallel to the
simple wall of $s$, the operator $H_s+1$ on its path trace is
pullback and sum along the finite etale family of Frobenius-fixed
rank-one refinement flags, of degree $q_x+1$. Consequently
$H_s$ takes the sum over the other flags.
\end{proposition}
\begin{proof}
The kernels in use are rigid. Standard simple kernels are
invertible with inverse the corresponding costandard kernel;
length-zero kernels are invertible as well. Finite standard
filtrations, cones, and retracts preserve dualizability. The
projection kernels have the pullback and proper-direct-image
adjunctions of the smooth proper flag projection.
The character formula is multiplicative under convolution:
the coevaluation for a product is the composite of its two
coevaluations, and the cyclic and interchange identities move
the two factors successively. Applying the triangle identities
then gives the product of their characters on cohomology. This
uses only the homotopy hexagon and tensor identities specified
in Theorem~\ref{cat:central}. For additivity, fix the central
label $L$ and apply its chosen natural equivalence of exact
enhanced functors to the cofiber diagram of the test kernels.
This supplies a morphism of that cofiber diagram, not merely
unrelated maps on its three objects. On a mapping-cone model
the interchange is triangular with respect to the two summands, while
coevaluation and evaluation pair equal summands. The two
diagonal contributions are the character of the second term
and the negative of the character of the first term. This
also explains why shifts give their usual alternating signs.
For the quadratic relation use the unshifted closed kernel
$C_s=\mathbf k_{P_s/I}$, where $P_s/I\simeq\mathbf P^1$.
Convolution through the partial level gives
\[
C_s\star C_s\simeq
R\Gamma(\mathbf P^1,\mathbf k)\otimes C_s.
\]
The open--closed triangle gives $\chi_{C_s}=H_s+1$.
The Weil trace of $R\Gamma(\mathbf P^1,\mathbf k)$ at $x$
is $1+q_x$. Thus $(H_s+1)^2=(1+q_x)(H_s+1)$, the asserted
relation. Length-additive convolution of open cells is an
isomorphism, giving the length and braid relations. For a
degree-$d_x$ place, all kernels are external products on its
geometric legs with cyclic Frobenius. The trace of a graded
cycle on $C^{\otimes d_x}$ is the trace of the composite
Frobenius on $C$; hence the parameter is $q^{d_x}$.
The same identity shows that a shifted simple or translation
kernel contributes its single-factor sign
$(-1)^{\ell(t^a)}$, not a spurious sign depending on the
number of geometric legs. This proves
Equation~\eqref{cat:translation-sign}. Length parity is a
character of the translation lattice, so multiplicativity is
preserved.
At $L=\mathbf1$ the path stack is the Frobenius-fixed bundle
groupoid. On it, the unit, counit, and Weil structure in
Equation~\eqref{cat:character} sum the stalk traces over the
Hecke correspondence, with its stack multiplicities. This is
the ordinary action on compactly supported functions on the
rational bundle groupoid. The construction for other central
labels uses the same diagrams and their central interchange,
so the verified relations apply there as well.
For general $P$ repeat the closed-kernel computation with the
full flag $P/I$. Connected transition groups act trivially
on its cohomology, so its cohomology is the constant Tate
complex with Frobenius trace
$c_P=\sum_{w\in W_P}q_x^{\ell(w)}$. The composite of the
two transfers at the $P$-level is multiplication by $c_P$;
the opposite composite is $\sum_{w\in W_P}H_w$.
Since $c_P\ne0$, the two normalized transfers give the
stated idempotent identification. Theorem~\ref{cat:central}
identifies the projected central kernel with Satake at a
hyperspecial level. Trace additivity on its Wakimoto
filtration then gives the usual Bernstein central character,
which proves compatibility of the spherical action.
We spell out the assertion about parallel walls, which is
needed for a geometric correspondence later. For a translation
$t^a$ parallel to the $s$ wall, its character on the rank-one
root subgroup has valuation zero. Thus the exact partial
relative position identifies the rank-one refinement flag at
the two ends. On the Frobenius path stack the allowed flags
are the fixed flags of that semilinear identification. They
form a finite etale family $Z\to Y$ of degree $q_x+1$.
The closed kernel factors through the partial level, and
its character factors through the two transfer maps.
Those maps are the pullback and sum for $Z\to Y$.
Indeed, smooth proper base change reduces the unit/counit
calculation to the diagonal of $\mathbf P^1$ and the graph
of its twisted Frobenius. Their intersections are transverse:
Frobenius has zero tangent map and the diagonal contributes
the identity. Every fixed flag therefore has coefficient
one. Etale locally on $Y$, where $Z$ is a disjoint union of
sections, the resulting maps are the diagonal map and the
sum map between constant coefficient complexes. They agree
on overlaps and hence descend. This calculation is unchanged
after tensoring with the coefficient pulled back from the
partial path stack, or after retaining automorphisms of the
bundles. Subtracting the diagonal, by the open--closed
triangle, gives the other-flags operator $H_s$.
Finally, naturality of full rotation makes Frobenius commute
with every Weil Hecke action. A slide at an auxiliary disjoint
place commutes with this action by its common disjoint-leg
diagram. Dividing the full slide by the other, invertible
slides proves the same assertion for the selected-leg slide.
Projection transfers use those same diagrams. The graded
translation assertion follows from the map compatibility in
Theorem~\ref{cat:central}. More explicitly, those filtered
interchanges and rotation maps commute with the inclusion,
quotient, and connecting maps of each successive filtration
triangle. Applying cohomology gives the asserted equivariant
long exact sequences. This is all the filtration compatibility
needed here; it does not require an action on a derived
completion of the Iwahori trace.
\end{proof}
Whenever an unqualified equality between a translation slide
and Frobenius is used below, the large translation labels are
chosen divisible by an even integer. Their length parity then
vanishes. The weights of a representation with such an
extremal label have the same even parity, since their
differences are roots of the dual group. Filter translations
remain arbitrary and always use the explicit normalization
in Equation~\eqref{cat:translation-sign}.
\subsection{The algebra of a twisted loop}
The final local calculation is given in terms of dg algebras,
so positive cohomological polynomial degrees cause no
connectivity assumption. It applies to the entire compact
cohomology functor, including fixed disjoint kernels.
Its enhanced input is derived spherical Satake and the coherent
path functor, independently of the homotopy central functor at
Iwahori level.
\begin{proposition}[Spectral form of a Frobenius trace]\label{cat:loops}
At a hyperspecial closed-place leg of degree $d$, the functor
$\mathsf K$ on perfect spherical labels has the form
\begin{equation}\label{cat:loop-pairing}
\mathsf K(P)\simeq
\left(\mathcal A\otimes_{B_{q^d}}
(B_{q^d}\otimes_{S^{\otimes d}}P)\right)^{\Gd},
\end{equation}
where $\mathcal A$ is an equivariant dg module, with no
coherence or boundedness assumption. After eliminating
intermediate factors, the algebra is
\begin{equation}\label{cat:loop-algebra}
B_r=\mathcal O(\Gd)\otimes
\operatorname{Sym}(\gd^*[-2]\oplus\gd^*[-1]),
\qquad r=q^d.
\end{equation}
Write $s$ for its degree-zero holonomy variable and $e$ for
its degree-two linear variable. The second summand supplies
odd Koszul generators $\eta_\alpha$ of degree $1$, with
\begin{equation}\label{cat:holonomy}
d\eta_\alpha=
\langle\alpha,\operatorname{Ad}(s)e-q^d e\rangle.
\end{equation}
The group acts by simultaneous conjugation.
The $d$-fold selected-leg slide on representation labels is
the action of $s$ on each factor. Spherical Hecke characters
act by the corresponding invariant characters of $s$.
On cohomology these descriptions agree with the transfers in
Proposition~\ref{cat:hecke}. A degree-two morphism linear
in the selected $e$ variable multiplies the corresponding
slide eigenvalue by $q^d$. All statements are natural in
perfect labels and their dg morphisms.
\end{proposition}
\begin{proof}
Put $\mathcal C=\operatorname{Mod}_S
(D(\operatorname{Rep}_{\rm rat}(\Gd)))$ and let $\Phi_r$
be extension of scalars by $\alpha\mapsto r\alpha$.
The free objects $S\otimes V$ are compact, dualizable
generators: rational invariants of a reductive group in
characteristic zero are exact and commute with direct sums
\cite[Remark~15.13 and Proposition~15.16]{MilneAlgebraicGroups}.
Thus this category is the Ind-completion of its perfect
objects.
We first calculate the universal trace for one factor.
Let $\mathcal C^e=\mathcal C\otimes\mathcal C^{\rm rev}$.
Use the regular right action
$L\cdot(P,Q)=Q\otimes L\otimes P$ and the twisted left
action $(P,Q)\cdot M=\Phi_r(P)\otimes M\otimes Q$.
The pairing $(L,M)\mapsto\mathsf K(M\otimes L)$ is
coherently balanced: the balancing map is precisely
\[
\mathsf K(M\otimes Q\otimes L\otimes P)
\xrightarrow{\rho}
\mathsf K(\Phi_r(P)\otimes M\otimes Q\otimes L).
\]
Proposition~\ref{cat:cyclic} verifies all the bar identities.
Consequently $\mathsf K$ extends to a continuous functional
on the relative tensor product of these two module
categories.
Here is an explicit algebra calculation of that relative
tensor product. It is the derived intersection of the
diagonal and the graph $(\Phi_r,\operatorname{id})$ in the
product presentation of $[\operatorname{Spec}S/\Gd]$.
This language abbreviates a module-algebra computation:
\cite[Proposition~4.1]{BFN} identifies the tensor product of
module categories with modules over the derived tensor
product of their defining algebra objects. It applies in
the stable presentable category of equivariant modules,
without a connectivity condition.
An object of this intersection has a diagonal coordinate
$x$, a graph coordinate $y$, and two group identifications
\[
a:x\longrightarrow r y,
\qquad b:x\longrightarrow y.
\]
Resolve the equations
$\operatorname{Ad}(a)x-r y=0$ and
$\operatorname{Ad}(b)x-y=0$ by their degree-one Koszul
generators. Their algebra, before group descent, is the
polynomial algebra on $x,y$ and the two group variables
with those Koszul generators. These resolutions are
$K$-flat: their exterior-degree filtrations have free
graded polynomial quotients. Eliminate $y$ and its
Koszul generator using the second equation. This is an
invertible linear substitution followed by removal of
contractible Koszul pairs. Trivialize $b$ by the product
group action, put $e=y$ and $s=a b^{-1}$, and transport
the remaining generator by the same change of basis.
The residual group is the diagonal $\Gd$, its action is
conjugation, and the remaining differential is
\[
d\eta_\alpha=
\langle\alpha,\operatorname{Ad}(s)e-r e\rangle.
\]
This is exactly $B_r$. In particular the specified
null-homotopies of the equation have been retained.
The universal trace sends a label $P$ to $B_r\otimes_S P$.
The comparison from the second-coordinate representation
to the first-coordinate representation in the preceding
diagram is $a b^{-1}:y\to r y$. It is the last-to-first
slide of Equation~\eqref{cat:last-first}. Hence its action
on a representation is $V(s)$, with the asserted
orientation. Coevaluation and evaluation then give
$\operatorname{Tr}_V(s)$ for a rigid character transfer.
For a cyclic degree-$d$ leg, retain $d$ coordinates and
the $d$ twisted identifications. Trivializing all but one
of the group variables eliminates the intermediate
coordinates and their Koszul pairs. The remaining
transport is their ordered product $s$, and its equation
is $\operatorname{Ad}(s)e=q^d e$. Carrying a representation
label around this full cycle acts by $V(s)$. For an
external tensor label every factor has this same full
transport; a single partial turn also performs the
appropriate graded cyclic permutation. This explains
both the $d$-fold slide and the map from the original
$S^{\otimes d}$ to the loop algebra in
Equation~\eqref{cat:loop-pairing}.
It remains to justify that the continuous functional is
pairing with a module, including when it is not coherent.
Let $\mathcal T=\operatorname{Mod}_{B_r}
(D(\operatorname{Rep}_{\rm rat}(\Gd)))$ and write the
functional as $L:\mathcal T\to D(\mathbf k)$.
Again $B_r\otimes V$ are compact dualizable generators.
The exact contravariant functor on $\mathcal T^c$
given by $P\mapsto L(P^\vee)$ is represented by an
object $\mathcal A\in\operatorname{Ind}(\mathcal T^c)
=\mathcal T$. Tensor duality gives, for compact $M$,
\[
L(M)\simeq
\operatorname{Hom}_{\mathcal T}(M^\vee,\mathcal A)
\simeq(\mathcal A\otimes_{B_r}M)^{\Gd}.
\]
Both sides preserve colimits, proving the formula for
all $M$, and hence Equation~\eqref{cat:loop-pairing}.
The character calculation is natural in labels and
their evaluation and coevaluation maps. The projection
transfer of Proposition~\ref{cat:hecke} is therefore the
same character operation on cohomology in this model. Finally, for a
linear degree-two morphism $u:P\to Q[2]$, cyclic
naturality and Theorem~\ref{cat:satake} give
\[
\sigma_Q[2]\,\mathsf K(u)
=q^d\,\mathsf K(u)\,\sigma_P,
\]
where $\sigma$ is the full selected-leg slide. This
proves the stated change in eigenvalue. The degree is
even, so no additional Koszul sign occurs.
\end{proof}
In particular $Z_{\rm sph}=\mathcal O(\Gd)^{\Gd}$ acts as an
actual degree-zero central dg algebra through $B_r$. Here is the
precise spectral-projector consequence. For an external Satake label
with representation $V^{\otimes d}$, let
$\rho=V^{\otimes d}$ and let $\sigma$ be its $d$-fold selected-leg
slide. Before tensoring with $\mathcal A$ and taking invariants,
$\sigma$ is the matrix $\rho(s)$ on the finite free $B_r$-module
$B_r\otimes V^{\otimes d}$. For any positive integer $N$,
\[
p(z)=\det\bigl(z-\rho(s)^N\bigr)\in Z_{\rm sph}[z]
\]
annihilates $\sigma^N$ strictly by Cayley--Hamilton. The identity
persists after tensoring with $\mathcal A$ and taking invariants.
If a flat $Z_{\rm sph}$-algebra $R$ gives a coprime factorization
$p=p_0p_1$, choose $a p_0+b p_1=1$. Then
$b(\sigma^N)p_1(\sigma^N)$ is an actual chain idempotent on the
spherical complex after this base change. It selects the $p_0$
factor and gives its dg direct summand, regardless of boundedness.
This is the enhanced projector used after transferring a
cohomological Iwahori summand to hyperspecial level.
Two consequences will be used repeatedly. First, imposing
finitely many equations in invariant functions of $s$ is
ordinary derived tensor with a finite Koszul complex and
commutes with every operation just described. Second,
after such a specialization, an invertible coefficient
of a nonresonant linear equation permits its variable
and Koszul generator to be eliminated as a contractible
pair. Thus the action of the original polynomial
variables on a resonant slice is their ordinary
restriction, with cohomological degree still $2$.
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/02-categorical.tex ====
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/03-amplitude.tex ====
\section{A uniform upper bound for the extreme slide}
\label{sec:amplitude}
We bound the cohomological degrees in the part of a spherical trace
whose slide eigenvalues have maximal absolute value, as the highest
weight grows along a fixed rational ray. The geometric input is a
uniform upper bound for the costandard tests of
Proposition~\ref{cat:springer}. After semisimple specialization, these
tests control the degrees of fibers of the resonant module, with a
correction determined by compatible nilpotent triples. A finite orbit
decomposition then turns the fiber bounds into the upper estimate of
Theorem~\ref{amp:amplitude}. The module may be neither coherent nor
bounded; the uniformity of the tests is what permits this passage.
We use cohomological grading: $H^i(C[a])=H^{i+a}(C)$.
An upper bound $b$ for a complex means that $H^i=0$ for $i>b$;
it does not assert finite-dimensionality or a lower bound.
Throughout this section all bundle levels, all kernels at other places,
and the number of variable kernels are fixed. Constants may depend on
these data. They will not depend on the dominant weights in a nef family.
For a reductive group we either bound the abelian degree or divide by a
fixed lattice of central modifications with cocompact image in the free
abelian degree lattice. Such a lattice acts freely on that lattice.
The assertions below apply to either convention.
\subsection{Bundle geometry and costandard transforms}
Fix a norm on the semisimple part of the rational cocharacter space.
The instability of a bundle is the norm of its dominant
Harder--Narasimhan type. The size of a modification is any fixed norm
bound for its relative positions, summed over the geometric legs.
Changing either norm changes only the constants.
\begin{proposition}[Bounded geometry]
\label{amp:geometry}
For a fixed split connected reductive group and fixed finite level,
the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item The number of rational isomorphism classes of bundles of
instability at most $b$ is bounded by $C(1+b)^N$, for fixed $C,N$.
\item There is a locally finite constructible stratification of the
bundle stack such that, on a stratum where the automorphism group has
dimension $r$, the stratum has dimension at most $C-r$.
The same constant works for all Harder--Narasimhan types.
\item Suppose that a sequence of bundles $\mathcal E_n$ has
meromorphic isomorphisms to its point-Frobenius transforms, or
meromorphic automorphisms, with the $n$th modification of size at
most $b_n$. If the ratio of the instability of $\mathcal E_n$ to
$1+b_n$ is unbounded, a subsequence has a coarsening of the canonical
reduction to one fixed maximal parabolic that these isomorphisms
carry to its point-Frobenius transform, or to itself, respectively.
The assertion holds over geometric points and is
compatible with descent and iteration of the Frobenius isomorphism.
\end{enumerate}
All statements allow an arbitrary effective level divisor, including
nonreduced divisors, and any fixed list of parahoric flags.
\end{proposition}
\begin{proof}
We recall precisely the principal-bundle input. In arbitrary
characteristic the canonical parabolic reduction exists and is unique;
its Levi bundle is semistable. Semistable Levi bundles in a fixed
component form a quasicompact stack. The reduced Harder--Narasimhan
strata are of finite type, and the stack of canonical reductions maps
finitely and radicially onto its stratum. These are the assertions of
\cite[\S\,2.3.1, Theorem~2.3.3(a),(c), and \S\,2.4.1]{Schieder}.
We use the finite radicial description, not an assertion that this map
is an isomorphism in every characteristic.
For each of the finitely many standard Levis $M$, central cocharacters
cover the free degree lattice of $M$ up to finite index; see
\cite[\S\,2.1.3]{Schieder}.
After tensoring by central bundles, its semistable bundles consequently
belong to finitely many bounded families. Degree-zero central bundles
are included in these families, since the corresponding Picard stacks
are of finite type. Fix all the representations of these Levis that
occur in the root filtrations and in highest-weight realizations of
the finitely many maximal flag varieties. On the bounded families,
the maximal and minimal slopes of these associated vector bundles
are bounded. Restoring a central translation changes each central
weight quotient by its prescribed scalar degree. It follows that a
quotient of central weight $\beta$ has all slopes in
\begin{equation}
\label{amp:slope-interval}
[\langle\beta,\nu\rangle-C_0,
\langle\beta,\nu\rangle+C_0],
\end{equation}
where $\nu$ is the canonical type and $C_0$ is independent of $\nu$.
There are only finitely many representations here, so one constant
suffices. This argument uses boundedness of families; it does not
use preservation of semistability by a representation in positive
characteristic.
Choose a fixed large number $A>C_0+2g_X-2$, enlarged for the finitely
many weight filtrations if necessary. Coarsen the canonical parabolic
by retaining exactly the simple gaps greater than $A$.
Write this parabolic as $P=LU$. In its induced Levi bundle the
remaining semisimple gaps are bounded by $A$. It therefore belongs,
modulo central translation in $L$, to a bounded family: its canonical
Levi is one of the semistable Levis considered above, and its remaining
degree gaps range over a finite set of rational lattice points.
Use the relative-root height filtration of $U$, whose successive
quotients are vector groups with linear $L$-action. This parabolic
filtration and its commutator inclusions hold in every characteristic;
see \cite[Theorem~5.4.3]{ConradReductive}. Filtering them further
by the weights of the original canonical Levi, every resulting root
has a positive coefficient on a retained gap. Equation~\eqref{amp:slope-interval}
therefore gives minimal slope greater than $2g_X-2$.
Their $H^1$ vanishes by Serre duality. The $H^1$ of every successive
vector-group quotient vanishes as well, and the nonabelian cohomology
sequence, applied successively, shows that the $P$-bundle is induced
from its $L$-bundle. The cutoff was chosen from the original
semistable-Levi families before coarsening, so there is no circular
choice of a bound depending on $A$.
It follows that every bundle is induced from one of finitely many
bounded families for a Levi, followed by a central translation in
that Levi. The translations required in an HN ball are lattice
points in a ball of radius $O(1+b)$ and are polynomially many.
These translations can be chosen over the fixed finite field after
passing to a fixed finite-index degree lattice: choose a divisor of
positive degree over that field and use its line bundle. The
remaining degree cosets are absorbed into the bounded families.
The bounded principal-bundle families have finitely many rational
classes. To see this directly, fix a faithful representation and
a twist for which all the associated vector bundles are globally
generated and have vanishing $H^1$. A basis of global sections
rigidifies each bundle into one of finitely many finite-type framed
algebraic spaces. Every rational bundle admits such a basis over
the finite field, and each framed space has finitely many rational
points. The canonical
reduction of a rational bundle descends by uniqueness, and the
vanishing of $H^1(U)$ above holds over the finite field itself.
Thus every rational class is obtained from the same finite list of
rational Levi families. This proves (a).
Choose finite-type scheme atlases for these bounded families, with
dimensions bounded by $C$. Each central translate of each atlas
maps to the bundle stack. A geometric fiber of such a map over an
object with automorphism dimension $r$ has dimension at least $r$:
it parametrizes isomorphisms from a family member to that object.
The dimension formula bounds the image stratum by $C-r$.
Within each fixed HN stratum only finitely many translating
choices occur, and the HN stratification is locally finite.
Refining the images and the stabilizer dimensions thus proves (b)
with a locally finite stratification.
Universal homeomorphisms in the canonical-reduction description do
not change these dimensions or the constructible-sheaf calculation.
For (c), pass to a subsequence on which a simple gap tends to infinity
relative to $1+b_n$, and fix its maximal parabolic $P_i$.
Choose an actual $G$-equivariant projective embedding
$G/P_i\hookrightarrow\mathbb P(V_i)$ using a sufficiently ample
linearized line bundle with weight a divisible multiple of the
fundamental weight. In particular the scheme-theoretic stabilizer
of the base-point line is $P_i$; no assertion about the stabilizer
of an arbitrary simple highest-weight module in small characteristic
is required.
The line defining the coarsened canonical reduction is the unique
highest-weight line. Every other weight differs from this weight
by a positive combination of simple roots involving the selected
root. Equation~\eqref{amp:slope-interval} shows that the degree of this
line exceeds every slope of the quotient by the selected gap minus
a fixed constant. A modification of size $b_n$ in this fixed
representation has poles bounded by a divisor of degree $O(b_n)$.
The resulting homomorphism from the highest line to the target
quotient therefore vanishes when the gap is sufficiently large.
The meromorphic isomorphism preserves the reduction line and hence
its maximal-parabolic reduction. For point Frobenius, the target
canonical type and degrees are the same: point Frobenius acts on
the parameter field, not by Frobenius pullback along $X$.
Uniqueness of the canonical reduction gives the descent and
iteration assertions.
Finally, a full level structure has a space of choices of fixed
finite type and fixed dimension; over the finite field its possible
trivializations form a torsor under the fixed finite group
$G(\mathcal O_D)$. Parahoric flags have the same boundedness
property. They change all the preceding constants by fixed amounts.
No argument uses reducedness of $D$.
