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\title{Frobenius Structures on Tame Hecke Eigensheaves}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $n\ge 2$ and let $\F_q$ have characteristic $p>n$. Let $X_0$ be a smooth
projective geometrically connected curve of genus at least two over $\F_q$,
with a rational point $x_0$, and let $\rho\colon\pi_1(U_0)\to\PGL_n(L)$ be a
continuous arithmetic representation with dense geometric image and tame
regular-unipotent inertia at $x_0$, where $U_0=X_0\setminus\{x_0\}$ and
$L/\Q_\ell$ is finite. We construct, after extending scalars to a finite field
$\F_{q^m}$, a nonzero locally constructible perverse Weil sheaf $M_0$ on the
stack of $G$-bundles on $X_0$ with $B$-reduction at $x_0$, $G=\SL_n$. Its
entire multi-leg Hecke eigenstructure is defined over $\F_{q^m}$ and has
eigenvalue the restriction of $\rho$; its geometric singular support lies in
the parabolic nilpotent cone. The proof first realizes the geometric
eigenobject as the image of a spectral skyscraper, then makes it invariant
under Frobenius, and finally specializes back to the finite field.
\end{abstract}
\tableofcontents
\clearpage
\section{Introduction}\label{sec:intro}
A \emph{Hecke eigensheaf} turns a local system on a curve into a sheaf on the
moduli stack of bundles, compatible with all Hecke correspondences. Over a
finite field, a local system $\rho$ carries information its geometric
restriction forgets: the action of Frobenius. An automorphic realization of
the whole arithmetic datum must respect this extra structure.
\subsection{The result}
Fix $G=\SL_n$ with upper-triangular Borel $B\subset G$ and dual group
$\Gd=\PGL_n$. For a pointed smooth projective curve $(X,x)$, let
$\Bun_{G,B,x}(X)$ parametrize $G$-bundles on $X$ together with a $B$-reduction
at $x$. A complex on this stack is \emph{locally constructible perverse} when
its pullback to each finite-type smooth chart, shifted by the relative
dimension of that chart, is bounded constructible and perverse; bounds may
depend on the chart. The \emph{parabolic global nilpotent cone}
$\Lambda_{\mathrm{par}}$ consists of generically nilpotent Higgs fields with
at most a simple pole at $x$ and residue in the nilradical of the chosen
Borel. Singular support is defined chartwise on those smooth charts.
For a finite set $I$ and representations $V_i$ of $\Gd$, $i\in I$, write
$\Hecke_{I,\boldsymbol V}$ for the multi-leg Hecke functor with legs in
$(X\setminus\{x\})^I$. We use the \emph{relative Satake normalization} of
Section~\ref{sec:conventions}: its tensor unit is exterior product with the
constant $\E$-sheaf, and it carries no moving-leg shift.
A \emph{Weil structure} over $\F_{q^m}$ on a geometric complex $M$ is an
isomorphism between the $q^m$-Frobenius pullback of $M$ and $M$; the
eigenisomorphisms of $M$ must respect this structure and the Frobenius
structure on the parameter.
\begin{theorem}\label{thm:main}
Let $n\ge 2$, let $\F_q$ have characteristic $p>n$, let $\ell\ne p$, and put
$E=\E$. Let $X_0/\F_q$ be a smooth projective geometrically connected curve of
genus at least two with $x_0\in X_0(\F_q)$, and put $U_0=X_0\setminus\{x_0\}$.
Suppose
\[
\rho\colon\pi_1^{\mathrm{et}}(U_0)\longrightarrow\PGL_n(L)
\]
is continuous for a finite extension $L/\Q_\ell$, its geometric restriction is
Zariski dense, and its inertia at $x_0$ kills wild inertia and has the form
\begin{equation}\label{eq:regular-monodromy}
\rho(\gamma)=\exp\bigl(t_\ell(\gamma)N\bigr)
\end{equation}
for a regular nilpotent $N$, where compatible $\ell$-power roots of unity
identify the tame quotient $\Z_\ell(1)$ with $\Z_\ell$ and define $t_\ell$.