\end{proof}
\begin{lemma}[The diagonal and the Frobenius graph]
\label{amp:trace-bound}
Let $\mathcal B$ be one of the bundle stacks in
Proposition~\ref{amp:geometry}. The object
$\Delta_!\mathbf k$ on $\mathcal B\times\mathcal B$ has a uniform
upper perverse bound. Moreover, for a constructible complex
$\mathcal F$ on $\mathcal B\times\mathcal B$ with upper perverse bound
$N$, pullback to the point-Frobenius graph followed by compact
cohomology has upper bound $N+C_1$, where $C_1$ is independent of
$\mathcal F$.
\end{lemma}
\begin{proof}
We use middle perversity with smooth descent, as in
\cite[Lemma~4.1 and Theorems~4.2, 5.1]{LaszloOlssonPerverse}, so that
$\mathbf k[-r]$ is perverse on the classifying stack of a smooth
group of dimension $r$. Stratify further by stabilizer dimension
and by the isomorphism relation. Over a pair of isomorphic bundles
with stabilizer dimension $r$, the stalk of $\Delta_!\mathbf k$
is the compact cohomology of their isomorphism scheme and has upper
degree $2r$. The corresponding locus of isomorphic pairs has
dimension at most $C-2r$, by Proposition~\ref{amp:geometry}(b).
The support criterion for the upper perverse truncation gives a
bound independent of $r$.
At a point of the Frobenius graph the stabilizer dimension in
$\mathcal B\times\mathcal B$ is $2r$. Every stratum through it has
dimension at least $-2r$. Thus the stalk upper bound of
$\mathcal F$ is at most $N+2r$. On the graph the relevant stratum
has dimension at most $C-r$. Compact cohomology of a sheaf on a
finite-type algebraic stack of dimension $a$ vanishes in degrees
above $2a$; see \cite[Theorem~1.4]{SunStacks}.
The resulting bound is
\[
(N+2r)+2(C-r)=N+2C.
\]
Stratification and the spectral sequence for cohomology sheaves give
the same estimate for complexes. Apply the calculation on
finite-type opens and then take the defining filtered colimit.
The bounds are independent of the open, so they persist in that
colimit. The central-degree convention is treated componentwise.
\end{proof}
\begin{proposition}[Uniform geometric tests]
\label{amp:tests}
In the hyperspecial cyclic calculation, a fixed list of perfect
spherical labels has bounded-above cohomology. The same upper bound
property holds uniformly when the list also contains a fixed number
of Springer pushforwards
\[
R\pi_*\mathcal O_{\widetilde{\mathcal N}}(\eta_j),
\qquad
\pi:\widetilde{\mathcal N}\longrightarrow\gd,
\]
whose line bundles are independently nef. Fixed additional twists
or fixed perfect factors are allowed. The number of factors and
the additional twists are fixed; their nef weights may vary
independently without changing the upper bound.
\end{proposition}
\begin{proof}
Compute the trace by applying the Hecke transforms to one factor
of $\Delta_!\mathbf k$ and then applying
Lemma~\ref{amp:trace-bound}. Every fixed bounded kernel has finite
upper perverse amplitude. A costandard kernel has a bound
independent of its length, as follows.
Let $\mathcal H_w$ be its exact-position Hecke stratum, with source
and target maps $p,q$. The map $p$ is smooth of relative dimension
$\ell(w)$ and $q$ is affine; these facts are checked after smooth
local trivialization, where their fibers are the affine Schubert
cell. In the closure correspondence the target map is proper.
Here is the coefficient comparison for an arbitrary locally bounded
constructible input. Choose a sufficiently deep jet subgroup acting
trivially on the chosen Schubert closure, and pass to its smooth
torsor of trivializations. The open and closed Hecke spaces then
become products over the source. For bounded constructible
complexes the formula
\[
R(\operatorname{id}\times j)_*(A\boxtimes B)
\simeq A\boxtimes Rj_*B
\]
follows by Verdier duality from the corresponding extension-by-zero
formula. It does not require $A$ to be lisse. Thus smooth base
change \cite[\S\,12.1]{LaszloOlssonSixII} and descent identify
the costandard coefficient with
$Rj_*p^*(-)[\ell(w)]$. Proper pushforward from the closure
identifies the costandard transform with
\[
Rq_*p^*(-)[\ell(w)].
\]
Smooth pullback with this shift preserves perversity, and affine
Artin vanishing says that $Rq_*$ preserves an upper perverse bound;
see \cite[\S\,3.2, paragraph preceding Theorem~3.2]{BeilinsonPerverse}.
Both assertions descend from the smooth scheme charts, since the
maps of the correspondence are representable. The resulting upper
bound is independent of $w$.
Passage between the fixed parahoric levels costs only a fixed
amplitude, because their fibers are fixed flag varieties.
Forgetting and then right-averaging torus equivariance has a
representable torus-torsor fiber. Its direct image has a finite
Postnikov filtration with constant cohomology pieces $H^i(T,\mathbf k)$:
translations act trivially on torus cohomology. This is cohomology
of $T$, not of $BT$, and no splitting of the filtration is needed.
It adds only fixed shifts. This is the averaging convention in
Proposition~\ref{cat:springer}.
By the same proposition, nef Springer line bundles correspond to
costandard Wakimoto translations, with shifts independent of their
weights. A bounded additional translation can be composed with a
nef translation at a fixed cost. Convolution of a fixed number of
these transforms preserves the uniform upper bound just proved.
The spherical polynomial action and the identification of their
products with derived tensor products are those of
Theorem~\ref{cat:satake} and Proposition~\ref{cat:springer}.
\end{proof}
\subsection{Specialization and uniform Levi tests}
Fix a closed leg of degree $d$, set $Q=q^d$, and choose a complex
realization of $\mathbf k$. Let $t\in\Gd$ be semisimple and put
\[
H=Z_{\Gd}(t),\qquad
E_t=\{e\in\gd:\operatorname{Ad}(t)e=Qe\}.
\]
In a torus containing $t$, decompose $\log_Q|t|$ into its
connected-central and semisimple components, using the rational
direct sum of the connected-center and derived-torus cocharacter
spaces. Throughout this section $\lambda$ denotes the
\emph{semisimple component}, and we assume that $\lambda$ is
rational. No rationality is required of the connected-central
component. This convention applies to every grading and
cocharacter below, as well as to metric pairings.
The direction $\lambda$ is central in $H$: conjugation fixing $t$
also fixes its absolute-value direction and its semisimple
projection. Roots annihilate the discarded central component;
therefore every vector in $E_t$ still has $\lambda$-weight one.
Replacing $t$ by $tz$, for $z\in Z(\Gd)^\circ$, leaves
$\lambda$ unchanged.
Specialization at the class of $t$ always means the following finite
operation. Choose once algebra generators $a_1,\ldots,a_r$ of
$\mathbf k[\Gd]^{\Gd}$, and tensor the cyclic calculation with
the Koszul complex of $a_1-a_1(t),\ldots,a_r-a_r(t)$. These functions
generate the maximal ideal of $[t]$. No regularity of the adjoint
quotient is assumed. The generators have degree zero, and the number
and degrees of the Koszul factors are independent of $t$. We retain
all the derived structure produced by this operation.
\begin{lemma}[The specialized test module]
\label{amp:specialization}
After this specialization, the cyclic calculation is a pairing
with an $H$-equivariant dg module $M$ over
$\operatorname{Sym}(E_t^*[-2])$. The module need not be coherent
or bounded. The action of a spherical polynomial at any one
geometric factor is restriction of its linear functions to $E_t$.
The uniform tests of Proposition~\ref{amp:tests} remain valid.
They also allow fixed equivariant perfect factors on the flag
varieties, in particular restriction to fixed closed
$H^\circ$-orbits of Borels.
\end{lemma}
\begin{proof}
By Proposition~\ref{cat:loops}, the holonomy satisfies the derived
linear equation $\operatorname{Ad}(s)e=Qe$.
The fiber of the adjoint quotient over $[t]$ consists of elements
conjugate to $tu$, with $u$ unipotent in $H$.
Indeed the adjoint quotient is the Weyl quotient of a maximal torus
\cite[Theorem~4.1\textup{(2)}]{LeeAdjointQuotients}, and representation
characters are unchanged by the commuting unipotent factor in Jordan
decomposition.
Here is an algebraic slice description, with no completion of the
coefficient module. Let $W$ be the sum of the nonresonant
$\operatorname{Ad}(t)$-eigenspaces in $\gd$. Choose an
$H$-stable affine open $U\subset H^\circ$ containing the unipotent
locus on which $1-\operatorname{Ad}(tu)$ on
$\gd/\operatorname{Lie}H$ and $\operatorname{Ad}(tu)-Q$ on
$W$ are invertible. Both determinants are nonzero at every
unipotent $u$. There are only finitely many $H$-classes of
semisimple $u$ for which $tu$ is conjugate to $t$, since their
torus representatives come from the finite Weyl orbit of $t$.
Invariant functions allow us also to remove all these classes
except $u=1$.
The conjugation map
\[
\Gd\mathbin{\times}^{H}(tU)\longrightarrow\Gd
\]
is etale: its transverse differential is the first displayed
invertible operator. After the removal of the other sheets,
uniqueness of the Jordan semisimple part identifies every
conjugating ambiguity over the class fiber with $H$.
The restriction to the schematic class fiber is therefore etale
and universally bijective, hence an isomorphism.
Being etale, it is an isomorphism over every infinitesimal
thickening of that fiber as well. Flat pullback and descent
therefore identify the supported quasi-coherent complexes on
these neighborhoods, equivariantly. This assertion applies to
arbitrary complexes: on affine charts it holds for modules killed
by each power of the fiber ideal, hence for their filtered unions;
flatness computes pullback on cohomology in every degree.
The finite Koszul specialization has cohomology annihilated by
the fiber ideal, so it is covered by this argument. In particular,
all equivariance remains ordinary rational $H$-equivariance.
Over $\mathcal O(U)$, the invertible nonresonant block of
$\operatorname{Ad}(tu)-Q$ gives a change of odd Koszul generators
for which their differentials are precisely the coordinates of
$W$. Eliminating these contractible pairs is an actual dg-algebra
quasi-isomorphism over $\operatorname{Sym}(E_t^*[-2])$.
Solve successively for the transported variables at the other
geometric factors in the same way. Push forward the unipotent
and additional derived variables along their affine projection.
They remain in $M$, including all invariant Koszul generators and
the residual equation $\operatorname{Ad}(u)e=e$.
The polynomial-action assertion follows by solving the equations
starting at the chosen geometric factor. The description is
$H$-equivariant.
Tensoring with the finite Koszul complex changes upper bounds by
only a fixed amount. For a fixed finite-dimensional representation
of a reductive subgroup of $\Gd$, every irreducible summand occurs
in a restriction from $\Gd$. One proof uses the surjection
$\mathbf k[\Gd]\to\mathbf k[H]$, takes a finite-dimensional
$\Gd$-subrepresentation containing lifts of its matrix coefficients,
and then uses complete reducibility. Thus all fixed $H$-tests are
available. Induction from $H^\circ$ to $H$ is finite, so it also
allows $H^\circ$-invariants and fixed $H^\circ$-tests.
These retracts are taken after specialization on the slice; their
maps need not extend equivariantly before specialization.
Here is the perfect-generation detail for the flag factors.
On the smooth projective flag variety choose an ample equivariant
line bundle $A$. Equivariant Serre evaluation gives surjections
from finite sums of $V\otimes A^{-n}$ onto any equivariant coherent
sheaf. For an explicit finiteness argument, let the flag variety have
dimension $a$. More than $2a$ successive evaluations give an
extension whose class vanishes, since the local-to-global Ext
spectral sequence and exact reductive invariants give
$\operatorname{Ext}^{n}_{H^\circ}(F,K)=0$ for coherent $F,K$ and
$n>2a$. The truncated evaluation complex therefore contains $F$
as a retract. Applying finite cohomology filtrations shows that
every perfect complex is in the finite thick closure of these
bundles. Apply this statement to its dual and dualize: the same
perfect complex is in the finite thick closure of bundles
$V\otimes A^n$, $n\geq0$. All representations, twists, cones,
and shifts in this construction are fixed once the perfect complex
is fixed. Tensoring a variable nef line with these positive twists
keeps it nef. The structure sheaf of a fixed closed smooth
$H^\circ$-flag orbit is perfect on the ambient flag variety, so the
assertion applies to its restriction test. This proves the last
claim with constants independent of the variable nef weights.
\end{proof}
The specialized module $M$ satisfies the geometric tests, but need
not be coherent or bounded. Bounds for each fixed representation
test alone would not control cohomology
whose representation types vary with the degree. We next obtain
bounds uniform under character twists on each Levi. In
Proposition~\ref{amp:fiber}, these twists and a finite list of fixed
representations will detect every representation constituent, turning
the invariant test bounds into a bound on the whole fiber.
For a cocharacter $\gamma$ commuting with $t$, define
\[
L=Z_{\Gd}(\gamma),\qquad H_L=H^\circ\cap L,
\qquad E_L=E_t\cap\operatorname{Lie}L,
\qquad M_L=M\otimes^{\mathbf L}_{\mathbf k[E_t]}\mathbf k[E_L].
\]
The group $H_L$ is connected reductive, by the structure of torus
centralizers \cite[Theorem~17.38 and Corollary~17.59]{MilneAlgebraicGroups}.
All coordinate rings in
this notation carry the degree-two shearing. A representation
test is placed in ordinary cohomological degree zero.
Since $\lambda$ is central in $H$, it commutes with $\gamma$
and is a rational central cocharacter of $H_L$; its weight
on $E_L$ is one.
\begin{proposition}[Uniform central-character tests]
\label{amp:levi}
For every such $L$ and every fixed finite-dimensional rational
$H_L$-representation $U$, there is a constant $C(U,L)$ such that
\begin{equation}
\label{amp:levi-bound}
H^i\bigl((M_L\otimes U\otimes\eta)^{H_L}\bigr)=0
\quad\text{for }i>C(U,L),\qquad \eta\in X^*(L).
\end{equation}
Invariants denote invariant global sections. The constant is
independent of the character $\eta$.
\end{proposition}
\begin{proof}
Choose Borels containing a reference torus with $t,\gamma$.
First order their roots by $\gamma$ and $-\gamma$, respectively;
within $L$, order them positively on $\lambda$ and complete the
order arbitrarily on the zero-weight roots. Their
$H^\circ$-orbits are complete flag varieties. Use two Springer
tests restricted to these orbits, as permitted by
Lemma~\ref{amp:specialization}.
Let $P_\gamma,P_{-\gamma}$ be the two opposite parabolics.
Choose an integral character $\chi$ strictly dominant for the first
partial flag variety and zero on the coroots of $L$.
The extremal pairing in mutually dual representations gives an
invariant section of the product of two nef line bundles on the
partial flags. Project the selected complete-flag orbits to these
partial-flag varieties. On the product of the two resulting
$H^\circ$-orbits, the section's nonvanishing locus is the locus
of opposite partial flags, which is
$H^\circ/H_L$. Indeed the ambient big-cell Gauss factorization
is unique and is preserved by conjugation by $t$. If an element
of $H^\circ$ admits that factorization, its unipotent and Levi
factors centralize $t$ as well and lie in the corresponding
parabolics of $H^\circ$. Thus ambient oppositeness restricts to
exactly the opposition locus for $H^\circ$. Localizing by that section is the filtered colimit
of simultaneous positive twists by the two nef lines.
The section and its localization maps have degree zero.
The pairs of complete flags form a fibration over this opposition
locus. Over the reference opposite partial-flag pair, its fiber is
the product of two complete flag varieties for $H_L$. The resonant
parts of the two ambient nilradicals are
\[
E_{\gamma\geq0}\quad\text{and}\quad E_{\gamma\leq0}.
\]
Indeed the zero $\gamma$-part $E_L$ has positive $\lambda$-weight
and belongs to both nilradicals, independently of the remaining
flags; the roots of $H_L$ have $\lambda$-weight zero.
The displayed subspaces span $E_t$ and intersect in $E_L$, so
\[
\mathbf k[E_{\gamma\geq0}]
\otimes^{\mathbf L}_{\mathbf k[E_t]}
\mathbf k[E_{\gamma\leq0}]
\simeq \mathbf k[E_L].
\]
Thus their derived intersection over $E_t$ has no excess.
For completeness, the other torus eigenspaces do not obstruct
splitting off this intersection. Decompose the ambient Lie
algebra as $E_t\oplus W$ by $t$-weights. The projectors are
polynomials in $\operatorname{Ad}(t)$, so this also splits the
universal nilradical subbundles globally and equivariantly. Its Koszul base change to $E_t$ has zero differential
on the factors from $W$, so these factors are finite exterior
algebras of equivariant vector bundles. Their exterior-degree-zero
summand is canonical; no splitting of the varying nonresonant
subbundle inside $W$ is required. Taking this summand in both tests leaves
exactly $\mathbf k[E_L]$ on the opposite-pair slice. Tensoring with
$M$ therefore gives $M_L$. The remaining
complete flags contribute
$R\Gamma(\mathcal O)=\mathbf k$. By equivariant descent from
$H^\circ/H_L$, the resulting invariant test is the expression in
Equation~\eqref{amp:levi-bound}.
We check the uniformity in $\eta$. Give the first partial-flag line
fiber character $\eta$. After the $n$th localization twist the two
fiber characters are, with the opposite-Borel conventions,
\[
\eta-n\chi,\qquad n\chi.
\]
Both lines are nef for every sufficiently large $n$, since $\chi$
is strictly positive on precisely the omitted simple coroots and
$\eta$ vanishes on the Levi coroots. The starting value of $n$ may
depend on $\eta$. It changes neither the localization colimit nor
the bound: every term in this cofinal tail is in the same uniformly
bounded nef family of Proposition~\ref{amp:tests}.
The fixed resolutions in Lemma~\ref{amp:specialization} introduce
only fixed additional positive twists and fixed shifts.
The character $\eta$ and the localization characters are trivial
on the derived subgroup of $L$, so the residual complete flags still
push their structure sheaves to constants.
If $L=\Gd$, the character line is underlying-trivial and already
nef; this also covers the case with no partial flags.
A fixed $U$ is obtained from a fixed representation test by the
restriction and summand argument in
Lemma~\ref{amp:specialization}. Finally, global sections on these
fixed flag varieties have finite cohomological dimension,
reductive invariants are exact, and filtered colimits of vector
spaces are exact. All the operations and the direct-summand
projection above are therefore valid for arbitrary dg $M$.
They preserve a uniform upper bound, proving the proposition.
\end{proof}
\subsection{Orbits, corrected fibers, and local cohomology}
For $e\in E_L$, choose a Jacobson--Morozov triple $(e,h_e,f)$
inside $\operatorname{Lie}L$ such that
\[
\operatorname{Ad}(t)(e,h_e,f)=(Qe,h_e,Q^{-1}f).
\]
We use $h_e$ for the integral neutral cocharacter as well as its
differential. Conjugate it into the reference torus of $H_L$ when
computing pairings.
\begin{lemma}[Resonance orbits]
\label{amp:orbits}
Every element of $E_L$ is nilpotent and admits such a compatible
triple. There are finitely many $H_L$-orbits in $E_L$.
For $e\in E_L$, every $h_e$-weight in the normal space
\[
N_e=E_L/[e,\operatorname{Lie}H_L]
\]
is nonpositive.
\end{lemma}
\begin{proof}
Choose a faithful algebraic representation
$\rho:L\hookrightarrow\operatorname{GL}(V)$.
The relation $\operatorname{Ad}(t)e=Qe$ says that
$d\rho(e)$ is conjugate to $Q\,d\rho(e)$.
Its finite multiset of eigenvalues is therefore preserved by
multiplication by $Q$. Since $Q>1$, every eigenvalue is zero.
Thus $d\rho(e)$, and hence $e$, is nilpotent.
Start with the Jacobson--Morozov
theorem in characteristic zero
\cite[Theorems~3.1 and~3.2]{BMYJM}.
On the torsor of triples with first member $e$, let $(t,Q)$ act by
\[
(e,h,f)\longmapsto
(Q^{-1}\operatorname{Ad}(t)e,\operatorname{Ad}(t)h,
Q\operatorname{Ad}(t)f).
\]
The closure of the subgroup generated by $(t,Q)$ is diagonalizable.
The torsor is under the unipotent radical of the centralizer of $e$
\cite[Proposition~3.3]{BMYJM};
successive vector-group quotients have vanishing first cohomology
for a diagonalizable group in characteristic zero. It therefore
has a fixed point, which is the required compatible triple.
Put $c=tQ^{-h_e/2}$. It commutes with the triple.
Decompose $\operatorname{Lie}L$ into simultaneous $c$-eigenspaces
and irreducible $\mathfrak{sl}_2$-modules $V_n$.
A vector of $h_e$-weight $k$ belongs to $E_L$ precisely when its
$c$-eigenvalue is $Q^{1-k/2}$.
Its predecessor of weight $k-2$, when present, has $t$-eigenvalue
one and lies in $\operatorname{Lie}H_L$.
The raising map is onto this vector unless $k=-n$ is the lowest
weight. Hence every normal weight is of the form $-n\leq0$.
For finiteness, conjugate $h_e$ into the fixed torus by $H_L$.
Its central projection in $L$ is zero, because the triple comes
from $\mathrm{SL}_2$. Its root pairings are integral and bounded
in absolute value by $\dim\operatorname{Lie}L$: they are weights
of an $\mathfrak{sl}_2$-representation of that dimension.
There are consequently only finitely many possibilities in the
fixed cocharacter lattice. Fix one such $h$. On the $c=1$ part of the same calculation the
raising map gives a surjection
\[
\operatorname{Lie}Z_{H_L}(h)
\longrightarrow E_L\cap(\operatorname{Lie}L)_2(h).
\]
Thus every $e$ admitting this $h$ has an open
$Z_{H_L}(h)$-orbit in that vector space. An irreducible vector
space has at most one open orbit. The finite list of $h$'s
therefore gives finitely many $H_L$-orbits.
\end{proof}
We make the grading used for a nonzero fiber explicit.
Let $a=2\lambda$ and first deshear by adding the $a$-weight to
cohomological degree. A linear function on $E_L$ has original
degree $2$ and $a$-weight $-2$, so the coordinate ring now has
degree zero. Take the derived fiber at $e$ in this grading.
The rational cocharacter
\[
\rho_e=h_e-a
\]
fixes $e$, and hence belongs to its stabilizer $S_e=(H_L)_e$.
Add its weight to the fiber degree. The result is called the
\emph{corrected fiber}, denoted $F_e^{\mathrm{corr}}$.
Equivalently, the total correction is by $h_e$.
Here is a literal implementation for rational $\lambda$.
Choose $N$ for which $N\lambda$ is integral, and decompose into the
finitely many sectors of $\lambda$-weights $w$ modulo $\mathbb Z$.
In the sector represented by $r\in[0,1)\cap N^{-1}\mathbb Z$,
replace degree $i$ by the integral degree $i+2(w-r)$ and retain
the numerical offset $2r$. The actual shifts are even integers,
so all dg signs are unchanged. A linear function changes
$(i,w)$ to $(i+2,w-1)$ and has new degree zero.
Tensor products add offsets, with an even integral carry.
Thus deshearing is a finite collection of honest complexes with
rational numerical offsets. Write $W_{e,r}$ for the ordinary derived
fiber in the sector represented by $r$. We use $H^j(W_e)$ for
sectorwise cohomology in numerical degree $j$: a class in
$H^n(W_{e,r})$ has $j=n+2r$. The actual complexes retain integral
degrees. The assertion that the corrected fiber has upper bound $B$
means precisely
\begin{equation}
\label{amp:corrected-fiber}
H^j(W_e)_b=0\quad\text{whenever }j+b>B,
\qquad b=\text{$\rho_e$-weight}.
\end{equation}
All nonzero-point pullbacks are taken after this deshearing.