Then there are an integer $m\ge 1$ and a nonzero $E$-adic Weil sheaf $M_0$ on
$\Bun_{\SL_n,B,x_0}(X_0)_{\F_{q^m}}$ whose geometric pullback $M$ is locally
constructible perverse and has $\SS(M)\subset\Lambda_{\mathrm{par}}$. If
$\rho_m$ denotes $\rho$ restricted to $\pi_1^{\mathrm{et}}(U_{0,\F_{q^m}})$,
then for every finite $I$ and every $V_i\in\Rep_E(\PGL_n)$ there are
isomorphisms of Weil sheaves
\begin{equation}\label{eq:main-eigenmaps}
\Hecke_{I,\boldsymbol V}(M_0)\simeq M_0\boxtimes
\bigboxtimes_{i\in I}(V_i)_{\rho_m},
\end{equation}
natural in the representations and coherent for the unit, convolution,
permutations, and all fusion diagonals.
\end{theorem}
Write $k=\overline{\F}_q$, $X=X_0\times_{\F_q}k$, $x=(x_0)_k$, and
$U=U_0\times_{\F_q}k$ for the geometric curve and its punctured complement.
The finite extension in the theorem arises when a point on an auxiliary bundle
stack is made invariant under a power of Frobenius. Since we only use Weil
structures, a compatible action of the infinite cyclic group suffices; no
profinite descent is required.
\subsection{The two obstructions}
The geometric construction of \cite{CT} already yields a locally constructible
perverse eigenobject with fusion-compatible eigenstructure and parabolic
nilpotent singular support. Two new problems must be solved to make it
\emph{arithmetic}:
\begin{enumerate}[(O1)]
\item An invariant parameter does not by itself make a geometrically chosen
eigenobject visibly invariant. Some choice of eigenobject must be recognized
from internal structure rather than from an uncontrolled equivalence.
\item At a separating node used for degeneration, the boundary parameters of
the two components face opposite orientations. The identifications that glue
them must commute with Frobenius and must be realized by a compact
conjugator, so that the resulting parameter is continuous with compact
image.
\end{enumerate}
Sections \ref{sec:conventions}--\ref{sec:final} address these in order.
\subsection{Strategy of the proof}
For a dense unramified parameter in characteristic zero, the relevant
component of the restricted spectral stack is a smooth formal space supported
at one geometric point. Semilinear transport preserves it. The inverse image
of the skyscraper on that component has scalar endomorphisms and no negative
self-extensions; these two properties identify its transport up to shift
(Lemma~\ref{lem:point-object}). An invariant polarization removes the shift on
a bounded open, giving an isomorphism with the transport. Density of the
parameter and triviality of $Z(\PGL_n)$ then force compatibility with the full
eigenstructure. This is Section~\ref{sec:unramified}.
The arithmetic lift is built in Section~\ref{sec:parameters}. A smooth
projective pointed lift of $(X_0,x_0)$ over $W(\F_q)$ carries a tower of finite
torsors obtained by extending $\rho$'s finite quotients across root stacks; the
tower recovers $\rho$ together with all of its arithmetic structure. At the
puncture the Frobenius matrix $A$ and the regular nilpotent logarithm $N_1$
satisfy $\Ad(A)N_1=qN_1$. A cocharacter commuting with $A$ scales $N_1$; a
suitable power makes the conjugator lie in a fixed compact open, though it
commutes with $A$. A ramified cover of controlled degree multiplies boundary
monodromy by an integer and thereby produces an auxiliary component whose
boundary acts by the inverse of the first. Gluing on a balanced twisted nodal
model then lifts to a dense unramified parameter on the smooth generic fiber.