Differentials retain degree one and preserve $b$.
Apply the same deshearing to every representation test
$U\otimes\eta$. On each of its simultaneous weight spaces,
the desheared degree plus the $\rho_e$-weight is the
$h_e$-weight, since $a+\rho_e=h_e$.
In particular, the possible $a$-weight of a character
$\eta\in X^*(L)$ cancels its $\rho_e$-weight: its $h_e$-weight
is zero because the triple lies in the derived group of $L$.
On $H_L$-invariants the total $a$-weight is zero, so deshearing
does not change the output cohomological degrees.
\begin{lemma}[Contributions of an orbit]
\label{amp:support}
Suppose $F_e^{\mathrm{corr}}$ has upper bound $B_e$.
For every finite-dimensional $H_L$-representation $V$, the
contribution of the orbit $H_Le$ to invariant sections of
$M_L\otimes V$ has upper bound
\begin{equation}
\label{amp:orbit-bound}
B_e+C_e+\max\{\text{$h_e$-weights of }V\},
\end{equation}
where $C_e$ is independent of $V$.
This assertion uses local cohomology in an open subset where the
orbit is closed and is valid for arbitrary dg $M_L$.
\end{lemma}
\begin{proof}
Deshear as above. In an invariant open subset where the orbit is
closed it is a smooth regular immersion into the smooth space
$E_L$. If $\mathcal I$ is its ideal, local cohomology is
\[
R\Gamma_{H_Le}(M_L)
=\operatorname*{colim}_{p}
R\mathcal Hom(\mathcal O/\mathcal I^p,M_L).
\]
Each quotient is perfect on the regular ambient space.
Its finite filtration has graded pieces the symmetric powers of
the conormal bundle. Regular-immersion duality therefore expresses
the corresponding support terms as
\[
i_*\bigl(Li^*M_L\otimes\det N\otimes
\operatorname{Sym}^{a}N\bigr)[-c],
\qquad c=\operatorname{codim}(H_Le),
\]
where $M_L$ is read in the desheared grading and $i$ denotes the
regular immersion. The shift $[-c]$ raises degrees by $c$;
the normal symmetric powers, rather than the conormal ones, are
the factors relevant to the upper bound.
The action of $\rho_e$ on a normal vector of $h_e$-weight $k$
has weight $k-2$. By Lemma~\ref{amp:orbits} this is at most $-2$.
Consequently arbitrary normal powers cannot increase the corrected
upper bound. The determinant and the codimension shift cost a
constant depending only on the orbit. This establishes a bound
uniform in $p$ before passage to the filtered colimit.
Equivariant sections on the orbit are rational group cohomology
of $S_e$. Its unipotent radical is computed by the finite exterior
Lie-algebra cochain complex, and invariants for a reductive Levi
are exact. The triple centralizer is a Levi in the centralizer
of $e$ \cite[Proposition~3.3]{BMYJM}; taking the fixed points of
the compatible diagonalizable action gives the same decomposition
for $S_e$. The cocharacter $\rho_e$ lies in, and is central in,
this triple-centralizer Levi. If $\mathfrak u$ is the Lie algebra
of the unipotent radical, a possible additional cost is
\[
\max_{0\leq a\leq\dim\mathfrak u}
\left(a+\max\operatorname{wt}_{\rho_e}
\bigwedge^a\mathfrak u^*\right).
\]
The exterior complex has finitely many columns, so it is valid
for unbounded coefficients without an infinite-totalization issue.
Finite component invariants are exact in characteristic zero.
The test $V$ is desheared too: its $a$-weight-$\alpha$ space
is placed in numerical degree $\alpha$. Its numerical degree
plus its $\rho_e$-weight is therefore its $h_e$-weight.
Taking invariants of the reductive triple-centralizer Levi forces
total $\rho_e$-weight zero. On the original $H_L$-invariants
the total $a$-weight is zero as well, so the resulting numerical
degree is the original cohomological degree.
This gives Equation~\eqref{amp:orbit-bound}.
No finite-generation hypothesis has been used. Tensoring with the
perfect quotients and their finite filtrations is legitimate for
unbounded dg modules. The operation displayed above is a filtered
colimit, not an inverse limit. Exactness of filtered colimits
preserves the upper bound uniform in $p$. Finally there are
finitely many orbits, so their support triangles give a finite
devissage.
\end{proof}
\begin{proposition}[Bounded corrected fibers]
\label{amp:fiber}
For every allowed Levi $L$ and every $e\in E_L$, the corrected
fiber $F_e^{\mathrm{corr}}$ is bounded above.
The bound is a bound on the whole complex; its cohomology need
not be finite-dimensional.
\end{proposition}
\begin{proof}
Induct on the semisimple rank of $L$, using
Proposition~\ref{amp:levi} for all its smaller allowed Levis.
If the centralizer of the triple in $H_L$ has a torus noncentral
in $L$, choose a cocharacter in that torus noncentral in $L$.
Its centralizer is a proper Levi $L'$ of $L$ containing the triple.
It is still allowed: combine this cocharacter with $\gamma$,
using a sufficiently large multiple to specify the same successive
centralizers. Derived restriction is transitive, so the derived
fiber obtained from $M_{L'}$ is the same as that from $M_L$.
Induction proves the assertion for this orbit.
Consider an orbit for which no such torus exists. The direction
$\lambda-h_e/2$ centralizes the triple, so it is central in $L$.
Since $\lambda$ is central in $H_L$, this implies that $H_L$
centralizes $h_e$. In the centralizer of $h_e$, an element
centralizing $e$ also centralizes $f$, by uniqueness of the
compatible triple with $e,h_e$ fixed. Thus $S_e$ equals the
triple centralizer in $H_L$. It is reductive and finite modulo
$Z(L)^\circ$. Since $\lambda-h_e/2$ is central in $L$, every vector
in $E_L$ has $h_e$-weight two. Lemma~\ref{amp:orbits} gives only
nonpositive weights in the normal space, so that space is zero:
$H_Le$ is open in $E_L$. The irreducible space $E_L$ has only one
open orbit.
Every boundary orbit has already been handled by the proper-Levi
case. Apply Lemma~\ref{amp:support} to those finitely many orbits
with tests $U\otimes\eta$.
Every neutral cocharacter of a triple in $L$ annihilates
$\eta\in X^*(L)$. Their boundary contributions therefore have
upper bounds independent of $\eta$.
Together with Equation~\eqref{amp:levi-bound}, the localization
triangle shows that the open orbit alone satisfies the same type
of uniform bound for each fixed $U$.
We explain why these invariant bounds detect the entire corrected
fiber. The group $S_e$ is reductive, possibly disconnected, and
contains the central torus $Z(L)^\circ$ with finite quotient.
Restrictions of characters of $L$ form a finite-index sublattice
of $X^*(Z(L)^\circ)$. Modulo tensoring by these restrictions,
there are finitely many irreducible rational representations of
$S_e$: there are finitely many central-character cosets, and for each
fixed central character the irreducibles are representations of
a finite-dimensional twisted group algebra of the finite group
$S_e/Z(L)^\circ$. Choose finitely many fixed
$H_L$-representations $U_1,\ldots,U_r$ whose restrictions detect
the duals of representatives of these classes.
Such representations exist by the matrix-coefficient and
complete-reducibility argument used above.
Arbitrary rational representations of a reductive group in
characteristic zero are direct sums of finite-dimensional
irreducibles \cite[Remark~12.53\textup{(a)}, Theorem~22.42, and
Corollary~22.43]{MilneAlgebraicGroups}.
Invariants are exact, so every nonzero cohomology
class of the corrected fiber is detected by one of the tests
$U_j\otimes\eta$. The correction on that test has only the
$h_e$-weights of $U_j$, since $h_e$ annihilates $\eta$.
More explicitly, an irreducible constituent of $H^j(W_e)$ of
$\rho_e$-weight $b$ has a nonzero invariant pairing with one of
these tests. If the chosen test component has $h_e$-weight $a$,
that pairing has ordinary degree $j+b+a$: the total $\rho_e$-weight
is zero, and the test's numerical degree plus its $\rho_e$-weight
is its $h_e$-weight. Hence
\[
j+b\leq C'(U_j)-\min\operatorname{wt}_{h_e}(U_j).
\]
The maximum over the finite list gives the bound in
Equation~\eqref{amp:corrected-fiber}. This argument includes rank zero, where $E_L=0$ and all
representation types are characters. It starts the induction
and completes the proof.
\end{proof}
\subsection{The extreme amplitude estimate}
Choose a rational Weyl-invariant positive-definite metric and use
it to identify weight and cocharacter directions. Conjugate
$\lambda$ into the closed dominant chamber. For sufficiently
divisible positive integers $m$, the weight $\mu=m\lambda$ is
integral, dominant, and trivial on the connected center. Let
$V_\mu$ be the corresponding irreducible representation.
\begin{theorem}[Extreme upper amplitude]
\label{amp:amplitude}
In the finite Koszul specialization at $[t]$ of
$\mathsf K(V_\mu^{\boxtimes d})$, retain the part with maximal
absolute value for the $d$-fold slide at the chosen leg.
Its cohomological upper bound is
\begin{equation}
\label{amp:amplitude-bound}
C+\max_{e\in E_t}d\langle\mu,h_e\rangle.
\end{equation}
The constant is independent of $m$.
It is also uniform when $t$ is replaced by $tz$, with
$z\in Z(\Gd)^\circ$, after omitting the absolute-value central
part, and when one chooses among a fixed finite list of
hyperspecial variants. Fixed additional kernels at disjoint
places are permitted.
Unless $t$, modulo its absolute-value central part, is a compact
factor times a Jacobson--Morozov point for $Q$, the difference
between $2d\langle\lambda,\mu\rangle$ and the nonconstant term
in Equation~\eqref{amp:amplitude-bound} is positive and grows
linearly with $m$.
\end{theorem}
\begin{proof}
In the loop description, the representation label becomes
$V_\mu^{\otimes d}$ and the $d$-fold slide acts by its holonomy;
this is the slide convention of Proposition~\ref{cat:loops}.
Among the weights of $V_\mu$, pairing with $\lambda$ is uniquely
maximal at $\mu$. Indeed a weight lies in the convex hull of the
Weyl orbit of $\mu$, and the linear functional defined by $\mu$
attains its maximum on that orbit only at $\mu$ itself.
This argument also applies on chamber walls.
Thus the extreme part is the line of character $d\mu$.
It is $H$-stable; its character is trivial on the derived subgroup
of $H$, and the unipotent part of holonomy acts trivially on it.
Nilpotent thickenings in the Koszul specialization do not change
this semisimple eigenvalue separation.
Apply Proposition~\ref{amp:fiber} and
Lemma~\ref{amp:support} with this one-dimensional representation.
There are finitely many orbits by Lemma~\ref{amp:orbits}.
The maximum of their fixed costs is a single constant $C$,
and the representation contribution on the orbit of $e$ is
$d\langle\mu,h_e\rangle$. This proves
Equation~\eqref{amp:amplitude-bound}.
Compatibility of the triple implies that
$\lambda-h_e/2$ centralizes it. The invariant metric extends,
on the derived Lie algebra, to an invariant bilinear form;
therefore
\[
(h_e,\lambda-h_e/2)
=([e,f],\lambda-h_e/2)=0.
\]
Consequently
\begin{equation}
\label{amp:squared-gap}
2d\langle\lambda,\mu\rangle
-d\langle\mu,h_e\rangle
=2dm\|\lambda-h_e/2\|^2.
\end{equation}
All directions here are projected away from the connected center.
If the right side vanishes, the semisimple element
$tQ^{-h_e/2}$ centralizes the triple and has absolute values one
in the remaining directions. It is conjugate into a maximal
compact subgroup after removing the absolute-value central part
\cite[Theorems~3.6 and~3.7]{AdamsTaibiCohomology}.
This is exactly the required compact-times-Jacobson--Morozov form.
Conversely that form gives equality for its triple.
If equality occurs for no orbit, the finite list of squared norms
in Equation~\eqref{amp:squared-gap} has a strictly positive minimum.
This is the asserted uniform linear gap.
Finally, replacing $t$ by $tz$ leaves $H$, $E_t$, their orbits,
the allowed Levis, and every flag test unchanged. It also leaves
the semisimple direction $\lambda$, the rational deshearing,
and all corrected-fiber and representation-test gradings unchanged,
even when $\log_Q|z|$ is irrational. The only change
in the finite invariant Koszul test is in the scalar values
subtracted from a fixed list of invariant generators.
Its length, shifts, and the preceding geometric bounds are
unchanged. All constants in the slice, flag, support, and
finite-representation arguments can therefore be chosen uniformly
on this central family. The label $\mu$ is central-trivial, so
its extreme line is unchanged as well.
For finitely many hyperspecial variants take the maximum of their
constants. Kernels at other places were fixed throughout
Proposition~\ref{amp:tests} and do not affect this uniformity.
\end{proof}
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/03-amplitude.tex ====
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/04-rotation.tex ====
\section{Rotation and a trace identity}\label{sec:rotation}
This section compares two ways of closing a positive translation path.
One closes it over $\mathbb F_q$; the other closes it after moving its
initial segment past Frobenius. The comparison requires a connected
Picard group. We construct that group using a divisible norm lift on the
splitting cover of the generic centralizer. Its modulus retains the
entire level divisor, including its multiplicities.
\subsection{Levels, translations, and genericity}
Throughout this section $G$ may be split reductive. Fix a split maximal
torus $T$, its Weyl group $W$, and the translation lattice
$\Lambda=X_*(T)$. Put
\[
\Lambda_0=\{a\in\Lambda:\langle\chi,a\rangle=0
\text{ for every }\chi\in X^*(G)\}.
\]
The positive chamber is the chamber of standard Bernstein translations
chosen in Section~\ref{sec:categorical}. At distinct closed points
$x_1,\ldots,x_r$, outside the fixed level divisor $D$, impose Iwahori
level. Write $d_j=\deg(x_j)$ and $q_j=q^{d_j}$. Additional fixed
parahoric flags are allowed. All modifications below preserve the full
trivialization along $D$.
If $G$ has a split connected center, quotient by a rational free lattice
$\Xi$ of central modifications whose degree map is injective and has
cocompact image in the free abelian degree lattice. Such a lattice exists:
central cocharacters span the character-degree space over $\mathbb Q$,
and rational divisors on $X$ realize a finite-index subgroup of its degree
lattice. Choose a representative of every resulting degree coset.
Every path used below has degree zero. If it closes in the quotient,
its closing central modification has degree zero and is therefore the
identity. The same holds for Frobenius closure, since the lattice is
rational and point Frobenius preserves character degrees. Thus these
paths lift to the selected degree sectors before taking the quotient.
This observation also specifies the quotient version of all subsequent
constructions.
For a positive tuple $a=(a_j)$, set
\[
L_a=\sum_j d_j\ell(t^{a_j}),\qquad
T_a=\prod_j q_j^{-\ell(t^{a_j})/2}
1_{I_jt^{a_j}I_j}.
\]
Here the rational Iwahori has volume one. The $T_a$ extend
multiplicatively to the Bernstein torus. They are the
\emph{function-normalized} operators of Proposition~\ref{cat:hecke}.
In particular, at a closed point the character of the pure standard
kernel is $(-1)^{\ell(t^a)}T_a$, including when the geometric factors
are cyclically permuted by Frobenius. We retain this correction for
arbitrary translation labels; we do not restrict the Bernstein algebra
to an even sublattice.
Choose $u_j$ in the closed positive chamber and in
$2|W|\Lambda_0$. At the $d_j$ geometric points above $x_j$, repeat
$u_j$ identically. Choose positive integral multiples $\varpi$ of the
semisimple fundamental weights that belong to $X^*(T)$ and are trivial
on the connected center. Assume
\begin{equation}\label{rot:genericity}
\sum_jd_j\langle\varpi,w_ju_j\rangle\ne0
\qquad\text{for every such $\varpi$ and every }(w_j)\in W^r.
\end{equation}
When $G$ is a torus the condition is empty. We do not require each
$u_j$ to be regular.
Write $\mathrm{Fr}$ for point Frobenius, reserving $F$ for the function
field. Let $Y_u$ be the open translation shtuka: an object consists of a
bundle $\mathcal E$ with the specified levels and a meromorphic
isomorphism
\[
\phi:\mathcal E\longrightarrow\mathrm{Fr}(\mathcal E)
\]
of exact Iwahori positions $t^{u_j}$ at the geometric legs, preserving
all levels away from them. Equivalently one can keep the intermediate
bundles in the path. Modifications at distinct legs commute, and the
open convolution maps used below identify these descriptions. Put
\begin{equation}\label{rot:complex}
C(u)=R\Gamma_c(Y_u,\mathbf k),\qquad
M(u)=C(u)[L_u](L_u/2).
\end{equation}
All $\ell(t^{u_j})$ are even, so the shift and cyclic-permutation signs
in this normalization are trivial. We use the natural Frobenius on
$M(u)$; in terms of $C(u)$ it is $q^{-L_u/2}\mathrm{Fr}^*$.
The complex $M(u)$ is the cyclic complex of the repeated standard
translation kernels $J_{u_j}$. For a positive tuple $v$, their coherent
commutation with the kernels $J_{v_j}$ lets us apply the character
construction in Equation~\eqref{cat:character}. Denote the resulting
function-normalized endomorphism of $M(u)$ by $T_v$, with the parity
correction specified above.
\begin{theorem}\label{rot:trace}
Let $u$ satisfy the chamber, divisibility, degree-zero, and genericity
conditions above. Let $v$ be any fixed regular positive tuple in
$\Lambda_0^r$, without a parity restriction. For all sufficiently
large integers $n$,
\begin{equation}\label{rot:normalized-trace}
\sum_{[\mathcal E]}(T_{nu+v})_{\mathcal E,\mathcal E}
=\operatorname{Tr}_{\mathrm{alt}}
(\mathrm{Fr}^{n}T_v\mid M(u)).
\end{equation}
The diagonal sum uses ordinary matrix entries on the rational
isomorphism classes of bundles with level, and is finite. The assertion
retains arbitrary multiplicities in $D$ and the allowed central quotient.
\end{theorem}
We first bound $Y_u$ and identify its translation operators with finite
correspondences. The large-Frobenius trace formula then counts loops
whose rotation by the first $u$-segment is identified with Frobenius.
The ordinary
matrix trace instead counts rational loops. The main comparison uses
a connected Picard group to identify these two descent groupoids;
the proof of Theorem~\ref{rot:trace} then assembles the counts.
\subsection{Boundedness and separation}
We first record the elementary translation facts needed to apply the
bundle bounds of Proposition~\ref{amp:geometry}. If affine Weyl elements
$w,w'$ satisfy $\ell(ww')=\ell(w)+\ell(w')$, the open convolution map
\begin{equation}\label{rot:open-product}
IwI\mathbin{\times^I}Iw'I\longrightarrow Iww'I
\end{equation}
is an isomorphism. Indeed the root subgroups in the two inversion sets
are disjoint, and their ordered products give the root coordinates on
the inversion set of $ww'$. Thus a path of exact position $ww'$ has a
unique intermediate flag of the prescribed type. This statement, with
its intermediate isomorphisms, also holds for Hecke groupoids. In
particular, translations in one closed chamber have unique
length-additive subdivisions.
For an ordinary local automorphism $g$ with
$g^m\in It^{mb}I$ for all $m\ge1$, its dominant Newton vector is $b$.
Here the Newton vector means the valuations of its semisimple part,
viewed as a cocharacter modulo $W$. Here is a linear-algebra verification that also works on a wall.
In any highest-weight representation, an integral torus-weight lattice
is preserved by $I$. The growth rates of the operator norms of $g^m$
and its dual are consequently the extremal pairings of its weights
with $mb$. Over a finite splitting extension, put $g$ in Jordan form.
Its unipotent part has finite $p$-power order, and conjugation by the
fixed change-of-basis matrix changes logarithmic operator norms by a
bounded amount. The same growth rates are therefore those of the
valuations of its semisimple eigenvalues. As the highest weights vary,
these extremal pairings determine the dominant cocharacter, proving
that it is $b$. No semisimplicity of the representation in positive
characteristic is needed. If $g$ preserves a parabolic reduction, the valuation
of a character of that parabolic is its pairing with the Newton vector
in the corresponding Levi, hence with a Weyl conjugate of $b$ in $G$.
\begin{lemma}\label{rot:bounded}
Under Equation~\eqref{rot:genericity}, $Y_u$ is a separated
Deligne--Mumford stack of finite type, and $C(u)$ is a bounded complex
with finite-dimensional cohomology. For a fixed regular positive
$v\in\Lambda_0^r$ and all sufficiently large integers $n$, the ordinary
rational loop groupoid of exact position $nu+v$ has finitely many
isomorphism classes and finite mass. Only bundles of instability
$O(1+n)$ occur in it.
\end{lemma}
\begin{proof}
Suppose the instability in $Y_u$ were unbounded. Proposition~\ref{amp:geometry}
would give a maximal-parabolic canonical reduction preserved by $\phi$,
where preservation means that the reduction is carried to its
Frobenius image. Work first at a geometric point with all its data
defined over $\mathbb F_{q^b}$, enlarging $b$ to be divisible by every
$d_j$. Iterating $\phi$ $b$ times gives an ordinary meromorphic
automorphism $\gamma$ of the bundle. Equation~\eqref{rot:open-product}
puts all its powers, at every geometric point above $x_j$, in the
Iwahori cells of translations $mbu_j$.
Apply the preceding Newton calculation to the preserved reduction.
The valuation of its fundamental character at one geometric leg is
$b\langle\varpi,w_ju_j\rangle$ for some $w_j$. The original
Frobenius path intertwines the consecutive local automorphisms and
their reductions. Since the character of the split parabolic is also
preserved, its valuation is the same at all $d_j$ geometric points
above $x_j$. The product formula for the meromorphic character
therefore gives
\[
b\sum_jd_j\langle\varpi,w_ju_j\rangle=0,
\]
contrary to Equation~\eqref{rot:genericity}. Proposition~\ref{amp:geometry}
now bounds the instability, so boundedness of bundles and of the Hecke
positions gives finite type. This argument includes wall labels, which
merely allow more Weyl choices.
The shtuka condition makes inertia unramified. Infinitesimally an
automorphism compatible with $\phi$ equals its transported Frobenius
pullback; the latter has zero differential, so its tangent vector is
zero. Automorphism schemes of a bundle with bounded level are affine
of finite type. Thus the inertia is quasi-finite, and $Y_u$ is
Deligne--Mumford.
For separation, let two objects over a discrete valuation ring $R$ be
isomorphic over its fraction field. Choose a faithful representation
of $G$. At the generic point of the special fiber of $X_R$, both path
maps are integral isomorphisms: the legs are horizontal. If $m$ is the
smallest valuation of a matrix entry of the generic isomorphism,
compatibility with the paths gives $m=qm$, since base Frobenius
multiplies vertical valuations by $q$ and multiplication by an integral
invertible matrix preserves the smallest valuation. Hence $m=0$.
The same argument applies to the inverse. The isomorphism consequently
extends at that vertical generic point. All horizontal codimension-one
points are already covered by the generic-fiber bundle isomorphism.
Normality of the smooth surface $X_R$ and affineness of the isomorphism
torsor extend it across codimension two. The path identities extend by
density. So do the prescribed level identities: $D_R$ is flat over
$R$, including when $D$ is nonreduced. The same argument treats fixed
parahoric flags. This proves the valuative criterion for the
quasi-finite diagonal, which is therefore proper.
For completeness, finite-dimensional compact cohomology can be reduced
to the corresponding assertion for algebraic spaces. On this bounded
family, choose a sufficiently large effective divisor $N$ disjoint
from the legs and the existing level such that an automorphism equal to the identity on $N$
is the identity. A faithful representation reduces this to the
uniform vanishing of sections of a bounded family of endomorphism
bundles twisted by $-N$. Frobenius-compatible trivializations on $N$
form a finite \'etale torsor under
\[
H_N=\bigl(\operatorname{Res}_{N/\mathbb F_q}(G_N)\bigr)(\mathbb F_q).