Section~\ref{sec:nodal} applies the unramified result to this glued parameter,
specializes on the prescribed node type, restricts to the first component, and
removes the enhancement of the flag. Nonvanishing and equivariance are checked
separately; perverse cohomology supplies nilpotent singular support, and the
unipotence of the boundary monodromy controls the torus direct image.
Section~\ref{sec:final} specializes back to the finite field. A second
relative family over $R=W(k)$ lands on the original pointed curve, and the
field-level nilpotent-detection theorem of \cite[Theorem~4.1]{CT} produces the
parabolic cone from the eigenstructure.
\subsection{Relation to earlier work}
Drinfeld's rank-two work and Laumon's geometric formulation established central
parts of geometric Langlands \cite{Drinfeld,Laumon}. Frenkel--Gaitsgory--Vilonen
constructed unramified $\GL_n$ eigensheaves, including the finite-field Weil
setting \cite{FGV,GaitsgoryVanishing}. For general groups, the restricted
spectral stack and its action on automorphic sheaves with nilpotent singular
support were developed in \cite{AGKRRV}; the positive-characteristic
specialization is due to Gaitsgory--Raskin \cite{GR}. The geometric input of the
present paper is the companion note \cite{CT}, which constructs locally
constructible perverse eigensheaves with Borel level, parabolic nilpotent
singular support, and fusion-compatible eigenstructure. That note does not
supply Frobenius descent; the present paper provides it, for $\SL_n$, under
the hypotheses of Theorem~\ref{thm:main}.
In characteristic-zero de Rham sheaf theory, F\ae rgeman constructs coherent
tame eigenobjects for irreducible regular-singular parameters \cite{Faergeman},
conditional on stated inputs. We work in the geometric $\ell$-adic setting and
establish perversity and the finite-field Weil action outright, for the dense
parameters with regular-unipotent boundary monodromy above.
The nodal framework is part of the automorphic gluing of Nadler--Yun
\cite{NYAutomorphicGluing}; twisted curves and root stacks are used as in
\cite{AbramovichVistoli,OlssonTwisted,LieblichOlsson,Rydh}. Sheaf-theoretic
specialization is geometric nearby cycles
\cite{HansenScholze,GR,CT}.
\section{Equivariant Hecke systems and specialization}\label{sec:conventions}
We fix sheaf conventions and identify which parts of the geometric
specialization formalism carry a semilinear action. The compatibility of the
whole Hecke system, including collision diagonals of moving legs, is the
substance.
\subsection{Sheaves, level structures, and moving legs}
Write $E=\E$, $G=\SL_n$, and $\Gd=\PGL_n$, with split forms understood over
every base field. On a smooth stack $\mathcal B$ locally of finite type,
$D_{\lcc}(\mathcal B,E)$ denotes the category of geometric adic complexes
whose pullback to each smooth finite-type scheme chart is bounded and
constructible; the bounds may depend on the chart. A complex $F$ is perverse
if $b^*F[d]$ is perverse for every smooth chart $b:Y\to\mathcal B$ of constant
relative dimension $d$. These conventions agree with \cite[Section~2.1]{CT}.
For a pointed smooth projective curve $(X,x)$ put
\[
\mathcal A=\Bun_{G,B,x}(X),\qquad
\mathcal A^+=\Bun_{G,R_u(B),x}(X),\qquad T=B/R_u(B).
\]
An enhanced flag is a reduction to $R_u(B)$; forgetting it is a representable
$T$-torsor $\mathcal A^+\to\mathcal A$. The trace form identifies a cotangent
vector at ordinary level with a Higgs field
\[
\theta\in H^0\bigl(X,\ad(P)\otimes\omega_X(x)\bigr),
\qquad \operatorname{Res}_x\theta\in\operatorname{Lie}R_u(B_P),
\]
where $B_P$ is the chosen Borel in the fiber at $x$. The cone
$\Lambda_{\mathrm{par}}$ consists of the fields nilpotent over the function
field of $X$. On a smooth chart, singular support means the geometric adic
singular support of the pulled-back complex, contained in the smooth cotangent
pullback of this cone; see \cite[Section~3.1]{CT} and
\cite{Beilinson,Barrett}.
Let $U$ be the locus of allowed modifications, disjoint from the level point.
For a finite set $I$ the Hecke stack $\mathcal H_I$ has an input bundle map
$p_I$ and an output-and-leg map $o_I\colon\mathcal H_I\to\mathcal B\times U^I$.
If $\boldsymbol V=(V_i)_{i\in I}$ are representations of $\Gd$, define
\begin{equation}\label{eq:hecke-normalization}
\Hecke_{I,\boldsymbol V}(F)
=o_{I,*}\bigl(F\,\widetilde\boxtimes\,
\operatorname{Sat}_I(\boldsymbol V)\bigr).
\end{equation}
The twisted product is ordinary exterior product in input frames, descended
using equivariance of the Satake kernel. All pushforwards are taken on finite
Schubert bounds, where the output maps are proper. On a fixed-point Schubert
orbit of dimension $d_\lambda=\langle2\rho_G,\lambda\rangle$, the simple
kernel is normalized by
\begin{equation}\label{eq:satake-normalization}
\IC_\lambda|_{\Gr^\lambda}=E[d_\lambda](d_\lambda/2).
\end{equation}
Here $\rho_G$ is the half-sum of positive roots. The cocharacter lattice of
$G$ is its coroot lattice, so $d_\lambda$ is even and all twists in
\eqref{eq:satake-normalization} are integral. There is no moving-leg shift in
\eqref{eq:hecke-normalization}. We use geometric Satake with its fusion
tensor structure \cite{MV}.
For a surjection $a\colon I\twoheadrightarrow J$, let
$\delta_a\colon U^J\to U^I$ be the collision diagonal and set
$W_j=\bigotimes_{i\in a^{-1}(j)}V_i$. The relative fusion convention is
\begin{equation}\label{eq:fusion-convention}
(1\times\delta_a)^*\Hecke_{I,\boldsymbol V}(F)
\simeq\Hecke_{J,\boldsymbol W}(F),
\end{equation}
ordinary pullback, without a codimension shift.
\begin{definition}\label{def:eigenstructure}
For a $\Gd$-local system $\vartheta$ on $U$, a coherent Hecke eigenstructure
on $F$ consists of isomorphisms
\begin{equation}\label{eq:eigenstructure}
\Hecke_{I,\boldsymbol V}(F)
\simeq F\boxtimes\bigboxtimes_{i\in I}(V_i)_\vartheta
\end{equation}
for all finite $I$, natural in the representations, and compatible with the
unit, successive modifications and convolution, permutations, and all
collision maps \eqref{eq:fusion-convention}, including their iterated
compatibilities. Here $(V)_\vartheta$ means the lisse sheaf obtained by
evaluating $V$ on $\vartheta$.
\end{definition}
Let $\sigma$ be a semilinear automorphism of the ground field and the curve,
preserving the split group and the level data. An equivariant sheaf has an
isomorphism $\sigma^*F\simeq F$; an equivariant eigenstructure requires
\eqref{eq:eigenstructure} to commute with this isomorphism and with the
isomorphism $\sigma^*\vartheta\simeq\vartheta$. We use the group generated by
$\sigma$, so the one isomorphism determines all its iterates and inverses.
Transport acts on the base, not on the coefficient field $E$. At a finite-field
fiber the ground-field action $a\mapsto a^q$ gives the arithmetic convention;
inverting the generator everywhere gives the usual geometric Frobenius
convention for Weil sheaves.
\subsection{Satake transport and its Weil normalization}
\begin{lemma}\label{lem:satake-transport}
Semilinear isomorphisms of pointed curves identify the relative Satake and
Hecke systems, compatibly with all the operations in
Definition~\ref{def:eigenstructure}. In representation labels the required
symmetric tensor comparison is unique. The same assertion holds for extension
between algebraically closed characteristic-zero fields.