\]
Indeed the jet group $\operatorname{Res}_{N/\mathbb F_q}(G_N)$ is smooth
and connected, and its Lang map is finite \'etale and surjective.
The resulting rigidified shtuka is a separated algebraic space of
finite type with a finite-group quotient equal to $Y_u$. Compact
cohomology is therefore bounded and finite-dimensional; taking
$H_N$-invariants is exact over $\mathbf k$. The same reduction applies
to weights. A finite-type algebraic space over a field admits a finite
stratification by schemes, obtained by successively choosing a dense
open subspace that is a scheme. Excision transfers the scheme weight
bounds to this algebraic space, and exact finite-group invariants retain
them. Thus the pure constant coefficient in
Equation~\eqref{rot:complex} has the usual integral-weight compact
cohomology bounds, without any appeal to a weight theorem for arbitrary
Artin stacks.
Finally, $a=nu+v$ is regular positive, and for each fixed pairing in
Equation~\eqref{rot:genericity} the corresponding pairing for $a$ is a
nonconstant affine function of $n$. Thus $a$ satisfies the same
nonvanishing condition for all sufficiently large $n$. A rational
ordinary loop of type $a$ with excessive instability has, by
Proposition~\ref{amp:geometry}, a preserved maximal-parabolic reduction.
The same character and product-formula argument excludes it. The
quantitative bound is $O(1+n)$. There are finitely many rational bundles
in that range, and each admits only finitely many rational
meromorphic automorphisms with these pole bounds: in a faithful
representation they lie in a fixed finite-dimensional space of
sections over a finite field. The automorphism group of every such
rational loop has finitely many rational points. This proves the
last assertion.
\end{proof}
\subsection{The finite correspondence underlying a translation}
For a positive tuple $v$, let $\mathcal C_v$ classify a commuting square
\[
\begin{tikzcd}[column sep=large,row sep=large]
\mathcal E \arrow[r,"h", "v"'] \arrow[d,"\phi"'] &
\mathcal E' \arrow[d,"\phi'"] \\
\mathrm{Fr}(\mathcal E) \arrow[r,"\mathrm{Fr}(h)"'] &
\mathrm{Fr}(\mathcal E').
\end{tikzcd}
\]
The vertical paths have type $u$, and $h$ has type $v$. Denote the
initial and final shtuka projections by $p$ and $q$ respectively.
The direction used on cohomology is $p_!q^*$, pullback towards the
initial vertex.
\begin{lemma}\label{rot:finite-correspondence}
The two projections of $\mathcal C_v$ are finite \'etale after passing
to perfections. On $C(u)$ the function-normalized categorical
translation is
\begin{equation}\label{rot:geometric-operator}
T_v=q^{-L_v/2}p_!q^*.
\end{equation}
These correspondences compose according to addition in a closed
chamber. For $v=u$ the correspondence is the graph of point
Frobenius, so $T_u$ acts as Frobenius on $M(u)$.
\end{lemma}
\begin{proof}
We first work at one geometric leg. Take a reduced gallery for $t^v$.
After a prefix $b$, cyclic rotation changes the shtuka label to the
translation $\tau=b^{-1}t^ub$. Since $u$ and $v$ lie in one closed
chamber,
\[
\ell(t^ub)=\ell(t^u)+\ell(b).
\]
For the next simple reflection $s$, one has
$\ell(\tau s)=\ell(\tau)+1$. A translation has opposite left and right
length changes at a wall to which its vector is not parallel.
Consequently either $s$ is an initial letter of $\tau$, or $\tau$
is parallel to the $s$-wall and commutes with $s$.
In the first case write $\tau=sw$. Unique open subdivision identifies
the step with the cyclic rotation from the path $sw$ to the path $ws$.
It is an isomorphism on perfections. The cancellation of the inverse
simple kernel against that initial factor in the categorical action
is precisely the cyclic isomorphism of Proposition~\ref{cat:cyclic}.
Thus its unnormalized action is pullback along this geometric step.
For a closed leg of degree $d$, perform this rotation on the whole block
of $d$ geometric factors. Cyclicity applied to
$\Delta_s^{\boxtimes d}\star\Delta_w^{\boxtimes d}$ changes the repeated
label $\tau=sw$ to the repeated label $ws=s\tau s$; unique reduced
subdivisions identify both ends with the stated path stacks.
In the parallel case, pass from the alcove to its $s$-panel. The partial
relative position identifies the two rank-one flag varieties: on the
root groups for that panel, translation conjugation has exponent zero.
A refinement of the partial shtuka is therefore a flag fixed by the
resulting Frobenius-semilinear identification. Over a degree-$d$ closed
leg, these refinements form a finite \'etale family of $q^d+1$ flags.
To check this over a general base, trivialize the rank-one torsor
\'etale locally and use the Lang map for its connected rank-one group;
the fixed flags then become $\mathbb P^1(\mathbb F_{q^d})$.
If $\pi$ denotes the map forgetting this refinement, the closed simple
correspondence acts by $\pi^*\pi_!$, and the open simple correspondence
acts by
\begin{equation}\label{rot:panel-action}
\pi^*\pi_!-\operatorname{id}.
\end{equation}
Here Equation~\eqref{rot:panel-action} is also an equality with the
categorical transfer, rather than only an equality of trace functions.
Use the unshifted closed rank-one kernel, which factors through the
panel projection. Its unit and counit give the diagonal and projection
maps for the proper flag variety. After restriction to the twisted
trace, the Frobenius graph meets the diagonal transversely: its
linearized fixed-point equation has derivative $-1$. Every fixed flag
therefore has coefficient one in the unit/counit transfer. The
projection formula identifies the resulting operator with sum and
pullback over the finite family of fixed flags. This is the explicit
rank-one calculation in Proposition~\ref{cat:hecke}. Subtracting the
identity kernel by the standard open/closed triangle gives
Equation~\eqref{rot:panel-action}. In particular no compact cohomology
shift of the flag variety remains in this finite transfer.
At a closed leg the same calculation is made in the product of its
$d$ geometric flag varieties, with Frobenius cyclically permuting the
factors. A fixed tuple is determined by one
$\mathbb F_{q^d}$-flag. The same transverse intersection proves the
coefficient-one assertion and gives the parameter $q^d$.
Graded cyclic permutation produces the single-factor parity in the
pure standard kernel. The correction defining $T_v$ removes that
parity even when $v$ is odd. Thus normalization gives the factor
$q^{-L_v/2}$ in Equation~\eqref{rot:geometric-operator}.
Length-zero gallery steps identify the relevant paths. Composing the
two kinds of simple steps proves finiteness on perfections and the
operator assertion. Unique subdivision also proves compatibility of
composition and of commuting-translation shuffles. When $v=u$, the
two initial steps in the commuting square are subdivisions of the same
path of position $2u$ and hence coincide. The square is therefore
$h=\phi$, $\mathcal E'=\mathrm{Fr}(\mathcal E)$, and
$\phi'=\mathrm{Fr}(\phi)$. Its action is $\mathrm{Fr}^*$ on $C(u)$;
after the normalization in Equation~\eqref{rot:complex} it is
Frobenius on $M(u)$.
\end{proof}
\subsection{The Frobenius trace and the shifted loop condition}
We use the following precise form of the large-Frobenius trace theorem.
If $A\xleftarrow{c_2}C\xrightarrow{c_1}A$ is a correspondence of
separated finite-type schemes over $\mathbb F_q$, with $c_1$ proper and
$c_2$ quasi-finite, then, for all sufficiently large $n$, its
Frobenius-twisted fixed-point set is finite and the alternating compact
trace equals the sum of the naive local traces.
This is \cite[Theorem~2.3.2(a),(b) and Remark~2.3.3(c)]{Varshavsky},
with the open subset there equal to all of $A$. The same proof applies
to compactifiable algebraic spaces, as stated in the final paragraph
of that paper's introduction. In our application both projections
are finite and the coefficient is constant, so each point counted by
the pullback/sum correspondence has local trace one.
Here is the reduction that avoids invoking such a theorem directly
for a stack. Use the rigidifying divisor $N$ from the proof of
Lemma~\ref{rot:bounded}. Every Hecke step at the legs transports its
$N$-trivialization uniquely. Hence the correspondence lifts to the
rigidified algebraic space $\widetilde Y_u$, commutes with the constant
finite group $H_N$, and has finite projections. Exact averaging gives
\begin{equation}\label{rot:averaging}
\operatorname{Tr}_{\mathrm{alt}}(\mathrm{Fr}^{n*}U_v\mid C(u))
=\frac1{|H_N|}\sum_{h\in H_N}
\operatorname{Tr}_{\mathrm{alt}}
(h\,\mathrm{Fr}^{n*}\widetilde U_v\mid
R\Gamma_c(\widetilde Y_u,\mathbf k)),
\end{equation}
where $U_v=p_!q^*$ is unnormalized. Apply the theorem to the finitely
many lifted correspondences. The fixed-point groupoid downstairs is
the quotient of the disjoint union of their twisted fixed points by
$H_N$, with $H_N$ acting by conjugation on the twisting element.
Its weighted cardinality is exactly the right side of
Equation~\eqref{rot:averaging} after replacing traces by counts.
This proves the stack formula with all automorphism factors.
If a gallery rotation uses an inverse purely inseparable map, choose a
fixed Frobenius power that clears its denominator, and do the same for
the finitely many maps in the correspondence. The resulting
correspondence has finite algebraic models. Composing it with
$\mathrm{Fr}^{n-b}$ gives the original correspondence composed with
$\mathrm{Fr}^n$. Since compact \'etale cohomology and fixed geometric
points are invariant under the intervening universal homeomorphisms,
the preceding argument applies once $n$ exceeds this fixed $b$.
The Frobenius operators in this procedure are transported with the
correspondence; none is replaced by the identity.
Let $\mathcal L_a$ denote the geometric groupoid of ordinary meromorphic
loops of exact position $a$: its objects are $(\mathcal E,\gamma)$,
where $\gamma$ is a self-modification preserving the levels. For
$a=nu+v$, let $R_u$ rotate the first $u$-segment to the end. It is
invertible when $a$ is regular, as will also follow from the torus
apartment description below. Write
\[
\mathcal L_a^{\mathrm{sh}}
=\{(\mathcal E,\gamma,\iota):
\iota:R_u(\mathcal E,\gamma)
\xrightarrow{\sim}\mathrm{Fr}(\mathcal E,\gamma)\}.
\]
Arrows in this notation must respect $\iota$.
\begin{lemma}\label{rot:concatenation}
For sufficiently large $n$, the fixed-point groupoid for
$\mathrm{Fr}^{n*}U_v$ on $Y_u$ is equivalent to
$\mathcal L_{nu+v}^{\mathrm{sh}}$. Consequently
\begin{equation}\label{rot:shifted-trace}
\operatorname{Tr}_{\mathrm{alt}}(\mathrm{Fr}^{n*}U_v\mid C(u))
=|\mathcal L_{nu+v}^{\mathrm{sh}}|.
\end{equation}
\end{lemma}
\begin{proof}
With the direction $p_!q^*$ used above, the fixed-point identification
is $p=\mathrm{Fr}^nq$. Thus its data consist of a shtuka
$(\mathcal E,\phi)$ and a closing step
$h:\mathrm{Fr}^n\mathcal E\to\mathcal E$ of type $v$, compatible with
$\phi$. Concatenate
\[
\mathcal E\xrightarrow{\phi}\mathrm{Fr}\mathcal E
\longrightarrow\cdots\longrightarrow
\mathrm{Fr}^n\mathcal E\xrightarrow{h}\mathcal E.
\]
The total position is $nu+v$ by Equation~\eqref{rot:open-product}.
Compatibility with $h$ identifies rotation of the initial $u$-step
with point Frobenius.
Conversely, subdivide a loop of position $nu+v$ into $n$ successive
$u$-steps and one $v$-step. The given identification of the first
rotation with Frobenius turns the first step into
$\phi:\mathcal E\to\mathrm{Fr}\mathcal E$; subsequent steps are its
Frobenius translates. If $\gamma$ is the total loop, the last step is
$h=\gamma(\phi^{(n)})^{-1}$. The identity
$\phi\gamma=\mathrm{Fr}(\gamma)\phi$ gives the required commuting
square for $h$. Unique open subdivision makes these constructions
inverse on objects, intermediate isomorphisms, and arrows. The trace
formula just proved gives Equation~\eqref{rot:shifted-trace}.
\end{proof}
\subsection{Regular centralizers and connected norm lifts}
We will compare the shifted count with the ordinary rational loop
count. Only their fixed-point groupoids need be finite; the geometric
groupoid $\mathcal L_a$ itself need not be bounded.
For a torus $G$, the centralizer is already split and all the
degree-zero labels are zero, so the following apartment assertion is
empty. Otherwise, for a regular positive $a_j$, an ordinary local automorphism $g$ of
position $t^{a_j}$ has all powers in the positions $t^{ma_j}$ by
Equation~\eqref{rot:open-product}. We explain why this forces both
strong regularity and the apartment assertion needed for rotation.
Work over a finite extension over which the object and its local flag
are defined, and use the reduced affine building of $G$ there. Let
$A_0$ be the base alcove. Its Iwahori fixes it pointwise
\cite[Sections~1.2 and~3.7]{TitsReductiveLocal}. For every
point $z\in A_0$, writing $g^m=i_1t^{ma_j}i_2$ gives
\[
d(z,g^mz)=d(z,t^{ma_j}z)=m\|a_j\|.
\]
Equality in the triangle inequality for $m=2$ shows that the segments
$[g^kz,g^{k+1}z]$ concatenate to a geodesic axis. Its direction is
regular, since its vector distance is $a_j$. After taking a power to
remove the finite permutation of building types, the minimum set of
this regular-axis isometry is a unique apartment; this is
\cite[Lemma~2.3]{CapraceCiobotaru}. The original $g$ preserves this
apartment because it commutes with its power. Its finite Weyl part
fixes the regular translation vector and hence is trivial. Thus $g$
acts on the apartment by a translation. An apartment corresponds to a
maximal split torus $S$, and its stabilizer is $N_G(S)$
\cite[Section~2.1]{TitsReductiveLocal}. The trivial finite Weyl part
therefore puts $g$ in $Z_G(S)=S$, since $G$ is split. The central
Euclidean factor for reductive $G$ can simply be restored; the labels
under consideration have zero character degree.
Every point of $A_0$ lies on such an axis, so the original flagged
lattice belongs to that torus apartment. The valuations of its root
characters are the root pairings with $a_j$, all nonzero. In particular
no root value is one. Moreover, a Weyl element centralizing $g$ would
fix its regular valuation vector and must be the identity. Hence $g$
is strongly regular semisimple and this split torus is its full
centralizer. Unique subdivision in the apartment now
identifies $R_u$ with local modification by the ordered cocharacter
$u_j$. Modification by $-u_j$ is its inverse. This description is
unchanged by further extension of the residue field.
The generic conjugacy invariant of a loop fixed by either ordinary or
shifted Frobenius is rational: rotation does not change that invariant.
Fix one such invariant $c$. It determines a canonical torus $S_c/F$
with a tautological element $\gamma_c\in S_c(F)$. This construction is
obtained by descending the split torus along the Weyl torsor above $c$.
For any realization of $c$ as a strongly regular automorphism, its
centralizer is canonically identified with $S_c$: two conjugating
choices differ by an element of the same commutative centralizer.
The preceding argument first splits this torus after a finite residue
extension. Over the original completed field at a closed leg, local
Galois acts on its split labels through $W$ and fixes the valuation
vector of the rational tautological element. That vector is regular,
so its Weyl stabilizer is trivial. Thus $S_c$ is already split over the
original completed field. The valuations distinguish the ordering,
so the rotation displacements are rational there as well.
\begin{lemma}\label{rot:anisotropic}
If $a$ satisfies Equation~\eqref{rot:genericity}, with $a$ replacing $u$,
then $S_c$ has no split cocharacter directions outside the connected
center of $G$.
\end{lemma}
\begin{proof}
A noncentral invariant cocharacter can be placed in the closure of a
Weyl chamber. Its stabilizer is the Weyl group of a proper standard
Levi. Choose a fundamental character for a simple root missing from
that Levi, multiplied to be integral. It is fixed by the arithmetic
monodromy of $S_c$, and so defines a rational character $\chi$ of that
torus. The value $\chi(\gamma_c)$ is a rational function on $X$.
Away from the legs it is a unit: any realization of the loop is an
integral automorphism there, as is its inverse, and hence all its
torus-character valuations vanish. At a leg its valuation is
$\langle\varpi,w_ja_j\rangle$ for a Weyl choice $w_j$. The product
formula contradicts the assumed nonvanishing. This argument uses
the tautological torus element and therefore applies even before a
rational bundle realizing $c$ has been chosen.
\end{proof}
\begin{lemma}\label{rot:picard}
Suppose $a=nu+v$ is regular and satisfies the nonvanishing condition,
and suppose the two fixed-point groupoids in question are finite.
Then ordinary and shifted Frobenius descent on $\mathcal L_a$ have
the same weighted count:
\begin{equation}\label{rot:descent-count}
|\mathcal L_a(\mathbb F_q)|=|\mathcal L_a^{\mathrm{sh}}|.
\end{equation}
The assertion holds with arbitrary multiplicities in $D$ and after
the central quotient described above.
\end{lemma}
\begin{proof}
We give the comparison separately for each rational invariant $c$.
First we construct a connected group acting on suitable full-level
subgroupoids of loops. We then realize rotation as a rational point
of that group and use Lang's theorem to compare the two descent
conditions.
Let $Y\to X$ be the normalization in a connected arithmetic Galois
splitting field of $S_c$. Its group $\Gamma$ is a subgroup of $W$;
write $N=|\Gamma|$. This cover need not be geometrically connected.
It is split and unramified at the regular legs. Let
$\mathsf M=X_*(T)$ be the abstract split cocharacter lattice used on
the cover. There is a norm homomorphism of generic tori
\begin{equation}\label{rot:norm}
\nu:\operatorname{Res}_{F(Y)/F}
(\mathbb G_m\otimes_{\mathbb Z}\mathsf M)\longrightarrow S_c,
\end{equation}
which multiplies Galois conjugates with their descent action on
$\mathsf M$. We will use a connected source for this norm. Neither
surjectivity nor separability of the norm itself is required.
Choose a finite rational set $Z\subset X$ disjoint from the legs and
containing the support of $D$, the ramification points, the points
where $\gamma_c$ is not regular integral, and all other prescribed
level points. Add one auxiliary closed point outside the legs.
For an effective rational modulus $B$ supported on the full inverse
image of $Z$, with positive multiplicity at every such point, put
\begin{equation}\label{rot:picard-source}
Q_B=\operatorname{Pic}^{\underline 0}(Y,B)
\otimes_{\mathbb Z}\mathsf M.
\end{equation}
The superscript means degree zero on \emph{each geometric component}.
The modulus meets every component, so a line bundle with its specified
$B$-trivialization has no scalar automorphisms. Consequently $Q_B$ is
an actual smooth connected commutative algebraic group. One can see
this from the generalized-Picard exact sequence: over the algebraic
closure it is an extension of the product of the component Jacobians
by the products of the local truncated unit groups, modulo one
constant multiplicative group per component. A truncated unit group
is smooth and connected, including for nonreduced moduli in positive
characteristic. This is also the precise representability and
connectedness statement in \cite[Section~1.4, Proposition~1.6 and the
following paragraph]{JordanRibetScholl}. The auxiliary point rigidifies this source
only; it adds no level to the original loop problem.
For the descent comparison, it suffices to construct a coherent action
of $Q_B(\overline{\mathbb F}_q)$ on geometric objects and arrows,
compatible with Frobenius. We now impose the integral conditions that
retain the full level. Let $\mathcal L_{a,c,B}$ be
the full subgroupoid on objects for which norms of split congruence
units at $B$ preserve the local lattices, all flags, and the complete
$D$-trivialization. Every object belongs to such a subgroupoid after
increasing the multiplicities of $B$. Indeed the generic torus
embedding and the norm are continuous for the local valuation
topologies; sufficiently deep units map into the prescribed principal
congruence stabilizer. Away from $Z$ and the legs the regular integral
centralizer is an unramified torus, and its integral units preserve the
lattice. At the legs this follows from the centralizer-apartment
description above.
Represent an element of $Q_B(\overline{\mathbb F}_q)$ by an
$\mathsf M$-valued divisor disjoint from $B$. Such a representative
exists by choosing meromorphic
sections agreeing with its trivializations to the specified finite
orders. Modify the bundle at the image of its support by the norm
cocharacters in Equation~\eqref{rot:norm}: change the local
trivializations at those points and glue them to the unchanged torsor
away from the support. This uses formal gluing for smooth affine
torsors on a curve, applied at finitely many points; see
\cite[arXiv version, Section~5.a, Lemma~5.1]{PappasZhu}.
At $Z$ keep the original lattice and its level.
All modifications commute with $\gamma_c$, so the result remains a
loop with the same invariant and the same relative positions at the
legs. If two representatives differ by the divisor of a rational
split-torus function $h$ with $h|_B=1$, the meromorphic automorphism
$\nu(h)$ identifies the two results. At $Z$ it extends and preserves
the full nonreduced level by the defining stabilizer condition;
elsewhere it is the usual change of lattice trivialization.
The normalized function $h$ is unique: the ratio of two choices is
constant on each geometric component and equals one along $B$.
The ratios of three representatives multiply, so these isomorphisms
satisfy their cocycle identities. Tensor product therefore gives a
coherent action of $Q_B(\overline{\mathbb F}_q)$ on the groupoid.
An arrow between loops identifies their canonical centralizers and
carries each norm modification to the corresponding one. Thus the
action is natural in all arrows, including every target automorphism.
The action preserves $\mathcal L_{a,c,B}$. Namely a torus modification
does not change which elements of the commuting norm-unit subgroup
stabilize a lattice. The same observation proves stability under
$R_u$. All these constructions are Frobenius-compatible, since the
cover and the modulus are rational.
We next exhibit rotation in the connected source
Equation~\eqref{rot:picard-source}. At a geometric point $y$ of $Y$
over the leg $x_j$, let $\nu_y\in\mathsf M\otimes\mathbb Q$ be the
valuation vector of the tautological element $\gamma_c$ in the split
coordinates on $Y$. It belongs to the Weyl orbit of $a_j$.
Since $a_j$ is regular, there is a unique $w_y\in W$ with
$\nu_y=w_ya_j$. Define the $\mathsf M$-valued divisor
\[
\Delta=\sum_j\ \sum_{y\mapsto x_j}w_y(u_j/N)[y],
\]
where the inner sum runs over all geometric points above $x_j$.
These cocharacters are integral because $u_j\in2|W|\Lambda_0$ and
$N$ divides $|W|$. The valuation vectors are transported by the deck
action and preserved by Frobenius on the abstract constant lattice
$\mathsf M$. Uniqueness of $w_y$ therefore makes $\Delta$ both rational
and $\Gamma$-equivariant with its descent action on the labels.
Under the ordered local identification, every lift contributes
$u_j/N$ to its norm. There are $N$ lifts over each geometric leg, so
the norm is exactly the rotation modification by $u_j$.
The support of $\Delta$ avoids $B$, giving its canonical
trivialization along the full modulus.
We verify its multidegree. Let $\Gamma_{\mathrm{geom}}$ be the
geometric monodromy subgroup, and write $e_C$ for the vector degree of
$\Delta$ on a geometric component $C$. Deck equivariance gives
\[
e_{\gamma C}=\gamma e_C,
\]
so $\Gamma_{\mathrm{geom}}$ fixes $e_C$. Frobenius preserves the labels
of the abstract split torus, and rationality gives
$e_{\mathrm{Fr}C}=e_C$. Choose $\sigma\in\Gamma$ inducing the same
permutation of geometric components as Frobenius. Then
\[
\sigma e_C=e_{\sigma C}=e_{\mathrm{Fr}C}=e_C.