\end{lemma}
\begin{proof}
Transport preserves the split Schubert orbits, their intersection complexes,
convolution diagrams, and fusion diagrams. It thus gives a symmetric tensor
equivalence of the spherical Satake categories preserving all highest-weight
labels. Under Satake this is a symmetric tensor autoequivalence of
$\Rep_E(\PGL_n)$ with trivial outer automorphism. Over the algebraically closed
coefficient field it is tensor-isomorphic to the identity: the fiber-functor
torsor is trivial and the remaining group automorphism is inner. Two such
isomorphisms differ by a tensor automorphism of the identity functor, i.e.\ an
element of $Z(\PGL_n)$, which is trivial.
On separated legs this is the exterior-product comparison. Fusion
extensions, proper convolution maps, and their exchange transformations are
transported together, so their compatibility maps agree. In the kernel
categories the fusion extensions are middle extensions with the usual perverse
leg shifts; a comparison on the separated locus determines them. Applying the
corresponding natural sheaf operations gives the Hecke comparison. Algebraically
closed base extension has the same properties: it preserves the simple kernels
and convolution and is an equivalence on spherical Satake. Uniqueness again
identifies its representation labels.
\end{proof}
\begin{lemma}\label{lem:normalized-weil-satake}
Over a finite field, the split Weil structures on
\eqref{eq:satake-normalization} give a symmetric tensor Satake system, with
ordinary coefficient multiplicity spaces carrying trivial action. Its
Frobenius comparison is the one of Lemma~\ref{lem:satake-transport}.
\end{lemma}
\begin{proof}
Purity alone does not specify the Frobenius scalars, so we verify the
normalization. Resolve a split affine Schubert variety by a Bott--Samelson
resolution using the Iwahori stratification; its iterated projective-bundle
structure is defined over the finite field. The projective-bundle formula gives
scalar geometric Frobenius $q^j$ on $H^{2j}$ and zero odd cohomology. After the
shift and twist $[d_\lambda](d_\lambda/2)$, the scalar on degree $i$ is
$q^{i/2}$.
In perverse degree zero of the proper direct image, the full-support
intersection complex occurs with multiplicity one. The decomposition theorem
\cite[Theorems~5.3.8 and~5.4.5]{BBD} makes its isotypic summand geometrically
well defined and hence Frobenius stable, with the normalization fixed on the
open orbit. The Frobenius-stable perverse Leray filtration, whose spectral
sequence degenerates by the same theorem, exhibits each cohomology group of
this summand as a subquotient of the corresponding scalar cohomology group.
Thus
\[
H^i(\Gr,\IC_\lambda)\text{ has Frobenius as on }E(-i/2)
\quad(i\text{ even}),\qquad H^i=0\quad(i\text{ odd}).
\]
The cohomological convolution comparison is graded and Frobenius compatible:
it is constructed by K\"unneth on separated legs and fusion over the split
affine line \cite[Lemma~6.1, Equations~(6.2a)--(6.2c), and Section~14]{MV}.
The cohomology system is lisse across collisions and geometrically constant; it
is constant on the separated locus in coordinate trivializations, and
lisseness extends this constancy. Comparison between rational configurations
therefore commutes with Frobenius; binary tensor comparison uses $(0,1)$ and
$(0,0)$, available over every finite field. The scalar rule is consequently
the same for convolution and for the corresponding tensor product. Every
graded tensor comparison on total cohomology commutes with these scalars.
Faithfulness of the cohomological Satake fiber functor shows that the kernel
tensor maps do as well. In particular, the induced Frobenius action on the
geometric multiplicity space $\Hom(\IC_\nu,\IC_\lambda*\IC_\mu)$ is trivial:
source and target cohomology have the same scalar in each degree, and the
cohomological fiber functor is faithful. Fusion extends the compatibility to
the relative system, and uniqueness in Lemma~\ref{lem:satake-transport}
identifies this Weil comparison with the transport comparison.
\end{proof}
\subsection{The specialization input with an action}
The geometric result we need is stronger than one-point Hecke commutation. It
is the combination of Propositions~2.1 and~5.1 of \cite{CT} with the following
relative setup. Let $S=\Spec R$ be an excellent complete strictly henselian
discrete valuation trait with algebraically closed residue field and with
$\ell$ invertible. Fix a geometric generic point $\bar\eta$ and write $s$ for
the closed point. Let $\mathcal B/S$ be a smooth locally finite-type bundle
stack and $U/S$ a smooth scheme of relative dimension one of allowed legs.
Bounded Hecke correspondences have proper representable output maps, and in
coordinates and input frames they have the split Schubert local models with
their spherical kernels. The setup includes ordinary and enhanced level
disjoint from the legs, and the twisted nodal families used below.
\begin{proposition}[Geometric specialization and transport]
\label{prop:equivariant-specialization}
In this setup, geometric nearby cycles preserve local bounded constructibility
and are perverse exact. They admit natural isomorphisms
\begin{equation}\label{eq:hecke-specialization}
\Hecke^s_{I,\boldsymbol V}(\Psi F)
\xrightarrow{\ \sim\ }
\Psi\bigl(\Hecke^{\bar\eta}_{I,\boldsymbol V}(F)\bigr)
\end{equation}
compatible with the unit, convolution, permutations, and every fusion
diagonal. Suppose $F$ has a coherent eigenstructure with parameter
$\vartheta_{\bar\eta}$. For each representation, suppose its evaluation sheaf
has a lisse lattice whose finite reductions extend lisse after finite trait
extensions, compatibly on further extensions, and with the reductions of a
special-fiber tensor system $\vartheta_s$. Then $\Psi F$ has a coherent
eigenstructure with eigenvalue $\vartheta_s$.
If the models, $F$ with its given coherent eigenstructure, and these
tensor-specialization data carry a semilinear generator with a compatible lift
to $\bar\eta$, all the comparisons and the resulting eigenstructure are
equivariant. For the finite-stage lattice models one may use compatible
invariant trait extensions. No single finite extension is required for all
reductions.
\end{proposition}
\begin{proof}
The geometric assertions are exactly \cite[Propositions~2.1 and~5.1]{CT}; the
nearby-cycle formalism is described in \cite[Section~4.1]{GR} and
\cite[Theorems~4.1 and~6.7]{HansenScholze}. These are geometric nearby cycles
without inertia invariants, and the input need not descend over one finite
extension of the fraction field.
Equivariance is additional. An isomorphism of traits and models, together with
its lift to the geometric generic point, identifies the defining nearby-cycle
diagrams. The pullback, proper-pushforward, and tensor exchange maps therefore
commute with its transport. In input frames the Hecke comparison is the natural
exterior-product exchange with a Satake kernel, followed by proper-pushforward
exchange; this is the construction of \cite[Section~5]{CT}, including its
comparison on collision diagonals. The Satake-label identification of
\cite[Lemma~5.3]{CT} also commutes with transport: the two composites are
tensor comparisons and hence agree by Lemma~\ref{lem:satake-transport}.
On the eigenvalue side, the lisse tensor comparison is natural at every finite
lattice reduction and every further trait extension; passing to the adic
system preserves this naturality. Thus \eqref{eq:hecke-specialization}
transports the equivariant eigenmaps and all their diagrams. The convolution
and nested diagonal identities are identities of the same exchange
transformations and require no separate choices. At a finite-field fiber their
Satake structures are the normalized Weil structures of
Lemma~\ref{lem:normalized-weil-satake}.