\]
The coset of $\sigma$ generates the cyclic group
$\Gamma/\Gamma_{\mathrm{geom}}$. Together with invariance under
$\Gamma_{\mathrm{geom}}$, this proves
\begin{equation}\label{rot:component-degree}
e_C\in(\mathsf M\otimes\mathbb Q)^\Gamma.
\end{equation}
By Lemma~\ref{rot:anisotropic}, this vector is central. Every coefficient
of $\Delta$ lies in the rational subspace annihilating $X^*(G)$, and so
does $e_C$. That derived subspace meets the rational central
cocharacter subspace in zero. Thus $e_C=0$ on every component.
This is the point at which arithmetic, rather than merely geometric,
monodromy is needed. We have obtained an element
$a_B\in Q_B(\mathbb F_q)$ whose action is $R_u$.
We can now compare descent. Lang's theorem applies to the smooth
connected source $Q_B$; see \cite[Theorem~1 and its corollary]{Lang}. In this
commutative case its surjectivity is especially direct:
$\mathrm{Fr}-1$ has derivative $-1$, hence has open image, and an open
subgroup of a connected algebraic group is the entire group. Choose
$b\in Q_B(\overline{\mathbb F}_q)$ such that
\[
\mathrm{Fr}(b)-b=a_B.
\]
If $\mathsf t_b$ denotes its action, Frobenius compatibility supplies
a natural identification
\[
R_u^{-1}\mathrm{Fr}\,\mathsf t_b
\simeq\mathsf t_b\,\mathrm{Fr}.
\]
Hence $\mathsf t_b$, with inverse $\mathsf t_{-b}$, gives an equivalence
of descent groupoids
\begin{equation}\label{rot:lang-equivalence}
\operatorname{Fix}(\mathrm{Fr},\mathcal L_{a,c,B})
\simeq
\operatorname{Fix}(R_u^{-1}\mathrm{Fr},\mathcal L_{a,c,B}).
\end{equation}
The identity in the source is an isomorphism of rigidified line
bundles, with no scalar ambiguity. Equation~\eqref{rot:lang-equivalence}
therefore includes the Frobenius isomorphisms and their compatible
arrows. It preserves automorphism groups, not only geometric
isomorphism classes.
We also spell out effectivity of the descent used here. An object and
a Frobenius isomorphism are defined over a finite extension. After
iterating by that extension degree, the resulting automorphism belongs
to the rational points of an affine finite-type group over a finite
field, hence has finite order. A further iterate is the identity.
The datum is thus finite Galois descent, which is effective for bundles,
levels, and their meromorphic isomorphisms. Ordinary Frobenius descent
is precisely the rational-point groupoid in
Equation~\eqref{rot:descent-count}. The shifted version can either be
read from Equation~\eqref{rot:lang-equivalence} or from the explicit
concatenation construction of Lemma~\ref{rot:concatenation}.
The subgroupoids $\mathcal L_{a,c,B}$ exhaust the geometric loop
groupoid as the multiplicities grow. Both fixed-point groupoids have
finitely many isomorphism classes by hypothesis. Choose $B$ large
enough to contain representatives of both. The equivalence then
proves equality of their entire weighted counts for $c$. Summing over
the finitely many contributing rational invariants proves
Equation~\eqref{rot:descent-count}.
Finally, a norm modification arising from componentwise degree-zero
source line bundles changes no character degree of $G$. Its action
commutes with central modifications and therefore descends to the
quotient by $\Xi$. The initial degree-lifting observation identifies
its closed paths with the ones just considered. This proves the
central-quotient assertion as well.
\end{proof}
\subsection{Proof of the trace identity}
\begin{proof}[Proof of Theorem~\ref{rot:trace}]
With $U_v=p_!q^*$ on the unnormalized complex, the identity to prove is
equivalently
\begin{equation}\label{rot:raw-trace}
\sum_{[\mathcal E]}(T_{nu+v})_{\mathcal E,\mathcal E}
=q^{-(nL_u+L_v)/2}
\operatorname{Tr}_{\mathrm{alt}}
(\mathrm{Fr}^{n*}U_v\mid C(u)).
\end{equation}
Set $a=nu+v$. For $n$ large it is regular and satisfies the
nonvanishing condition. Lemma~\ref{rot:bounded} makes the ordinary
loop count finite, and the large-Frobenius formula makes the shifted
count finite. For a fixed rational bundle $\mathcal E$, closing a
Hecke modification whose target is isomorphic to $\mathcal E$ amounts
to choosing such an isomorphism, modulo its automorphisms. Consequently
its diagonal matrix entry is the corresponding loop mass. Summing
gives
\[
\begin{aligned}
\sum_{[\mathcal E]}(T_a)_{\mathcal E,\mathcal E}
&=q^{-L_a/2}|\mathcal L_a(\mathbb F_q)|\\
&=q^{-L_a/2}|\mathcal L_a^{\mathrm{sh}}|\\
&=q^{-L_a/2}\operatorname{Tr}_{\mathrm{alt}}
(\mathrm{Fr}^{n*}U_v\mid C(u)).
\end{aligned}
\]
The first equality includes the stack multiplicities; no extra reciprocal
automorphism factor is to be inserted in the matrix trace.
The second equality is Lemma~\ref{rot:picard}, and the third is
Lemma~\ref{rot:concatenation}. Since lengths add in the positive chamber,
$L_a=nL_u+L_v$, proving Equation~\eqref{rot:raw-trace}.
Equation~\eqref{rot:geometric-operator} and the even shift in
Equation~\eqref{rot:complex} give
Equation~\eqref{rot:normalized-trace}. The operator $T_v$ here is
function-normalized; the parity correction for an arbitrary $v$ was
kept in Lemma~\ref{rot:finite-correspondence}.
\end{proof}
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/04-rotation.tex ====
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/05-extraction.tex ====
\section{Extraction from the discrete spectrum}\label{sec:extraction}
We prove a local restriction on the discrete automorphic spectrum.
All absolute values in this section refer to one fixed complex
realization. We transport the coefficient field and the sheaf
constructions to that realization as abstract characteristic-zero linear
algebra. The eigenvalues coming from mixed compactly supported
cohomology retain their Weil weights under this transport.
Let $G$ be split connected reductive and let $T$ be a split maximal
torus. An \emph{ordered Bernstein weight} of a local representation
at a place $x$ is a character $t_0:X_*(T)\to\mathbf k^\times$
occurring in its Iwahori invariants,
before passage to its Weyl orbit. Put $q_x=q^{\deg(x)}$ and define its
exponent by
\[
\langle\lambda_0,a\rangle=\log_{q_x}|t_0(a)|.
\]
We call this exponent dominant if
$\langle\lambda_0,\alpha^\vee\rangle\ge0$ for every positive root
$\alpha$ of $G$, with the positive-translation convention of
Section~\ref{sec:categorical}.
\begin{theorem}\label{ext:discrete}
Let $G$ be split connected reductive, let $x$ be a closed point, and
impose Iwahori level at $x$ and arbitrary fixed compact open level
away from $x$. Let $\pi$ be a unitary discrete automorphic
representation with nonzero invariants at that level. If an ordered
Bernstein weight $t_0$ of $\pi_x$ has dominant exponent
$\lambda_0=\log_{q_x}|t_0|$, then
\begin{equation}\label{ext:local-form}
t_0\sim
\phi\begin{pmatrix}q_x^{1/2}&0\\0&q_x^{-1/2}\end{pmatrix}c,
\end{equation}
where $\phi\colon\operatorname{SL}_2\to\Gd$ is a homomorphism and
$c$ is semisimple and conjugate into a maximal compact subgroup of
$Z_{\Gd}(\phi)$.
\end{theorem}
The proof is an induction on semisimple rank. For a hypothetical
violating weight, a polynomial spectral filter and the rotation trace
identity force its occurrence in compact cohomology of translation
shtukas. Integral Frobenius weights then make $\lambda_0$ rational,
allowing integral highest weights $\mu=m\lambda_0$ along its exact ray.
The Wakimoto filtration isolates the repeated highest-weight term,
and hyperspecial projections transfer its high-degree cohomology to
the spherical calculation. A separate spherical dg model supplies
an actual chain projector onto this summand. A finite Koszul test
preserves its highest nonzero cohomological degree, which the
Frobenius lower bound and Theorem~\ref{amp:amplitude} then bound
incompatibly as $m$ grows. A fixed auxiliary place makes the translation
shtukas bounded even when the ray lies on a chamber wall.
\subsection{Dominant ordered weights and hyperspecial projections}
We first isolate the local algebra needed for passage from Iwahori to
hyperspecial level. Throughout this subsection, $Q>1$ is the residue
cardinality. Let $L=X_*(T)$ be the cocharacter lattice of a split connected
reductive group, let $W$ be its finite Weyl group, and let
\[
\mathcal H=\mathcal H(L,Q),\qquad
A=\mathbf k[L],\qquad Z=A^W
\]
be its Iwahori Hecke algebra, Bernstein subalgebra, and center.
We use the group-theoretic Bernstein presentation in
\cite[Lemmas~1.6.1 and~1.7.1, Remark~1.7.2, and
Equation~(1.15.2)]{HainesKottwitzPrasad}.
The coefficient field $\mathbf k$ is algebraically closed of characteristic
zero. The unnormalized finite simple generators, denoted here by
$\mathsf T_s$, satisfy
\[
(\mathsf T_s-Q)(\mathsf T_s+1)=0.
\]
We write $\theta_a$ for the Bernstein element corresponding to $a\in L$.
In the positive chamber of standard translations it is
\[
\theta_a=Q^{-\ell(t^a)/2}\mathsf T_{t^a}=T_a.
\]
Thus the trivial representation has positive real Bernstein exponent.
With $\alpha$ a positive simple root of the group, the Bernstein relation
in these conventions is
\begin{equation}\label{ext:bernstein-relation}
\theta_a\mathsf T_s-\mathsf T_s\theta_{sa}
=(Q-1)\frac{\theta_a-\theta_{sa}}{1-\theta_{-\alpha^\vee}}.
\end{equation}
The quotient on the right belongs to $A$.
For a character $t\colon L\to\mathbf k^\times$, choose a complex
realization of its values and write
$\lambda=\log_Q|t|\in\operatorname{Hom}(L,\mathbf R)$.
We call $t$ dominant if
\begin{equation}\label{ext:dominant-character}
\lambda(\alpha^\vee)\ge0\qquad(\alpha>0).
\end{equation}
This condition places no restriction on the central part of $\lambda$.
Put $c=Wt\in\operatorname{Spec}(Z)$ and
\[
R=\widehat{Z}_{c},\qquad B=\mathcal H\otimes_Z R.
\]
Since $A$ is finite over $Z$, its completed scalar extension decomposes as
\[
A\otimes_Z R\simeq\prod_{r\in Wt}\widehat A_r.
\]
Let $p_t$ be the idempotent of this product selecting the factor at $t$.
On a finite-dimensional module with central character $c$, $p_t$ selects
the generalized $A$-weight space at $t$.
Choose a base alcove in the reduced building. Let $\mathcal V$ be its
finite set of hyperspecial vertices and let $e_v\in\mathcal H$,
$v\in\mathcal V$, be their normalized
parahoric idempotents. In particular, at the base hyperspecial vertex,
\begin{equation}\label{ext:spherical-idempotent}
e=\frac{\sum_{w\in W}\mathsf T_w}
{\sum_{w\in W}Q^{\ell(w)}}.
\end{equation}
The denominator is nonzero. All these vertices are conjugate under the
length-zero alcove stabilizer for the adjoint semisimple root datum;
the corresponding conjugate idempotents belong to $\mathcal H$ even when
those length-zero elements do not belong to its original extended Weyl
group.
\begin{lemma}\label{ext:hecke}
Suppose that $t$ satisfies Equation~\eqref{ext:dominant-character}.
Then, in $B$,
\begin{equation}\label{ext:hecke-ideals}
Bp_tB=\sum_{v\in\mathcal V}Be_vB.
\end{equation}
Equivalently, a simple module at the central character $Wt$ contains the
ordered Bernstein weight $t$ if and only if at least one of the
hyperspecial idempotents acts nontrivially on it.
\end{lemma}
\begin{proof}
We first treat the full semisimple coweight lattice, then descend through
a finite lattice extension, and finally pass from simple modules to
idempotent ideals.
\smallskip
\noindent\emph{The full lattice.}
Suppose first that $L$ is the direct sum of the full coweight lattice of
the semisimple root system and a central lattice. The latter contributes
a central Laurent polynomial algebra and may be specialized at its
character. It therefore suffices to consider the semisimple factors.
For a character $r$ put
\[
I(r)=\mathcal H\otimes_A\mathbf k_r.
\]
Its spherical line is spanned by
$\mathbf 1_r=\sum_{w\in W}\mathsf T_w\otimes1$.
We use the following precise principal-series theorem: for the full
weight lattice of a root system, the spherical vector generates the
principal-series module if and only if
\begin{equation}\label{ext:spherical-cyclicity}
\prod_{\alpha>0}\bigl(1-Qr(\alpha^\vee)\bigr)\ne0.
\end{equation}
This is \cite[Proposition~2.11(b)]{Ram}, applied to the dual root system.
The parameter denoted by $q$ there is $Q^{1/2}$ here, and its finite
generator is $Q^{-1/2}\mathsf T_s$. Thus the exceptional root value in
that proposition is $Q^{-1}$, as in
Equation~\eqref{ext:spherical-cyclicity}. The result includes singular
characters. For a product of root systems it follows by tensor product.
Its characteristic-zero formulation over $\mathbf k$ follows by
extension of scalars: the finite matrices determining cyclicity are
defined over the field generated by the finitely many character values
and $Q^{1/2}$, which admits a complex realization.
If $r$ is dominant, every positive-coroot value has modulus at least
one, so Equation~\eqref{ext:spherical-cyclicity} holds. Consequently
$I(r)$ is generated by $\mathbf 1_r$. A simple module $S$ containing
the generalized weight $r$ contains an ordinary eigenvector of that
weight: a finite-dimensional module over a commutative algebra has a
common eigenvector in each nonzero generalized weight space. Inducing
such a vector gives a surjection $I(r)\to S$. The generating spherical
vector cannot map to zero, so $eS\ne0$.
Conversely, there is exactly one spherical simple at each central
character. For completeness, this follows directly from the spherical
Satake identity $e\mathcal He=Ze$
\cite[Sections~4.1 and~4.6, Equation~(4.6.1)]{HainesKottwitzPrasad}.
At a central character $c'$, the
module
\[
P=\mathcal He\otimes_Z\mathbf k_{c'}
\]
is finite-dimensional and satisfies $eP=\mathbf k e$. Every proper
submodule of $P$ has zero $e$-projection, since any submodule containing
$eP$ contains the generator of $P$. The sum of its proper submodules
is therefore proper and is its unique maximal submodule. Any simple
module $S$ with $eS\ne0$ and central character $c'$ is a quotient of
$P$: for $0\ne u\in eS$, simplicity gives $S=\mathcal H u$ and
$eS=(e\mathcal He)u=\mathbf k u$. This proves uniqueness.
At $c'=Wr$, every simple quotient of $I(r)$ is spherical by the preceding
paragraph. Hence its unique spherical simple is such a quotient and
contains $r$, since the inducing vector generates $I(r)$ and has a
nonzero image in every nonzero quotient. We have proved, for the full
lattice and every dominant $r$,
\begin{equation}\label{ext:full-lattice-detector}
S[r]\ne0\quad\Longleftrightarrow\quad eS\ne0,
\end{equation}
where $S[r]$ denotes the generalized weight space.
\smallskip
\noindent\emph{A finite enlargement for a general reductive lattice.}
Let $V=L\otimes_{\mathbf Z}\mathbf Q$. Write
\[
V=V_{\mathrm{ss}}\oplus V_{\mathrm c},
\]
where $V_{\mathrm{ss}}$ is the rational span of the coroots and
$V_{\mathrm c}$ is the common kernel of the roots. Let $Q^\vee$ and
$P^\vee$ be the coroot and full coweight lattices in $V_{\mathrm{ss}}$,
and put
\[
L_{\mathrm{ss}}=L\cap V_{\mathrm{ss}},\qquad
L_{\mathrm c}=\operatorname{pr}_{\mathrm c}(L),\qquad
L'=P^\vee\oplus L_{\mathrm c}.
\]
The projection is rational, so $L_{\mathrm c}$ is a lattice.
Root integrality gives
$\operatorname{pr}_{\mathrm{ss}}(L)\subset P^\vee$; hence $L\subset L'$.
The inclusion has finite index, and
\begin{equation}\label{ext:lattice-quotient}
\Gamma=L'/L\simeq P^\vee/L_{\mathrm{ss}}.
\end{equation}
Indeed the map from $P^\vee$ to $L'/L$ is surjective because
$L\to L_{\mathrm c}$ is surjective, and its kernel is
$L_{\mathrm{ss}}$. In particular $Q^\vee\subset L_{\mathrm{ss}}$.
Set $\mathcal H'=\mathcal H(L',Q)$ and $A'=\mathbf k[L']$.
The Weyl group acts trivially on $\Gamma$, since
$w a'-a'\in Q^\vee$ for $a'\in L'$. The Bernstein presentation
consequently grades $\mathcal H'$ by $\Gamma$, with degree-zero
subalgebra $\mathcal H$. Each component has an invertible homogeneous
element $\theta_{a'}$, which commutes with $A$. Thus
\begin{equation}\label{ext:strong-grading}
\mathcal H'=\bigoplus_{\gamma\in\Gamma}
\theta_{a'_{\gamma}}\mathcal H
\end{equation}
is a strong finite grading. No splitting of $\Gamma$ into a subgroup
of units is required.
We shall also use a different set of homogeneous units. The
length-zero stabilizer $\Omega'_{\mathrm{ss}}$ of the semisimple base
alcove is isomorphic to $P^\vee/Q^\vee$, and its map onto $\Gamma$ is
surjective by Equation~\eqref{ext:lattice-quotient}. Choose
$\omega_\gamma\in\Omega'_{\mathrm{ss}}$ in each degree and put
$U_\gamma=\mathsf T_{\omega_\gamma}$. Then
\begin{equation}\label{ext:hyperspecial-conjugates}
e_\gamma=U_\gamma^{-1}eU_\gamma\in\mathcal H
\end{equation}
is the idempotent of a hyperspecial vertex. To see the inclusion in
$\mathcal H$, one may either use its degree zero or note that conjugation
by $\omega_\gamma$ takes the finite Weyl subgroup at the base vertex
to the finite parahoric Weyl subgroup at that hyperspecial vertex.
Central translations fix the reduced building and give no further
idempotents. Taking all of $\mathcal V$, rather than just the vertices
arising from these representatives, is therefore harmless.
\smallskip
\noindent\emph{Descent of the simple-module detector.}
Let $S$ be a simple $\mathcal H$-module at the central character $c$.
It is finite-dimensional, since $\mathcal H$ is finite over $Z$.
Choose a simple quotient $S'$ of the finite-dimensional induced module
$\mathcal H'\otimes_{\mathcal H}S$. Adjunction gives a nonzero map
$S\to S'$, hence an embedding on restriction. We identify $S$ with
its image. Simplicity of $S'$ and the two choices of homogeneous units
give
\begin{equation}\label{ext:clifford-sums}
S'=\sum_{\gamma\in\Gamma}\theta_{a'_\gamma}S
=\sum_{\gamma\in\Gamma}U_\gamma S.
\end{equation}
Each summand is a simple $\mathcal H$-module, because each homogeneous
unit normalizes the degree-zero algebra. This also proves directly
the needed Clifford-theoretic assertion that the restriction is a
semisimple sum of conjugates. More particularly,
$\theta_{a'_\gamma}$ commutes with $A$, so all summands in the first
sum have the same $A$-weight support. Consequently
\begin{equation}\label{ext:restriction-support}
S[t]\ne0\quad\Longleftrightarrow\quad S'[t]\ne0,
\end{equation}
with the right-hand side interpreted as an $A$-weight space.
The second sum gives the complementary assertion
\begin{equation}\label{ext:restriction-spherical}
eS'\ne0
\quad\Longleftrightarrow\quad
e_\gamma S\ne0\text{ for some }\gamma\in\Gamma.
\end{equation}
Suppose first that $S[t]\ne0$. The nonzero $A$-weight space $S'[t]$
contains an $A'$-weight $t'$ restricting to $t$. Since $L$ has finite
index in $L'$, restriction determines the real absolute-value exponent
uniquely; thus $t'$ is dominant. Applying
Equation~\eqref{ext:full-lattice-detector} to $\mathcal H'$ gives
$eS'\ne0$. Equation~\eqref{ext:restriction-spherical} now gives a
nonzero hyperspecial projection on $S$.
Conversely, suppose that $e_vS\ne0$ for a hyperspecial vertex $v$.
The idempotent $e_v$ is conjugate to $e$ by a length-zero unit of
$\mathcal H'$, so $S'$ is spherical. Let $Wr'$ be its $A'^W$-central
character. Restriction of this central character to $Z$ is $Wt$;
hence, after replacing $r'$ by a Weyl conjugate, its restriction to
$L$ is exactly $t$. This representative $r'$ is dominant. The
full-lattice detector says that the spherical simple $S'$ contains
$r'$, and hence contains $t$ on restriction to $A$.
Equation~\eqref{ext:restriction-support} gives $S[t]\ne0$. We have
proved the asserted equivalence on all simple modules at $c$.
\smallskip
\noindent\emph{The completed ideals.}
The algebra $B$ is finite over the complete local ring $R$.
Every simple $B$-module is annihilated by the maximal ideal
$\mathfrak m_R$: it is finite over $R$, and centrality and simplicity
make $\mathfrak m_R S$ either zero or $S$, with the second possibility
excluded by Nakayama's lemma. The same assertion holds for every
quotient of $B$. Thus the simple $B$-modules are precisely the simple
modules of the residue fiber to which the preceding detector applies.
Put $J=\sum_{v\in\mathcal V}Be_vB$. In $B/J$, the image of $p_t$
annihilates every simple module by that detector. It belongs to the
Jacobson radical and is an idempotent, so it is zero. Therefore
$Bp_tB\subset J$. Conversely, in $B/Bp_tB$ every image of $e_v$
annihilates every simple module. These images are again idempotents
in the Jacobson radical and are zero. This proves the reverse
inclusion and Equation~\eqref{ext:hecke-ideals}.
\end{proof}
\begin{corollary}\label{ext:hecke-finite}
In the notation of Lemma~\ref{ext:hecke}, let
$N=\bigoplus_i N^i$ be a graded $B$-module whose $B$-action preserves
degree. No finite-generation hypothesis on $N$ is imposed.
If $p_tN$ is finite-dimensional, then $e_vN$ is finite-dimensional for
every $v\in\mathcal V$. Moreover,
\[
p_tN^i\ne0\quad\Longrightarrow\quad
e_vN^i\ne0\text{ for some }v\in\mathcal V.
\]
These assertions apply in particular to any invariant direct summand
selected by a projector commuting with $B$.
\end{corollary}
\begin{proof}
Equation~\eqref{ext:hecke-ideals} supplies, for each $v$, a finite
expression $e_v=\sum_j a_jp_tb_j$ with $a_j,b_j\in B$. Hence
$e_vN\subset\sum_j a_j(p_tN)$, proving finite-dimensionality.
Conversely, write $p_t$ as a finite sum of elements of the ideals
$Be_vB$. If all $e_vN^i$ were zero, this expression would give
$p_tN^i=0$. Finally, the image of a projector commuting with $B$ is a
$B$-submodule, so the same argument applies to it.