\end{proof}
\subsection{The characteristic restriction for \texorpdfstring{$\SL_n$}{SLn}}
The remaining geometric inputs of \cite{CT} have four Lie-theoretic
hypotheses: (H1) a nondegenerate invariant symmetric form, also nondegenerate
on every Levi's Lie-algebra center; (H2) Chevalley restriction for every Levi;
(H3) scheme-theoretic semisimple centralizers that are Levis; and (H4)
containment of every nilpotent element, over every field extension, in the
nilradical of a parabolic defined over that field. We verify them to make the
characteristic restriction explicit.
\begin{lemma}\label{lem:lie-hypotheses}
For $G=\SL_n$ over an algebraically closed field of characteristic zero or of
characteristic $p>n$, hypotheses (H1)--(H4) of \cite[Theorem~1.1]{CT} hold.
\end{lemma}
\begin{proof}
The form $\kappa(X,Y)=\operatorname{tr}(XY)$ is nondegenerate on
$\mathfrak{sl}_n$, since $n$ is invertible. For a block Levi with block sizes
$n_1,\ldots,n_r$, its restriction to the Lie-algebra center is the diagonal
form with entries $n_i$ on
\[
\{(z_i):\textstyle\sum_i n_i z_i=0\}.
\]
This hyperplane is the orthogonal complement of $(1,\ldots,1)$, whose squared
length is $n$. All $n_i$ and $n$ are invertible, so the restricted form is
nondegenerate. This is (H1).
For (H2), apply block diagonalization on the dense regular-semisimple locus.
Block-symmetric polynomials in the eigenvalues lift by the block
characteristic polynomials. The Weyl-group order divides $n!$, which is
invertible; averaging shows that invariants on the total-trace-zero hyperplane
are restrictions of invariants on the full diagonal space. Block
diagonalization may be performed in the determinant-one Levi, by adjusting the
determinant with an element of the diagonal centralizer.
For (H3), the scheme-theoretic centralizer of a semisimple Lie-algebra element
is its block Levi, as follows directly from the matrix commutation equations.
For (H4), over every field extension, vanishing of the positive-degree
invariant polynomials says that the matrix is nilpotent. A basis adapted to its
kernel filtration puts it in a strictly triangular Lie algebra defined over
that field. It therefore lies in the nilradical of a rational Borel, which is
the required rational parabolic.
\end{proof}
\section{An equivariant unramified eigencomplex}\label{sec:unramified}
The first specialization begins in characteristic zero with an unramified
eigencomplex on a smooth projective curve. The geometric existence theorem
supplies the complex, but not its semilinear action. We obtain that action by
choosing the particular eigencomplex attached to a spectral skyscraper: its
spectral component is transport-stable and the skyscraper is recognized
intrinsically. A second rigidity argument then shows that any transport
isomorphism must respect the full eigenstructure.
\begin{proposition}\label{prop:equivariant-unramified}
Let $C$ be a smooth projective connected curve of genus at least two over an
algebraically closed field $K$ of characteristic zero. Let $f$ be a semilinear
automorphism of $C$ with an ample line bundle $\mathcal L$ and an isomorphism
$f^*\mathcal L\simeq\mathcal L$. Let $G=\SL_n$ and $\Gd=\PGL_n$. Suppose
$\tau$ is a continuous $\Gd$-local system on $C$, defined over a finite
extension of $\Q_\ell$, with Zariski-dense image, and equipped with an
isomorphism $u\colon f^*\tau\xrightarrow{\sim}\tau$.
Then there is a nonzero locally bounded constructible complex $D_\tau$ on
$\Bun_G(C)$, with nilpotent singular support, a full coherent Hecke
eigenstructure of eigenvalue $\tau$, and an isomorphism
\[
b\colon f^*D_\tau\xrightarrow{\sim}D_\tau
\]
compatible with that eigenstructure and $u$. The object $D_\tau$ is compact in
the automorphic nilpotent category. The eigenstructure uses the relative
Satake normalization of Section~\ref{sec:conventions} and is compatible with
the unit, convolution, permutations, and every fusion diagonal. The
isomorphisms $b$ and $u$ define compatible actions of $\Z$ on the eigencomplex
and on its eigenvalue.