\end{proof}
\subsection{The finite-level analytic input}
We specify the analytic facts used below. Write $L=X_*(T)$ for a split
maximal torus and put
\[
L^0=\{a\in L:\langle\chi,a\rangle=0
\text{ for every }\chi\in X^*(G)\}.
\]
This is a saturated sublattice spanning the semisimple directions.
The annihilator of $L^0$ in the dual torus is $Z(\Gd)^\circ$.
Consequently two ordered parameters have the same character on $L^0$
if and only if they differ by an element of $Z(\Gd)^\circ$.
Choose a free group $\Xi$ of rational central modifications whose degree
image is cocompact in the free central degree lattice and intersects
degree zero trivially, as in Section~\ref{sec:rotation}.
For a compact open level $K$, let
\[
\mathcal L_K=L^2\bigl(G(F)\backslash G(\mathbb A_F)/(K\Xi)\bigr).
\]
The measure on its point basis includes the usual automorphism factors.
An arbitrary unitary automorphic central character can be treated in
this model after a unitary unramified twist: a character on the lattice
$\Xi$ extends to the degree group of $G$, in which its image has finite
index, since the circle group is divisible. Such a twist has zero real exponent and does not affect the
assertion we shall prove.
\begin{proposition}[Finite-level spectral facts]\label{ext:spectral}
At a fixed compact open level the following assertions hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The discrete summand $\mathcal D_K$ of $\mathcal L_K$ is
finite-dimensional. Its orthogonal complement admits a Hecke-equivariant unitary
realization as a closed invariant subspace of a finite sum of
direct integrals of normalized parabolic inductions
\[
\operatorname{Ind}_{P(\mathbb A_F)}^{G(\mathbb A_F)}
(\sigma\otimes\chi),
\]
where $P$ is proper, $\sigma$ belongs to the discrete spectrum of its
Levi $M$, and $\chi$ ranges over a compact torus of unitary unramified
characters, subject to the central condition imposed by $\Xi$.
All fixed-level fiber dimensions are bounded.
\item Let $Q_b$ be the orthogonal projection onto the point basis in an
HN ball of radius $b$. There are constants $C,c>0$ such that
\begin{equation}\label{ext:discrete-tail}
\|(1-Q_b)v\|\le C e^{-cb}\|v\|
\qquad(v\in\mathcal D_K, b\ge0).
\end{equation}
\item Suppose that the level at $x$ is Iwahori. The ordered Bernstein
weights of a fiber induced from $M$ have the form
\begin{equation}\label{ext:induced-weights}
w(t_M z),\qquad w\Phi_M^+\subset\Phi_G^+,
\end{equation}
where $t_M$ is an ordered Bernstein weight of $\sigma_x$ and $z$ is the
local value of $\chi$. There are finitely many such weight families,
and their generalized-weight lengths are uniformly bounded.
\end{enumerate}
\end{proposition}
\begin{proof}
We use the following precise form of Langlands' spectral theorem.
At a fixed compact open level, the discrete pairs $(M,\sigma)$ with
nonzero fixed-level induced vectors form finitely many classes under
Weyl association and unramified twisting. Their compact induced
models are finite-dimensional, their unitary twisting spaces are
compact real tori modulo finite stabilizers, and Eisenstein wave
packets on these spaces give the Hecke-equivariant Hilbert spectral
decomposition. This formulation is stated in
\cite[Theorem III.7 and the preceding definitions, pp.~33--36]{LLafforgueSpectral},
where it is identified with \cite[Section VI.2.1]{MW}; the
function-field treatment in the English edition is
\cite[Section VI.2.7]{MW}. The statement includes discrete inducing
representations. If the given level is not contained in the chosen
standard maximal compact subgroup, intersect it with that subgroup
first and then take invariants under the original compact level.
This replaces the full compact models by closed invariant subspaces
and preserves all the asserted finiteness and norm bounds.
We explain the effect of $\Xi$. Unramified characters are characters
of a free degree lattice. On the central degree lattice of $G$, the
condition that $\sigma\otimes\chi$ be trivial on $\Xi$ fixes finitely
many characters because the image of $\Xi$ has finite index.
For $M=G$ this leaves finitely many points of each twisting torus.
There are finitely many pairs by the quoted theorem and their
fixed-level models have finite dimension, so $\mathcal D_K$ is finite.
For a proper Levi $M$, the condition cuts its unitary parameter torus
into finitely many translates of a closed subtorus, hence leaves a
compact parameter space. The wave-packet construction allows finite
Weyl identifications between parameters; equivalently it realizes
the corresponding Hilbert space inside finitely many full induced
compact models. This proves (i) in precisely the form needed here.
Boundedness of fixed-level induction ranks can also be read directly
from the compact picture. The quotient of a maximal compact subgroup
by its intersection with the inducing parabolic and the prescribed
open level is finite. Its finitely many representatives replace the
level on $\sigma$ by finitely many compact open subgroups of $M$.
Their invariant spaces are finite-dimensional. Unitary unramified
twists are trivial on these compact subgroups, so the same dimension
bounds hold throughout each torus.
For (ii), use the exact constant-term assertion on deep
function-field Siegel sectors
\cite[Chapter I, Section 5.9, p.~107]{MorrisI}; see also
\cite[Section I.2.7]{MW}. Once the root coordinates outside a Levi
are sufficiently large, a function at the prescribed compact level
equals its constant term for that parabolic. The threshold is
uniform at a fixed level and on the bounded transverse data.
The constant-term exponent theorem says that translating an
automorphic form's normalized constant term in the split-center
degree variables gives a finite sum of characters times polynomials
\cite[Section 6.2, pp.~200--201, Equations (6.1)--(6.3)]{BernsteinLapid};
see also \cite[Sections I.3.1--I.3.2 and I.4.2]{MW}.
We need the strict decay consequence, and give its elementary proof
to include all boundary directions.
Partition a Siegel domain according to which simple-root coordinates
are above the constant-term threshold. The remaining roots define
the Levi; their bounded coordinates range over bounded data modulo
its center. After passing to finite-index degree lattices, finitely
many transverse values and translates of positive orthants suffice.
Take the image of each sufficiently deep sector in
$P(F)\backslash G(\mathbb A)/(K\Xi)$, with its quotient Iwasawa
measure $\delta_P(m)^{-1}\,du\,dm\,dk$.
The compact unipotent quotient $U(F)\backslash U(\mathbb A)$ has
volume one, and the function equals its constant term $f_P$ on
this sector. The deep-sector separation statement
\cite[Chapter I, Section 4.2, p.~101]{MorrisI} puts every additional
rational identification within a sector in $P(F)$, which has
already been quotiented out. The finitely many bounded transverse
charts of \cite[Chapter I, Section 4.6, pp.~102--103]{MorrisI}
therefore give bounded residual multiplicity for the map to the
automorphic quotient. The sector's weighted squared norm is
consequently at most a fixed multiple of the global squared norm.
This statement uses the parabolic quotient and its measure;
the integral-unipotent multiplicity in a raw subset of
$G(\mathbb A)$ need not be bounded.
Absorb the finitely many transverse measure factors into a norm
on a finite-dimensional space $V$. Multiplication by
$\delta_P^{-1/2}$ then normalizes the constant term to an
exponential-polynomial sequence
\[
F\colon\mathbf N^r\longrightarrow V,
\qquad \dim V<\infty,
\]
and the preceding norm comparison gives
$F\in\ell^2(\mathbf N^r,V)$.
Let $E$ be the span of all forward coordinate shifts of $F$.
It is finite-dimensional because $F$ is exponential-polynomial.
Each coordinate shift $S_j$ preserves $E$ and is a contraction for
its inherited $\ell^2$ norm. It has no eigenvalue of modulus one:
if $S_jh=zh$, $|z|=1$, and $h\ne0$, a nonzero slice in the other
coordinates gives a coordinate ray of constant nonzero norm,
contradicting square summability. Thus choose $\rho<1$ greater
than the moduli of every eigenvalue of every $S_j|_E$.
Finite-dimensional Jordan estimates and commutation of the shifts
give
\[
\|S_1^{n_1}\cdots S_r^{n_r}\|\le C\rho^{n_1+\cdots+n_r}.
\]
Evaluation at the origin is bounded in the $\ell^2$ norm, so this
proves exponential decay of $F$. Summing over the polynomially
many lattice points of a given radius gives an exponential squared
norm tail. Bounded sectors contribute none for large radius.
The reduction height and the HN radius are comparable up to fixed
constants, by the degree description of the canonical reductions.
This proves Equation~\eqref{ext:discrete-tail} for each discrete form.
A basis of the finite-dimensional space $\mathcal D_K$ makes the
constants uniform on that space.
Finally, compatibility of normalized parabolic induction with Iwahori
invariants identifies the Iwahori invariants of the local induced representation with the
corresponding induced Hecke module
\cite[Section~5.8, Theorem~8]{PrasadBernstein}.
We use left modules and the positive-translation convention of
Equation~\eqref{ext:bernstein-relation}; the conversion from the
source's right-module convention is described in its Section~7.
The Bernstein presentation now proves (iii). The ambient Hecke
algebra is free as a right module over the Levi Hecke algebra,
with basis indexed by the representatives $w$ for which
$w\Phi_M^+\subset\Phi_G^+$. Commuting a Bernstein element past these
basis elements by Equation~\eqref{ext:bernstein-relation} is triangular
in Bruhat order, and its diagonal characters are exactly
Equation~\eqref{ext:induced-weights}. The real exponent of a unitary
$\sigma$ is zero in the split central directions of $M$. The finite
list of inducing data and the bounded fiber dimensions from (i)
give the last assertions of (iii).
\end{proof}
We shall use the following elementary uniform bound to handle the fact
that Bernstein translations need not be normal operators.
\begin{lemma}\label{ext:matrix-bound}
Let $B,r_0>0$ and $R_0\in\mathbf N$ be fixed. For every $r>r_0$ there
is a constant $C$ with the following property: if $A$ is an operator
on a Hilbert space of dimension at most $R_0$, with
$\|A\|\le B$ and all eigenvalues of modulus at most $r_0$, then
\[
\|A^n\|\le Cr^n\qquad(n\ge0).
\]
\end{lemma}
\begin{proof}
On the circle $|z|=r$, the determinant of $z-A$ has absolute value at
least $(r-r_0)^{\dim A}$. In an orthonormal basis, its adjugate has a
bound depending only on $B,r,R_0$. The resolvent is therefore uniformly
bounded on that circle. The contour formula for $A^n$ gives the stated
estimate, after increasing $C$ to cover the zero-dimensional case.
\end{proof}
\subsection{A dominant weight forces high cohomology}
To set up the induction for Theorem~\ref{ext:discrete}, suppose that it
is known in smaller semisimple rank and that $t_0$ violates its conclusion.
We work at a level, and after a unitary central twist, for which $\pi$ is contained in
$\mathcal D_K$. Its central exponents are zero. In particular
$\lambda_0$ lies in the semisimple directions, and it is nonzero: an
exponent-zero semisimple parameter has the form in
Equation~\eqref{ext:local-form} with the trivial homomorphism $\phi$.
Choose a second closed point $y\ne x$ where $\pi$ is unramified, refine
its level to an Iwahori, and fix an ordered Bernstein weight $t_{0,y}$
there. Shrink the level away from $x,y$ to a product of principal
congruence subgroups. Every compact open subgroup contains such a
product, so the resulting level
$K=I_x\times I_y\times K^{x,y}$ still has $\pi^K\ne0$.
The factor $K^{x,y}$ is full level along an effective divisor disjoint
from $x,y$, as required by the geometric constructions.
All these levels now remain fixed.
We give the finite-level tensor argument ensuring simultaneous
occurrence of the two chosen local weights; the general tensor-product
theory is due to Flath \cite{FlathTensorProducts}.
The finite-dimensional space $V=\pi^K$ is simple over the product
Hecke algebra $\mathcal H_x\otimes\mathcal H_y\otimes\mathcal H^{x,y}$.
Indeed any nonzero vector generates $\pi$ under the adelic group by
irreducibility; averaging its translates over $K$ shows that it
generates $V$ under this Hecke algebra.
Let $A_x,A_y$ be the images of the two local Hecke algebras in
$\operatorname{End}(V)$. The subspace
$\operatorname{rad}(A_x)V$ is stable under the entire product algebra,
so is either zero or $V$. The second possibility contradicts
nilpotence of the radical; hence $A_x$ is semisimple.
Every element of its center commutes with the product algebra and
therefore acts as a scalar on $V$. Thus $A_x$ is a single matrix
algebra. The same argument applies to $A_y$.
The commuting matrix-algebra actions consequently give
\[
V\simeq U_x\otimes U_y\otimes W,
\]
where $U_x,U_y$ are their simple modules and $W\ne0$ is the
multiplicity space. The simple local types are independent of the
auxiliary level: refinement embeds the finite invariant spaces,
whose isotypic restrictions therefore have the same simple type.
Taking the union over auxiliary levels identifies these types with
the local Iwahori modules $\pi_x^{I_x}$ and $\pi_y^{I_y}$.
The tensor of their nonzero generalized weight spaces at $t_0$ and
$t_{0,y}$ is therefore nonzero. Thus $(t_0,t_{0,y})$ occurs on the
finite-dimensional discrete space at this refined level.
In the following, weights are compared on $(L^0)^2$ unless explicitly
stated otherwise.
\begin{lemma}\label{ext:spectral-gap}
There is a Laurent polynomial $P_0$ in the degree-zero Bernstein
translations at $x$, nonzero at $t_0$, and an open cone of dominant
regular directions arbitrarily close to $\lambda_0$, such that the
following holds. For every fixed auxiliary label $\zeta$ at $y$
and every sufficiently large label $\mu$ in this cone, every continuous
joint weight surviving $P_0$ has strictly smaller modulus on
$T_{(\mu,\zeta)}$ than the target joint weight.
\end{lemma}
\begin{proof}
Complexify the twisting torus of each family in
Proposition~\ref{ext:spectral}. Its ordered $x$-weights form a coset
of an algebraic subtorus. There are finitely many such cosets.
A coset not containing the target on $L^0$ can be annihilated by a
Laurent polynomial nonzero at the target. Raise that polynomial to
a power exceeding the uniformly bounded generalized-weight length,
and multiply over these cosets. The resulting $P_0$ kills their
actions, including generalized weights.
Consider a remaining family, induced from $M$ and shuffled by $w$.
Let $C_M$ be its real twisting-center subspace after that shuffle, and
let $\nu$ be the real exponent of its unitary contour. Unitarity of
the inducing central character gives $\nu\perp C_M$. Membership of
the target in its complexified coset gives
$\lambda_0-\nu\in C_M$. The possible ambiguity from working on $L^0$
is a connected-central factor of $G$ and is already contained in the
twisting center of $M$. Therefore
\begin{equation}\label{ext:orthogonal-projection}
\nu=\operatorname{pr}_{C_M^\perp}(\lambda_0),\qquad
(\lambda_0,\nu)=\|\lambda_0\|^2
-\|\operatorname{pr}_{C_M}(\lambda_0)\|^2.
\end{equation}
If the second subtracted term were zero, the twisting factor taking
the inducing parameter to $t_0$ would be unitary. Moreover,
$w\Phi_M^+\subset\Phi_G^+$ and dominance of $\lambda_0$ would make
the corresponding inducing weight dominant for $M$. The induction
hypothesis would give Equation~\eqref{ext:local-form} for that weight,
and hence for its unitary central twist and Weyl conjugate $t_0$.
This contradicts the choice of $t_0$. Thus every remaining family
has
\begin{equation}\label{ext:strict-projection}
(\lambda_0,\nu)<\|\lambda_0\|^2.
\end{equation}
The finite list of strict inequalities persists in a small open cone
of directions adjacent to $\lambda_0$ and inside the regular dominant
chamber. The $y$-exponents run through a finite list on the unitary
contours. Their pairings with the fixed $\zeta$ are bounded constants,
whereas the strict $x$-gap grows linearly with $\mu$. This proves the
assertion for all sufficiently large $\mu$ in a slightly smaller cone.
\end{proof}
Choose once and for all a sufficiently divisible regular positive label
$\zeta\in L^0$ such that all the finitely many parabolic pairings
involving $\zeta$ alone are nonzero. This is possible by avoiding a
finite collection of rational hyperplanes. The divisibility includes
the fixed multiples required in the rotation calculation and the even
multiple needed to remove convention signs. We impose the same fixed
divisibility on the varying $\mu$.
\begin{proposition}\label{ext:lower}
Under the induction assumption and the choice of a violating weight
above, let $u=(\mu,\zeta)$, where $\mu$ is sufficiently large in the
cone of Lemma~\ref{ext:spectral-gap} and $u$ satisfies
Equation~\eqref{rot:genericity}. Then $M(u)$ contains the joint
Bernstein character $(t_0,t_{0,y})$ on $(L^0)^2$. At some full ordered
extension $(t_x,t_y)$ of that character it has nonzero cohomology in a
degree $i$ satisfying
\begin{equation}\label{ext:degree-lower}
i\ge 2d_x\langle\lambda_0,\mu\rangle
+2\log_q|\zeta(t_{0,y})|.
\end{equation}
\end{proposition}
\begin{proof}
The complex $M(u)$ has finite-dimensional total cohomology by
Lemma~\ref{rot:bounded}. If the target character did not occur,
a Laurent polynomial in the two degree-zero Bernstein algebras would
annihilate its entire cohomology and remain nonzero at the target.
Multiply it by $P_0$ and by polynomial factors killing every other
discrete joint character. Powers larger than the finite generalized
weight lengths kill the corresponding generalized spaces. Denote
the resulting polynomial by $P$. It is nonzero at the target, kills
$M(u)$, and on the discrete spectrum retains only the target joint
generalized weight space $\mathcal D_0$.
Multiply $P$ by a Bernstein monomial with sufficiently large positive
labels at both places. This changes none of these properties, since
all Bernstein monomials are invertible. We can consequently write
\[
P=\sum_{v}c_vT_v
\]
with a finite set of regular positive degree-zero labels $v$. For
the remainder of this argument $u$ and $P$ are fixed and only the
positive integer $n$ varies. Apply Theorem~\ref{rot:trace} to each
term. The normalized version of that identity gives, for every
sufficiently large $n$,
\begin{equation}\label{ext:filtered-diagonal}
\sum_b(T_u^nP)_{b,b}
=\operatorname{Tr}_{\mathrm{alt}}
(\mathrm{Fr}^nP\mid M(u))=0.
\end{equation}
Here $b$ indexes the normalized point basis of $\mathcal L_K$;
normalization by the square root of its measure does not change
diagonal matrix entries. The translation $u$ is even; each insertion $T_v$ is in the
function normalization of that theorem, including its known parity
correction to the perverse trace. No evenness condition is imposed
on $v$. Only
this ordinary diagonal sum is used; the operator is not asserted
to be trace class on all of $\mathcal L_K$.
The support bound in Theorem~\ref{rot:trace} puts every contributing
diagonal entry in an HN ball of radius $Bn$ for a fixed $B$. Increase
$B$ if necessary and write $Q_n=Q_{Bn}$. The left side of
Equation~\eqref{ext:filtered-diagonal} is therefore
$\operatorname{Tr}(Q_nT_u^nP Q_n)$. By
Proposition~\ref{amp:geometry}, $\operatorname{rank}Q_n$ is polynomial
in $n$.
Put
\[
a=\mu(t_0)\zeta(t_{0,y}),\qquad R=|a|>0.
\]
Lemma~\ref{ext:spectral-gap} gives an $r_0<R$ bounding the moduli of
all continuous eigenvalues of $T_u$ on the image of $P$. Fixed Hecke
operators are uniformly bounded on all unitary fibers by their
$L^1$ norms. The image of $P$ in each fiber is invariant under $T_u$,
has bounded dimension, and inherits its Hilbert norm. Choose
$r_0<r<R$ and apply Lemma~\ref{ext:matrix-bound} on this image.
We obtain
\[
\|T_u^nP\|_{\mathrm{cont}}\le C r^n.
\]
This bound is uniform even where the dimension of the image changes.
Hence the continuous contribution to the compressed trace is at
most $\operatorname{rank}(Q_n)Cr^n=o(R^n)$.
On $\mathcal D_0$, simultaneous triangularization of the commuting
Bernstein operators gives
\begin{equation}\label{ext:discrete-leading-trace}
\operatorname{Tr}_{\mathcal D_K}(T_u^nP)
=\dim(\mathcal D_0)P(t_0,t_{0,y})a^n.
\end{equation}
The coefficient is nonzero; generalized-weight nilpotents do not
change the diagonal of these matrices. The norm of $T_u^nP$ on this
finite-dimensional space is at most $C(1+n)^N R^n$ for fixed $C,N$.
Equation~\eqref{ext:discrete-tail} therefore makes the difference
between its full trace and its $Q_n$-compressed trace $o(R^n)$.
Decomposing the Hilbert space into its orthogonal discrete and
continuous summands, Equations~\eqref{ext:filtered-diagonal} and
\eqref{ext:discrete-leading-trace} would imply
\[
0=\dim(\mathcal D_0)P(t_0,t_{0,y})a^n+o(R^n),
\]
an impossibility. Thus the target character occurs in $M(u)$.
Since the full Bernstein torus also acts on this finite-dimensional
space, the target character on $(L^0)^2$ has at least one full ordered
extension $(t_x,t_y)$ in its support. The cyclic-slide identity of
Proposition~\ref{cat:cyclic} identifies normalized Frobenius on that
generalized weight space with the translation $T_u$, up to the
already harmless convention sign. Every one of its Frobenius
eigenvalues has modulus
\[
|\mu(t_x)\zeta(t_y)|
=q^{d_x\langle\lambda_0,\mu\rangle
+\log_q|\zeta(t_{0,y})|}.
\]
Recall the weighted translation length
$L_u=d_x\ell(t^\mu)+d_y\ell(t^\zeta)$ and the normalization in
Equation~\eqref{rot:complex}. The reduction in the
proof of Lemma~\ref{rot:bounded} applies the compact-support weight
theorem \cite[Corollary~3.3.4]{DeligneII} to $Y_u$: the weights of
the constant sheaf's $H_c^j$ are integers at most $j$. Since
\[
H^i(M(u))=H_c^{i+L_u}(Y_u,\mathbf k)(L_u/2),
\]
the weights of this normalized group are integers at most $i$.
Comparing with the displayed Frobenius modulus proves
Equation~\eqref{ext:degree-lower}.
\end{proof}
\begin{corollary}\label{ext:rationality}
The semisimple exponent $\lambda_0$ of a violating weight in this
induction argument is rational.
\end{corollary}
\begin{proof}
The Frobenius weights of $M(u)$ are integers by the weight theorem
and normalization just used in Proposition~\ref{ext:lower}. Set $b=2\log_q|\zeta(t_{0,y})|$. The occurrence in
Proposition~\ref{ext:lower}, not merely its degree inequality, gives
\begin{equation}\label{ext:integral-pairings}
2d_x\langle\lambda_0,\mu\rangle+b\in\mathbf Z
\end{equation}
for every sufficiently large allowed label in the open cone.
Let $L_1\subset L^0$ be the fixed finite-index lattice imposing the
divisibility conditions. Given $a\in L_1$, choose $\mu$ sufficiently
deep in that cone that both $\mu$ and $\mu+a$ belong to it, and avoid
the finitely many affine hyperplanes excluded by
Equation~\eqref{rot:genericity} for both labels. Such labels exist:
a full-dimensional cone has lattice points arbitrarily far from any
prescribed finite collection of proper affine hyperplanes.
Subtracting the two instances of
Equation~\eqref{ext:integral-pairings} gives
$2d_x\langle\lambda_0,a\rangle\in\mathbf Z$. These pairings on the
full-rank lattice $L_1$ prove the assertion.
\end{proof}
\subsection{The extreme summand and specialization}
Use the rational metric to identify the exponent direction with a label
direction. By Corollary~\ref{ext:rationality} we can now take
\begin{equation}\label{ext:exact-ray}
\mu=m\lambda_0
\end{equation}
for arbitrarily large divisible integers $m$. These labels may lie on a
wall. The spectral gap proved above also holds on this exact ray by
Equation~\eqref{ext:strict-projection}. The fixed choice of $\zeta$
makes Equation~\eqref{rot:genericity} hold for every sufficiently large
such $m$: if a main-ray parabolic pairing vanishes, its auxiliary pairing
does not; if it does not vanish, the combined pairing has at most one
exceptional value of $m$. Thus Proposition~\ref{ext:lower} and its proof
apply along this ray as well.
\begin{lemma}\label{ext:unique-extreme}
For $\mu$ as in Equation~\eqref{ext:exact-ray}, the only weight $a$ of
the irreducible representation $V_\mu$ maximizing
$\langle\lambda_0,a\rangle$ is $a=\mu$, and its multiplicity is one.
Consequently the only maximizing tuple in $V_\mu^{\otimes d_x}$ is
$(\mu,\ldots,\mu)$.
\end{lemma}
\begin{proof}
Every weight has the form $\mu-\sum_\alpha n_\alpha\alpha$ with
$n_\alpha\ge0$ in the positive simple roots of the dual group.
Equality of its pairing with that of $\mu$ forces every root appearing
in this difference to be orthogonal to $\lambda_0$. The highest
weight $\mu$ has zero pairing with the coroots of that Levi root
system. Its highest line therefore generates the trivial
representation of the corresponding derived Levi. Applying only
its lowering operators produces no different weight. The maximizing
face consists of the highest line alone. Additivity of the pairing
proves the assertion about tuples.
\end{proof}
Let $(t_x,t_y)$ be one of the full extensions supplied by
Proposition~\ref{ext:lower} for the chosen $m$. It can vary with $m$,
but $t_x/t_0\in Z(\Gd)^\circ$. Put $d=d_x$ and consider the Iwahori
trace complex
\[
C_\mu=\mathsf K\bigl(Z(V_\mu)^{\boxtimes d};J_\zeta\bigr),
\]
where the semicolon records the fixed auxiliary label at $y$ and all
other fixed levels and kernels. Let $\mathcal H_x$ be the affine
Hecke algebra at $x$ with coefficients in $\mathbf k$, let $Z_x$ be
its center, and set
\[
R_x=\widehat{(Z_x)}_{Wt_x},\qquad
\widehat H_\mu=H^*(C_\mu)\otimes_{Z_x}R_x.
\]
This is flat scalar extension of a graded cohomology module.
No dg $Z_x$-module structure on the Iwahori complex is used.
The auxiliary label at $y$ remains fixed; no projection onto a
$y$-weight is needed in this step.
We use a different operator here from the full normalized Frobenius
in Proposition~\ref{ext:lower}. On the repeated highest $x$-tuple,
the $d$-fold $x$-label slide has character $d\mu$, whereas full
Frobenius on $M(\mu,\zeta)$ has joint character $(\mu,\zeta)$.
The selected-place slide separates Wakimoto terms; the lower degree
bound retains the single residue-degree factor $d_x$ in
Equation~\eqref{ext:degree-lower}.
\begin{proposition}\label{ext:extreme-transfer}
Fix a positive even integer $\kappa$ removing the slide convention
signs, and let $S_x$ be the $\kappa$th power of the $d$-fold
$x$-label slide. There is a polynomial projector
$\varepsilon_{\mathrm{ext}}$ on $\widehat H_\mu$, commuting with
$\mathcal H_x\otimes_{Z_x}R_x$, such that every
\[
N_v=e_v\varepsilon_{\mathrm{ext}}\widehat H_\mu
\]
has finite-dimensional total graded space. For some hyperspecial
vertex $v$, it is nonzero in a degree satisfying
Equation~\eqref{ext:degree-lower}. The transfer identifies $N_v$
with the extreme $S_x$-part of
$H^*(C_{\mu,v}^{\mathrm{sph}})\otimes_{Z_x}R_x$, where
$C_{\mu,v}^{\mathrm{sph}}
=\mathsf K_v(V_\mu^{\boxtimes d};J_\zeta)$.
\end{proposition}
\begin{proof}
Use the finite Wakimoto filtration of
Theorem~\ref{cat:central} and its cohomology spectral sequence.
Its first page has labels $\mathbf a=(a_1,\ldots,a_d)$ of weights
of $V_\mu$, with their weight-space multiplicities. The Bernstein
algebra and $S_x$ act on this spectral sequence. The commutation
statements of Propositions~\ref{cat:cyclic} and \ref{cat:hecke}
also make $S_x$ commute with the full Hecke action on its abutment.
Tensoring the pages and the abutment with the flat algebra $R_x$
gives a convergent spectral sequence with its finite filtration.
Write $\Theta_b$ for the Bernstein element of a label $b$.
Taking the $d$-fold slide first returns each tuple $\mathbf a$ to
its original ordering. For $b=a_1+\cdots+a_d$, the Wakimoto
interchange formula of Theorem~\ref{cat:central} identifies $S_x$
on this graded term with
\[
\Theta_b^\kappa\otimes U_{\mathbf a},
\]
where $U_{\mathbf a}$ acts on the finite tensor product of
weight-multiplicity spaces. The chosen power removes the parity
signs, and the trivial semisimplified Weil structures make
$U_{\mathbf a}$ unipotent. The two factors commute.
After specializing the Bernstein algebra at an ordering $r$,
its only possible eigenvalue is
\begin{equation}\label{ext:graded-slide}
\left(\prod_{j=1}^{d}a_j(r)\right)^\kappa.
\end{equation}
Here is an annihilator construction entirely on cohomology.
The completed Bernstein algebra is finite over $R_x$.
For each of the finitely many labels $\mathbf a$, put
\[
D_{\mathbf a}(z)=\prod_{w\in W}
\bigl(z-\Theta_{wb}^\kappa\bigr).
\]
Its coefficients are invariant, and hence lie in $R_x$.
One factor gives $D_{\mathbf a}(\Theta_b^\kappa)=0$.
Therefore $D_{\mathbf a}(S_x)$ lies in the ideal generated by
$1\otimes(U_{\mathbf a}-1)$ in the commutative algebra of these
operators. Choose $n_{\mathbf a}$ with
$(U_{\mathbf a}-1)^{n_{\mathbf a}}=0$. The monic polynomial
$Q_{\mathbf a}=D_{\mathbf a}^{n_{\mathbf a}}\in R_x[z]$
annihilates the entire first-page term, regardless of the size
of its cohomology. Its residual roots are exactly the values in
Equation~\eqref{ext:graded-slide} at the orderings of $t_x$.
Put $Q_0=\prod_{\mathbf a}Q_{\mathbf a}$. This annihilates the
entire first page, hence every later page and the associated graded
of the abutment. If the filtration has $b_\mu$ steps, then
$Q=Q_0^{b_\mu}$ annihilates $\widehat H_\mu$: each application of
$Q_0(S_x)$ lowers its filtration by at least one step. The same
$Q$ also annihilates every page.
The maximal modulus of these residual roots is independent of
the ordering of $t_x$, by Weyl invariance of the weight polytope.
Hensel factorization over $R_x$ gives
$Q=Q_+Q_-$, with $Q_+$ collecting all roots of that maximal modulus
and $Q_-$ all remaining roots. The factors are relatively prime.
Their Bezout identity defines an idempotent
$\varepsilon_{\mathrm{ext}}$, polynomial in $S_x$, on every page,
the filtered cohomology, and its abutment. On the abutment it
commutes with the full Hecke algebra, since $S_x$ does and its
polynomial coefficients are central.
Apply also the ordered idempotent $p_{t_x}$ from
Lemma~\ref{ext:hecke}. It commutes with the spectral-sequence
maps and with $\varepsilon_{\mathrm{ext}}$.
Lemma~\ref{ext:unique-extreme} shows that exactly one label tuple
survives on the projected first page: every entry is $\mu$.
Its multiplicity is one. The Weil normalization and interchange
compatibility in Theorem~\ref{cat:central} identify this term with
the $t_x$-ordered part of $H^*(M(\mu,\zeta))$.
Central factors cause no additional eigenvalues, since
$\mu\in L^0$ and $V_\mu$ is trivial on $Z(\Gd)^\circ$.
There is no other surviving filtration step to support a
differential. It follows that
$p_{t_x}\varepsilon_{\mathrm{ext}}\widehat H_\mu$ is
finite-dimensional and contains the nonzero high-degree piece
of Proposition~\ref{ext:lower}. Flat completion preserves this
finite-dimensional generalized central-character piece.
Apply Corollary~\ref{ext:hecke-finite} to the graded module
$\varepsilon_{\mathrm{ext}}\widehat H_\mu$.
Each $N_v$ is finite-dimensional, and some $N_v$ retains a nonzero
piece in the same high degree. The cohomology transfer in
Proposition~\ref{cat:hecke} identifies
$e_v\widehat H_\mu$ with
$H^*(C_{\mu,v}^{\mathrm{sph}})\otimes_{Z_x}R_x$ and intertwines
$S_x$ and the center. This proves the stated identification.
The finite set of hyperspecial variants is among those allowed
in Theorem~\ref{amp:amplitude}.
\end{proof}
The next step supplies the actual complex needed for specialization.
It uses the spherical dg model separately from the Iwahori
cohomology construction.
\begin{proposition}\label{ext:spherical-projector}
For each $v$, the genuine spherical $Z_x$-dg module
$C_{\mu,v}^{\mathrm{sph}}$ has, after flat scalar extension to
$R_x$, a dg direct summand $E_{\mu,v}$ with
$H^*(E_{\mu,v})=N_v$. Its chain projector is polynomial in the
spherical slide $S_x$ and commutes with finite Koszul tests over
$R_x$.
\end{proposition}
\begin{proof}
Use Proposition~\ref{cat:loops}. On the finite representation
bundle $V_\mu^{\otimes d}$, the $d$-fold slide is the degree-zero
closed matrix
\[
\rho_\mu(s)=V_\mu(s)^{\otimes d}
\]
over $\mathcal O(\Gd)$. Therefore
\[
P_\mu(z)=\det\bigl(z-\rho_\mu(s)^\kappa\bigr)
\in\mathcal O(\Gd)^{\Gd}[z]=Z_x[z]
\]
satisfies $P_\mu(S_x)=0$ as a literal identity of chain maps.
Indeed Cayley--Hamilton holds on the finite bundle before tensoring
with the arbitrary dg module $\mathcal A$ in that proposition;
tensoring and then taking exact rational invariants preserve the
identity. In particular this assertion requires no finiteness
assumption on $\mathcal A$ or on the whole spherical cohomology.
Now the completed complex
\[
\widehat C_{\mu,v}^{\mathrm{sph}}
=C_{\mu,v}^{\mathrm{sph}}\otimes_{Z_x}R_x
\]
is defined using that actual dg module structure.
The residual roots of $P_\mu$ are the products of weight values
in Equation~\eqref{ext:graded-slide} at $t_x$, with their tensor
multiplicities. Their set is Weyl invariant, so it is the same
root set used for $Q$.
Factor $P_\mu=P_+P_-$ over $R_x$ by Hensel's lemma, separating the
same maximal-modulus residual roots from the others. For a Bezout
identity $uP_++vP_-=1$, the chain map
\[
e_+=v(S_x)P_-(S_x)
\]
is idempotent: $e_+^2-e_+$ is a multiple of $P_\mu(S_x)=0$.
Its image $E_{\mu,v}$ is an actual dg direct summand.
The coefficients lie in $R_x$, so this projector commutes with
tensoring by any finite Koszul complex over $R_x$.
We verify that its cohomology is the transferred extreme part;
polynomial annihilators on cohomology alone would not have
sufficed for the preceding chain construction. On
$H^*(\widehat C_{\mu,v}^{\mathrm{sph}})$ both $Q(S_x)$ and
$P_\mu(S_x)$ vanish, and the two polynomial projectors commute.
On the intersection of the $Q_+$ part with the $P_-$ part,
$Q_+(S_x)$ and $P_-(S_x)$ vanish. Their residual roots have
different moduli, so these two polynomials are relatively prime
over $R_x$; their Bezout identity makes this intersection zero.
The same argument interchanging $+$ and $-$ kills the other
cross intersection. Thus $e_+$ and the transferred
$\varepsilon_{\mathrm{ext}}$ act identically on cohomology.
This argument applies to arbitrary modules, without finite
generation. Consequently $H^*(E_{\mu,v})=N_v$, as claimed.
\end{proof}
\begin{lemma}[Top cohomology under a Koszul test]\label{ext:koszul}
Let $(R,\mathfrak m)$ be a noetherian local ring and let $C$ be a
complex of $R$-modules bounded above in cohomology. Suppose that
$N$ is its highest nonzero cohomological degree and $H^N(C)$ is a
finite nonzero $R$-module. If $f_1,\ldots,f_r$ generate
$\mathfrak m$, then
\[
H^N\bigl(C\otimes_R K_R(f_1,\ldots,f_r)\bigr)\ne0,
\]
where the Koszul complex is placed in degrees $[-r,0]$.
No regular-sequence hypothesis is required.
\end{lemma}
\begin{proof}
The bounded Koszul complex is a complex of finite free modules, so its
tensor product has a convergent spectral sequence
\[
E_2^{p,q}=H^p\bigl(K_R(f;H^q(C))\bigr)
\ \Longrightarrow\ H^{p+q}(C\otimes_R K_R(f)).
\]
Its term at $(p,q)=(0,N)$ is
$H^N(C)/\mathfrak mH^N(C)$, which is nonzero by Nakayama's lemma.
No later differential enters this term, since its source would have
second index greater than $N$; none leaves it, since its target
would have positive Koszul degree. The term survives in total
degree $N$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{ext:discrete}]
For semisimple rank zero, the group is a split torus. Its unitary
local characters have zero absolute-value exponent, so the assertion
holds with trivial $\phi$. Assume the result in lower semisimple rank
and choose a violating weight $t_0$ as above. All the preceding
constructions are then available for arbitrarily large $m$ in
Equation~\eqref{ext:exact-ray}.
Use Proposition~\ref{ext:extreme-transfer} to choose $v$, and then
Proposition~\ref{ext:spherical-projector} to obtain the dg summand
$E_{\mu,v}$. Let $N$ be its highest nonzero cohomological degree.
It exists because $H^*(E_{\mu,v})=N_v$ has finite-dimensional total
graded space, and it satisfies
\begin{equation}\label{ext:top-lower}
N\ge 2dm\|\lambda_0\|^2+b,
\qquad b=2\log_q|\zeta(t_{0,y})|.
\end{equation}
Use the invariant algebra generators $a_1,\ldots,a_r$ fixed before
Lemma~\ref{amp:specialization}, and put $f_j=a_j-a_j(t_x)$.
These functions generate the maximal ideal of $Wt_x$ in $Z_x$,
and their images generate the
maximal ideal of $R_x$. Apply their finite Koszul complex to the
actual $R_x$-dg summand $E_{\mu,v}$.
Lemma~\ref{ext:koszul} preserves a nonzero class in degree $N$.
The chain projector of Proposition~\ref{ext:spherical-projector}
commutes with this tensor product, so the tested summand is a
direct summand of the tested completed spherical complex.
Completion can be removed from the latter test. Indeed it is
flat, and each $f_j$ acts null-homotopically on its Koszul complex.
The tested cohomology is therefore annihilated by the maximal
ideal, and extension of scalars from $Z_x$ to $R_x$ changes
none of its cohomology groups. The natural map from the
uncompleted spherical Koszul test to its completion is thus
a quasi-isomorphism. On this specialized cohomology the polynomial
coefficients of the chain projector reduce to their residue-field
values. Its image is exactly the maximal-modulus group for the
holonomy slide $S_x$, by its Hensel factorization. Raising an
invertible slide to the positive power $\kappa$ preserves the
ordering of eigenvalue moduli. Hence the surviving degree $N$
belongs to the extreme summand to which
Theorem~\ref{amp:amplitude} applies, giving
\begin{equation}\label{ext:top-upper}
N\le C+dm\max_{e\in E_{t_x}}
\langle\lambda_0,h_e\rangle.
\end{equation}
The restriction of $t_x$ to $L^0$ is that of $t_0$. Hence $t_x$ differs
from $t_0$ only by a connected-central factor. The adjoint eigenspace
$E_{t_x}$, its compatible neutral cocharacters, and the semisimple
exponent are consequently independent of that factor. Because
$t_0$ is a violation and its central exponent is zero, the strict part
of Theorem~\ref{amp:amplitude} gives
\[
\delta=2\|\lambda_0\|^2-
\max_{e\in E_{t_x}}\langle\lambda_0,h_e\rangle>0.
\]
The same theorem makes $C$ uniform under the possible connected-central
changes of $t_x$. Its hypotheses also allow the fixed auxiliary label
and all the finitely many hyperspecial variants; replace their
constants by their maximum. No level, auxiliary kernel, or
semisimple holonomy datum varies with $m$.
Subtracting Equation~\eqref{ext:top-upper} from
Equation~\eqref{ext:top-lower} now yields
$dm\delta\le C-b$, impossible for arbitrarily large $m$.
There is no violating weight. This completes the induction and the
proof of Theorem~\ref{ext:discrete}.
\end{proof}
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/05-extraction.tex ====
==== BEGIN Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/06-global.tex ====
\section{The global decomposition}
\label{sec:global}
We now assume that $G$ is semisimple, and put
$U=X\setminus\operatorname{supp}(D)$. Throughout this section,
Frobenius and the normalized Satake isomorphism have the conventions
fixed in the introduction. In particular, if $d_x=\deg(x)$, then
$q_x=q^{d_x}$.
Theorem~\ref{ext:discrete} gives a complex compact--Jacobson--Morozov
form at each good place and each complex realization. Three steps
remain. At each place we must obtain one algebraic factorization whose
remainder is compact at every complex embedding. We must then show
that its nilpotent orbit is independent of the good place, and descend
the resulting decomposition of joint Hecke eigenspaces to $\mathbb Q$.
\subsection{The rational cuspidal spectrum}
We first record precisely the finite-dimensional space to which the
argument applies. For a split group, the isomorphism classes of bundles
with full level $D$ are
\[
\Bun_{G,D}(\mathbb F_q)
=G(F)\backslash G(\mathbb A_F)/K_D.
\]
In general the bundle description has a disjoint union indexed by
$\ker^1(F,G)$; this kernel is trivial for split $G$
\cite[Section~2.1, Equation~(2.1), and Remark~2.2]{VLafforgue}.
Thus this identification does not assert the stronger, generally false
statement that every $G$-torsor over $F$ is trivial.
The finite-level cuspidal space is finite dimensional
\cite[Section~1.1]{VLafforgue}. The finite subscheme called $N$ there
can be precisely $D$, with all its multiplicities: its subgroup
$K_N$ is defined by the same kernel as $K_D$. Moreover, the central
lattice used there can be trivial here, since the center of a semisimple
group has compact adelic points. These statements therefore concern
the entire space $C_{D,\mathbb Q}$ in Theorem~\ref{main:decomposition}.
Write $C_{D,\mathbb C}=\mathbb C\otimes_{\mathbb Q}C_{D,\mathbb Q}$,
and let
\[
\mathbb T_{D,\mathbb Q}
\subset\operatorname{End}_{\mathbb Q}(C_{D,\mathbb Q})
\]
be the unital algebra generated by the good-place spherical Hecke
algebras $A_x$, for $x\in |U|$.
\begin{lemma}\label{glob:hecke-spectrum}
The algebra $\mathbb T_{D,\mathbb Q}$ is finite dimensional and reduced.
After extending scalars to $\overline{\mathbb Q}$, its action has a
decomposition
\begin{equation}\label{glob:joint-spectrum}
\overline{\mathbb Q}\otimes_{\mathbb Q}C_{D,\mathbb Q}
=\bigoplus_{\chi\in\Sigma_D} C_\chi
\end{equation}
into joint eigenspaces, where $\Sigma_D$ is finite. All good-place
Satake classes of these characters are algebraic.
\end{lemma}
\begin{proof}
Finite dimensionality of $C_{D,\mathbb Q}$ implies that of its Hecke
image algebra. On the complex cusp space use the positive automorphic
$L^2$ inner product. For a spherical function $a$, the adjoint of
convolution by $a$ is convolution by
$a^*(g)=\overline{a(g^{-1})}$. This function is again spherical and
its operator preserves the cusp space. The spherical operators at
one place commute, and operators at different places commute.
Consequently all their restrictions are commuting normal operators.
Simultaneous diagonalization follows in the finite-dimensional space,
and implies that the Hecke image algebra is reduced. A finite reduced
algebra over $\mathbb Q$ is a product of number fields, so it splits
over $\overline{\mathbb Q}$ and gives
Equation~\eqref{glob:joint-spectrum}. The normalized Satake
isomorphism is defined over $\mathbb Q(\sqrt{q_x})$; hence an algebraic
Hecke character determines an algebraic point of the dual adjoint
quotient, and therefore an algebraic semisimple conjugacy class.
\end{proof}
\subsection{A single triple at every complex embedding}
The definition of $R_{r,\mathcal O}$ requires one algebraic triple and
an algebraic remainder compact at every complex embedding. The local
theorem supplies complex triples separately at those embeddings. We
compare their neutral cocharacters in a fixed torus: the exact root
equations will force them to agree, after which polynomial equations
produce a triple over $\overline{\mathbb Q}$.
For a split maximal torus $T$ of $\Gd$ and a semisimple
$t\in T(\overline{\mathbb Q})$, define its exponent at an embedding
$\iota:\overline{\mathbb Q}\hookrightarrow\mathbb C$, relative to a
prime power $r>1$, by
\begin{equation}\label{glob:exponent}
\langle a,\lambda_\iota(t)\rangle
=\log_r|\iota(a(t))|,
\qquad a\in X^*(T).
\end{equation}
Thus $\lambda_\iota(t)\in X_*(T)\otimes_{\mathbb Z}\mathbb R$.
We first keep this vector in the fixed torus, without applying a
different Weyl element at each embedding.
\begin{lemma}\label{glob:embeddings}
Suppose that, at every embedding $\iota$, the element $\iota(t)$ is
conjugate to
\[
\phi_\iota\!\begin{pmatrix}\sqrt r&0\\0&\sqrt r^{-1}\end{pmatrix}
c_\iota,
\]
where $\phi_\iota:\operatorname{SL}_2\to\Gd_{\mathbb C}$ is an
algebraic homomorphism and $c_\iota$ is semisimple and conjugate into
a compact subgroup of
$Z_{\Gd}(\phi_\iota(\operatorname{SL}_2))(\mathbb C)$.
Then the following hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The vectors $\lambda_\iota(t)$ are all equal to a vector
$\lambda$, and $\mu=2\lambda$ belongs to $X_*(T)$.
\item There is a homomorphism
$\phi:\operatorname{SL}_2\to\Gd_{\overline{\mathbb Q}}$
with diagonal cocharacter $\mu$ such that
\[
t=\mu(\sqrt r)c,
\qquad
c\in Z_{\Gd}(\phi(\operatorname{SL}_2))
(\overline{\mathbb Q}),
\]
and the image of $c$ is compact at every complex embedding.
\item The nilpotent orbit of the raising element of $\phi$ is unique.
In particular, $[t]$ belongs to precisely one of the sets
$R_{r,\mathcal O}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Fix one embedding and conjugate its proposed decomposition so that
its product is $\iota(t)$ itself. Let $A$ be the image of the diagonal
torus of $\operatorname{SL}_2$. Its centralizer
$L=Z_{\Gd}(A)$ is a connected Levi subgroup
\cite[Theorem~17.38 and Section~25.25]{MilneAlgebraicGroups}.
The semisimple element
$c_\iota$ lies in $L$, so it lies in a maximal torus $S$ of $L$.
The central torus $A$ belongs to every maximal torus of $L$.
Therefore $S$ contains both $A$ and $\iota(t)$ and is a maximal torus
of $\Gd$. Both $S$ and $T_{\mathbb C}$ are maximal tori of the
connected group $Z_{\Gd}(\iota(t))^\circ$; maximal-torus conjugacy
\cite[Theorem~17.10]{MilneAlgebraicGroups} therefore puts the diagonal
cocharacter in $T$ without moving $\iota(t)$.
This argument never requires the full semisimple
centralizer, or the centralizer of the triple, to be connected.
After this alignment, let $\mu_\iota\in X_*(T)$ be the diagonal
cocharacter. The compact factor belongs to $T(\mathbb C)$, and the
closure of its cyclic subgroup is compact. Every torus character
therefore has modulus one on it. Equation~\eqref{glob:exponent}
gives
\begin{equation}\label{glob:integral-neutral}
2\lambda_\iota(t)=\mu_\iota.
\end{equation}
We prove that these integral cocharacters are independent of the
embedding. Choose a positive definite Weyl-invariant inner product
on $X_*(T)\otimes\mathbb R$, and extend it to an invariant complex
bilinear form on $\gd_{\mathbb C}$. Such a form is obtained by
choosing a positive multiple of the Killing form on each simple
factor. For two embeddings write $\lambda,\lambda'$ for their
exponents and $\delta=\lambda'-\lambda$. The aligned triple at the
first embedding has neutral element
$H_\mu=d\mu(1)=2\lambda$ under the usual identification of the
real cocharacter space with the real subspace of $\operatorname{Lie}T$.
If a root component $e_\alpha$ of its raising element is nonzero,
then
\[
\iota(\alpha(t))=r,
\qquad \alpha(\lambda)=1.
\]
Since $\alpha(t)$ is algebraic and $r$ is rational, this is the
algebraic identity $\alpha(t)=r$. At the second embedding it implies
$\alpha(\lambda')=1$. The corresponding argument for the lowering
element uses the identity $\alpha(t)=r^{-1}$. It follows that
$\delta$ centralizes both the raising and lowering elements, say
$e$ and $f$. Invariance gives
\[
(\delta,2\lambda)
=(\delta,[e,f])=([\delta,e],f)=0.
\]
Applying the same argument to the aligned triple at the second
embedding gives $(\delta,2\lambda')=0$. Subtracting proves
$(\delta,\delta)=0$, whence $\lambda'=\lambda$.
Equation~\eqref{glob:integral-neutral} proves \textup{(i)}.
The common cocharacter $\mu$ is defined over $\mathbb Q$, since $T$
is split. Put $H_\mu=d\mu(1)$. The equations in $e,f\in\gd$
\begin{equation}\label{glob:algebraic-triple}
\begin{gathered}
[H_\mu,e]=2e,\qquad [H_\mu,f]=-2f,\qquad [e,f]=H_\mu,\\
\operatorname{Ad}(t)e=re,\qquad
\operatorname{Ad}(t)f=r^{-1}f
\end{gathered}
\end{equation}
are polynomial equations over $\overline{\mathbb Q}$ and have a
complex solution by the alignment above. They consequently have a
solution over $\overline{\mathbb Q}$. In characteristic zero, the
resulting $\mathfrak{sl}_2$ homomorphism integrates to an algebraic
homomorphism $\phi:\operatorname{SL}_2\to\Gd$. Its diagonal
cocharacter is $\mu$, because cocharacters in characteristic zero
are determined by their differentials. This includes the zero
triple and the trivial homomorphism.
Choose an algebraic square root of $r$ and set
$c=t\mu(\sqrt r)^{-1}\in T(\overline{\mathbb Q})$.
Equation~\eqref{glob:algebraic-triple} says that $c$ centralizes the
triple, hence its connected algebraic image
$\phi(\operatorname{SL}_2)$. At every embedding and for every
$a\in X^*(T)$,
\[
|\iota(a(c))|
=r^{\langle a,\lambda\rangle}
r^{-\langle a,\mu\rangle/2}=1.
\]
Thus the cyclic subgroup generated by $\iota(c)$ has compact closure
in $T(\mathbb C)$. This closure is contained in the closed subgroup
$Z_{\Gd}(\iota\phi(\operatorname{SL}_2))(\mathbb C)$, and is
conjugate into a maximal compact subgroup of that reductive group
\cite[Theorems~3.6 and~3.7]{AdamsTaibiCohomology}.
This argument also applies to a disconnected triple centralizer and
proves \textup{(ii)}.
For uniqueness, first observe that any compact--Jacobson--Morozov
decomposition of $t$ has diagonal cocharacter $\mu$ after the same
torus alignment. Decompose the Lie algebra by $H_\mu$ as
$\gd=\bigoplus_j\gd_j$. For any triple $(e,H_\mu,f)$,
elementary $\mathfrak{sl}_2$ representation theory gives
\[
[\gd_0,e]=\gd_2:
\]
in every irreducible $\mathfrak{sl}_2$ summand having weight two,
raising maps weight zero onto weight two. The orbit of $e$ under
$Z_{\Gd}(H_\mu)^\circ$ is therefore open in the irreducible vector
space $\gd_2$. There can be only one such open orbit. Hence the
neutral cocharacter determines the nilpotent orbit, without an
exception for very-even orbits in type~$D$.
Finally conjugate $\phi$ over $\overline{\mathbb Q}$ to the selected
representative $\phi_{\mathcal O}$ of its orbit. The conjugated
remainder satisfies precisely the all-embeddings condition defining
$H_{\mathcal O}^c(\overline{\mathbb Q})$
\cite[Sections~3.1.1 and~3.4.1--3.4.2]{GLR}. This proves
\textup{(iii)}.
\end{proof}
\begin{remark}\label{glob:signs}
The sets $R_{r,\mathcal O}$ are pairwise disjoint by
Lemma~\ref{glob:embeddings}. They are Galois-stable with the rational
forms stipulated in their definition
\cite[Section~3.4.2]{GLR}. To see the square-root issue directly,
replacing $\sqrt r$ by $-\sqrt r$ inserts
$\phi_{\mathcal O}(-I)$ into the compact factor. This element is
central in the triple centralizer and has finite order, so preserves
compactness at every embedding. Multiplication of $t$ by any
finite-order element of $Z(\Gd)$ likewise preserves its orbit label
and membership in $R_{r,\mathcal O}$.
For later use, let $2\rho_G$ be the sum of the positive roots of $G$,
viewed as a cocharacter of the dual torus. In the usual square-root
presentation of normalized Satake, if $\tau$ changes the
chosen square root of $q_x$ by a sign $\epsilon_{\tau,x}$, then the
Satake class of the Galois-conjugate Hecke character is
\begin{equation}\label{glob:satake-galois}
[t_{\chi^\tau,x}]
=[(2\rho_G)(\epsilon_{\tau,x})\,\tau(t_{\chi,x})].
\end{equation}
Indeed, on a dominant coweight $a$ the change of the normalization
factor is $\epsilon_{\tau,x}^{\langle2\rho_G,a\rangle}$.
The element $(2\rho_G)(-1)$ is central in $\Gd$, since its pairing
with every dual root is even. It has order at most two. Thus
Equation~\eqref{glob:satake-galois} preserves all the assertions of
Lemma~\ref{glob:embeddings}. If the rational Satake form absorbs
this correction, the comparison is ordinary Galois equivariance.
In either presentation of the identification in
\cite[Equation~(3.2)]{GLR}, the comparison differs at most by the
finite central factor just computed; only this conclusion is used below.
\end{remark}
\subsection{Purity on the good open curve}
We next prove that the orbit found locally is independent of the
good place. Sawin and Templier's Theorem~11.7 gives an earlier
place-independence result for unramified temperedness using
V.~Lafforgue's parameters \cite{SawinTemplier}.
Here the global purity argument compares every nilpotent-orbit label.
We will use the following precise consequence of the
purity theorem, allowing ramification at every point of $X\setminus U$.
\begin{lemma}\label{glob:twist-purity}
Let $\mathcal V$ be a lisse $\overline{\mathbb Q}_\ell$-sheaf on the
smooth geometrically connected curve $U/\mathbb F_q$, defined over a
finite extension of $\mathbb Q_\ell$. Suppose that all its Frobenius
eigenvalues at closed points are algebraic over $\mathbb Q$.
For each complex embedding of these algebraic numbers, the multiset
of numbers
\[
\log_{q_x}|\alpha|,
\qquad \alpha\text{ an eigenvalue of }
\operatorname{Frob}_x\text{ on }\mathcal V,
\]
counted with multiplicity, is independent of $x\in|U|$.
\end{lemma}
\begin{proof}
Take the irreducible constituents $\mathcal W_j$ of a
Jordan--H\"older filtration of $\mathcal V$. After a finite extension
of coefficients they are defined over finite extensions of
$\mathbb Q_\ell$. At each closed point the characteristic polynomial
of $\mathcal V$ is the product of those of its constituents, so their
eigenvalues are still algebraic. It suffices to treat one constituent
$\mathcal W$ of rank $n$.
The rank-one sheaf $\det\mathcal W$ has finite geometric monodromy
\cite[Proposition~1.3.4]{DeligneII}. Fix a closed point $y\in|U|$
of degree $d_y$ and choose an algebraic number $b$ such that
\[
b^{n d_y}
=\det(\operatorname{Frob}_y\mid\mathcal W).
\]
Normalize the degree map on the Weil group by
$\deg(\operatorname{Frob}_x)=d_x$. The number $b$ is an
$\ell$-adic unit: the original representation has compact image,
so the eigenvalues of every group element, and in particular its
determinant, are $\ell$-adic units. The constant-field character
$b^{\deg}$ consequently extends continuously from the Weil group
to the arithmetic fundamental group. Set
$\mathcal W^0=\mathcal W\otimes b^{-\deg}$.
Its determinant still has finite geometric monodromy, and has value
one on $\operatorname{Frob}_y$. A power killing its geometric
monodromy factors through $\operatorname{Gal}(\overline{\mathbb F}_q/
\mathbb F_q)$; its value on an element of degree $d_y$ is one, so this
constant-field character has finite order. Thus
$\det\mathcal W^0$ has finite order. This is also the determinant
normalization described in \cite[Section~1.3.6]{DeligneII}.
Laurent Lafforgue's purity theorem applies to the irreducible lisse
sheaf $\mathcal W^0$ on the smooth curve $U$: its finite-order
determinant implies that all its Frobenius eigenvalues are algebraic
and have modulus one under every complex embedding
\cite[Theorem~VII.6\textup{(i)--(ii)}]{LLafforgue}. Therefore every
Frobenius eigenvalue $\alpha$ of $\mathcal W$ at $x$ satisfies
\[
\log_{q_x}|\alpha|=\log_q|b|.
\]
This value is independent of $x$. Taking the union over the fixed
list of constituents, with their ranks and multiplicities, proves
the assertion. Neither the rank-one theorem nor the purity theorem
requires $U$ to be proper.
\end{proof}
\begin{proposition}\label{glob:place}
Fix a joint character $\chi\in\Sigma_D$ and an embedding
$\iota:\overline{\mathbb Q}\hookrightarrow\mathbb C$.
Let $\lambda_{\chi,x}$ be the dominant exponent of its normalized
Satake class at $x\in|U|$, with base $q_x$ and absolute value induced
by $\iota$. Then $\lambda_{\chi,x}$ is independent of $x$.
\end{proposition}
\begin{proof}
Choose an embedding
$\overline{\mathbb Q}\hookrightarrow\overline{\mathbb Q}_\ell$ and
view $\chi$ as an $\ell$-adic character. V.~Lafforgue's global
parameter theorem gives a continuous semisimple parameter
\[
\sigma:\pi_1(U)\longrightarrow\Gd(\overline{\mathbb Q}_\ell),
\]
defined over a finite extension of $\mathbb Q_\ell$, for which
the semisimple class of $\sigma(\operatorname{Frob}_x)$ is the Satake
class of $\chi$ at every $x\in|U|$
\cite[Theorem~1.1]{VLafforgue}. Here is why one parameter can be
chosen for this joint character. Its nonzero joint eigenspace is
preserved by the commuting excursion operators. Their
finite-dimensional image has a character on that eigenspace.
The parameter attached to such an excursion character has the
required good-place Satake evaluations. We need neither uniqueness
of this lift of $\chi$ nor reducedness of the excursion algebra.
For an algebraic representation $V$ of $\Gd$, the composition
$V\circ\sigma$ is a lisse sheaf on $U$. Its Frobenius eigenvalues
are algebraic, since the corresponding semisimple Satake classes
are algebraic. Lemma~\ref{glob:twist-purity} shows that their
multiset of logarithmic absolute values, with base $q_x$, is
independent of $x$.
For completeness, this determines the exponent in the dual torus.
Let $V_a$ have dominant highest weight $a$. If
$\lambda_{\chi,x}$ is dominant, every weight $a'$ of $V_a$ satisfies
\[
\langle a',\lambda_{\chi,x}\rangle
\le\langle a,\lambda_{\chi,x}\rangle.
\]
The largest Frobenius exponent in $V_a\circ\sigma$ is therefore
$\langle a,\lambda_{\chi,x}\rangle$, and it is independent of $x$.
The dominant weights in the character lattice of the dual maximal
torus span its real weight space: a sufficiently divisible positive
multiple of each fundamental weight belongs to this lattice.
Their pairings determine
$\lambda_{\chi,x}$. This argument applies to every isogeny type
of the semisimple dual group.
\end{proof}
\subsection{Descent of the orbit summands}
\begin{proposition}\label{glob:descent}
For every effective divisor $D$, the rational cuspidal space has the
decomposition
\begin{equation}\label{glob:rational-decomposition}
C_{D,\mathbb Q}
=\bigoplus_{\mathcal O}C_{D,\mathcal O,\mathbb Q}
\end{equation}
with the spectral-support summands specified in the introduction.
\end{proposition}
\begin{proof}
If the cusp space is zero, the assertion is immediate. Otherwise
fix $\chi\in\Sigma_D$ and a good place $x$. At any complex
realization, a nonzero vector of $C_\chi$ occurs in the discrete
cuspidal automorphic spectrum. Choose a nonzero projection of such a vector to an irreducible
cuspidal constituent. This projection retains the joint Hecke
character and is spherical at $x$.
Lemma~\ref{ext:hecke} shows that its Iwahori invariants contain the
dominant ordering of its spherical Bernstein character.
Theorem~\ref{ext:discrete} therefore puts the spherical parameter
in compact--Jacobson--Morozov form.
The same statement applies to every Galois-conjugate character:
$C_{D,\mathbb Q}$ and its Hecke action are defined over $\mathbb Q$,
so every such character occurs again in
Equation~\eqref{glob:joint-spectrum}. By
Equation~\eqref{glob:satake-galois} and Remark~\ref{glob:signs}, every
embedding of an algebraic torus representative $t_{\chi,x}$ itself
has compact--Jacobson--Morozov form. Lemma~\ref{glob:embeddings}
then gives a unique orbit $\mathcal O_{\chi,x}$ with
\[
[t_{\chi,x}]\in R_{q_x,\mathcal O_{\chi,x}}.
\]
Proposition~\ref{glob:place} makes its dominant exponent independent
of $x$. The orbit uniqueness in Lemma~\ref{glob:embeddings} thus
makes $\mathcal O_{\chi,x}$ independent of $x$ as well; denote it
by $\mathcal O_\chi$.
We have partitioned the finite set $\Sigma_D$ into the subsets
$\Sigma_{D,\mathcal O}=\{\chi:\mathcal O_\chi=\mathcal O\}$.
They are Galois-stable by Remark~\ref{glob:signs} and
Equation~\eqref{glob:satake-galois}. Let $e_\chi$ be the primitive
idempotent of
$\overline{\mathbb Q}\otimes\mathbb T_{D,\mathbb Q}$ associated
with $\chi$. The sum
\[
e_{\mathcal O}
=\sum_{\chi\in\Sigma_{D,\mathcal O}}e_\chi
\]
is Galois-fixed, so belongs to $\mathbb T_{D,\mathbb Q}$.
These idempotents are orthogonal and sum to one. They yield a
rational direct sum decomposition with summands
$e_{\mathcal O}C_{D,\mathbb Q}$.
It remains to identify them with the given cyclic-support spaces.
For $f\in C_{D,\mathbb Q}$ write $f=\sum_\chi f_\chi$ after scalar
extension using Equation~\eqref{glob:joint-spectrum}.
At a fixed place $x$, the character support of
$\overline{\mathbb Q}\otimes A_x f$ is exactly
\[
\{\chi|_{A_x}:f_\chi\ne0\}.
\]
Indeed, the local eigenspace for a restricted character is the
direct sum of the joint eigenspaces with that restriction, and the
projection of $f$ to it is nonzero precisely when one of the
corresponding $f_\chi$ is nonzero. It follows that this cyclic
support is contained in $R_{q_x,\mathcal O}$ for every good $x$ if
and only if every $\chi$ with $f_\chi\ne0$ has
$\mathcal O_\chi=\mathcal O$. Here we use the pairwise
disjointness of the orbit sets. The condition is therefore
equivalent to $e_{\mathcal O}f=f$, proving
$e_{\mathcal O}C_{D,\mathbb Q}=C_{D,\mathcal O,\mathbb Q}$.
This is the spectral-support definition of
\cite[Section~3.4.3]{GLR}, with the closed points restricted to $U$.
Equation~\eqref{glob:rational-decomposition} follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{main:decomposition}]
Proposition~\ref{glob:descent} proves the rational assertion.
Tensor Equation~\eqref{glob:rational-decomposition} with
$\overline{\mathbb Q}_\ell$ over $\mathbb Q$. The resulting
summands are exactly $C_{D,\mathcal O,\ell}$ by their definition.
Scalar extension is exact, so their natural addition map to
$C_{D,\ell}$ is an isomorphism.
\end{proof}
\subsection{Generic cuspidal representations}
\begin{proof}[Proof of Corollary~\ref{main:generic-ramanujan}]
Choose the split model of $G$ over the constant field with the same
root datum. Its base change to $F$ is isomorphic to the given group.
Let $S$ be the finite set of places at which $\pi$ is ramified.
For a split adjoint group all hyperspecial maximal compact subgroups
at a place are conjugate, so unramifiedness is equivalent to having
$G(\mathcal O_x)$-fixed vectors for this model. Smoothness of the local
representations and their restricted tensor product allow an effective
divisor $D$ supported on $S$ with $\pi^{K_D}\ne0$: at each point of $S$,
take a principal congruence subgroup fixing a nonzero local vector.
The spherical Hecke algebras outside $S$ act on $\pi^{K_D}$ through the
joint character determined by the local components $\pi_x$.
This character occurs in the complex cusp space
$C_{D,\mathbb C}=\mathbb C\otimes_{\mathbb Q}C_{D,\mathbb Q}$.
By Lemma~\ref{glob:hecke-spectrum}, it and its Satake classes are
algebraic. Choose $\iota:\overline{\mathbb Q}\hookrightarrow\mathbb C$
realizing this character, and write $[t_x]$ for its algebraic normalized
Satake classes. The complex scalar extension of
Theorem~\ref{main:decomposition} and the
pairwise disjointness in Lemma~\ref{glob:embeddings} give a single orbit
$\mathcal O$ such that
\[
[t_x]\in R_{q_x,\mathcal O}
\qquad (x\notin S).
\]
We use positive real square roots in the complex normalized Satake
parameters of $\pi_x$. Their comparison with $\iota(t_x)$ has at most
the finite central correction in Remark~\ref{glob:signs}, which changes
neither the orbit nor the polar exponent.
At the place $v$, where $\pi_v$ is generic, choose an algebraic representative
$t_v\in\widehat T(\overline{\mathbb Q})$ in a maximal torus of $\Gd$.
Let $\nu_v\in X_*(\widehat T)\otimes\mathbb R$
be its polar exponent, so
$|\iota(a(t_v))|=q_v^{\langle a,\nu_v\rangle}$ for every
$a\in X^*(\widehat T)$. The torus alignment in
Lemma~\ref{glob:embeddings} identifies
\[
\nu_v=\tfrac12\mu_{\mathcal O},
\]
where $\mu_{\mathcal O}\in X_*(\widehat T)$ is the diagonal
cocharacter of a Jacobson--Morozov homomorphism for $\mathcal O$.
For every weight $a$ of an algebraic representation of $\Gd$,
\[
\langle a,\nu_v\rangle
=\tfrac12\langle a,\mu_{\mathcal O}\rangle
\in\tfrac12\mathbb Z.
\]
Thus the complex Satake parameter of $\pi_v$ is geometric in the sense of
\cite[Definition~6.3]{CiubotaruHarris}. Since $G$ is adjoint, $\Gd$ is
simply connected; in particular, the adjoint and fundamental
representations required in their Corollary~6.32 are algebraic
representations of $\Gd$ and satisfy the displayed condition.
The cuspidal representation $\pi$ occurs in the unitary cuspidal
$L^2$ spectrum, so each local component is unitarizable. The component
$\pi_v$ is therefore spherical, generic, and unitary. By
\cite[Corollary~6.32\textup{(1)}]{CiubotaruHarris}, its polar exponent
is zero. Hence $\mu_{\mathcal O}=0$. The $\mathfrak{sl}_2$ relations
then make the raising and lowering elements zero, so
$\mathcal O=\{0\}$. For this orbit the Jacobson--Morozov homomorphism is
trivial, so every class in $R_{q_x,\{0\}}$ has compact image under
$\iota$. A spherical representation with compact normalized Satake
parameter is tempered. This proves the assertion at every $x\notin S$.
Finally, a nonzero global Whittaker functional on $\pi$ induces a
nonzero local Whittaker functional at each place. Indeed, choose a pure
tensor on which the global functional is nonzero and fix all its factors
except the chosen local one. The resulting functional on that factor
has the required local equivariance and is nonzero. A globally generic
$\pi$ is therefore generic at every place and unramified outside the
finite set $S$, so it satisfies the hypothesis of the corollary.
\end{proof}
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//sections/06-global.tex ====
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note = {First posted 1995; locators refer to version 4}
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note = {Theorem locators refer to the author's version}
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note = {Locators refer to arXiv version 3; see also the errata in \cite[Section~12]{HainesPaving}}
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eprint = {1805.12231v4},
archivePrefix = {arXiv},
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author = {Ciubotaru, Dan and Harris, Michael},
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eprint = {2311.15300v1},
archivePrefix = {arXiv},
primaryClass = {math.NT},
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note = {Version 1, 26 November 2023; theorem locators refer to this version}
}
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eprint = {math/0309168v3},
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note = {Version 3; locators refer to this version}
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@incollection{FlathTensorProducts,
author = {Flath, Daniel},
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author = {Tits, Jacques},
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booktitle = {Automorphic Forms, Representations and {$L$}-Functions},
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@incollection{LeeAdjointQuotients,
author = {Lee, Ting-Yu},
title = {Adjoint quotients of reductive groups},
booktitle = {Autour des sch{\'e}mas en groupes. Vol. III},
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year = {2015},
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eprint = {1211.3559v1},
archivePrefix = {arXiv},
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@misc{AdamsTaibiCohomology,
author = {Adams, Jeffrey and Taibi, Olivier},
title = {Galois and {Cartan} cohomology of real groups},
year = {2016},
eprint = {1611.05956v2},
archivePrefix = {arXiv},
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year = {2017},
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eprint = {1511.01393v1},
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note = {Theorem and section locators refer to arXiv version 1}
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@article{JKLMBaseChange,
author = {Javanpeykar, Ariyan and Kr{\"a}mer, Thomas and Lehn, Christian and Maculan, Marco},
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}
==== END Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level//references.bib ====