\end{proposition}
Here $f^*$ denotes transport on the curve and on its bundle stack, with the
coefficient field $\E$ fixed. We prove the proposition in three steps: identify
the spectral summand and its behavior under transport, recognize the
transported object without a shift, and prove uniqueness of the coherent
eigenstructure on that object.
\subsection{The spectral skyscraper and transport of the action}
Write
\[
\mathcal S=\Loc_{\Gd}^{\mathrm{restr}}(C),\qquad
\mathcal D_C=\Shv_{\Nilp}(\Bun_G(C)).
\]
The first stack parametrizes local systems with restricted variation; the
second is the automorphic category with global nilpotent singular support. We
use the following characteristic-zero input from the proof of
\cite[Lemma~8.1]{CT}. The Gaitsgory--Raskin equivalence
\begin{equation}\label{eq:unramified-equivalence}
\mathcal D_C\simeq\IndCoh_{\Nilp}(\mathcal S)
\end{equation}
is linear for the $\QCoh(\mathcal S)$-action, and the inclusion of
$\mathcal D_C$ in the ambient automorphic category preserves compact objects
\cite[Main Theorem~1.3.9(ii), Lemma~1.3.7, and Theorem~1.1.7]{GR}. These
statements apply to geometric $\E$-adic sheaves over an arbitrary algebraically
closed characteristic-zero field \cite[Section~2.3]{GR}. The spectral action
agrees, through universal evaluation, with the coherent Hecke action
\cite[Main Theorem~14.3.2 and Section~14.3.3]{AGKRRV}. As in
\cite[Lemma~8.1]{CT}, the external Satake labels coincide with our
input--output convention once a tensor relabeling $\tau\circ\alpha^{-1}$ has
been absorbed; evaluation at the point below denoted $\tau$ is therefore
exactly $V\mapsto V_\tau$.
Let $i_\tau\colon\Spec\E\to\mathcal S$ denote this point, and put
\[
\delta_\tau=i_{\tau,*}^{\IndCoh}\E.
\]
The components of $\mathcal S$ are indexed by semisimple parameters; the
component of an irreducible parameter has only one isomorphism class of
geometric points \cite[Proposition~3.7.2 and Corollary~3.7.5]{AGKRRV}. Density
makes $\tau$ irreducible and gives
\begin{equation}\label{eq:unramified-centralizer}
\Aut(\tau)=Z_{\Gd}(\operatorname{im}\tau)=Z(\Gd)=1.
\end{equation}
The tangent complex of $\mathcal S$ at $\tau$ is
$\RGamma(C,\ad(\tau))[1]$ \cite[Section~21.2.1]{AGKRRV}. Its degree $-1$
cohomology is zero by \eqref{eq:unramified-centralizer}. The invariant
nondegenerate form on $\operatorname{Lie}(\Gd)$ and Poincar\'e duality give
\[
H^2(C,\ad(\tau))\simeq H^0(C,\ad(\tau))^\vee(-1)=0.
\]
Hence the open-and-closed component $\mathcal S_\tau$ containing $\tau$ has no
infinitesimal automorphisms or obstructions and is a smooth one-point formal
polydisc, with tangent space $H^1(C,\ad(\tau))$. For this formal component,
\[
\IndCoh(\mathcal S_\tau)\simeq\IndCoh(\mathbb A^d_{\E})_{\{0\}},
\qquad d=\dim_{\E} H^1(C,\ad(\tau)),
\]
where the right-hand category consists of objects set-theoretically supported
at the origin \cite[Sections~21.1.7--21.1.10]{AGKRRV}. Its compact objects are
bounded coherent complexes with this support: they are compact and generate the
supported category, as on the smooth affine space via the Koszul complex at the
origin. In particular their cohomology modules have finite length. The
skyscraper $\delta_\tau$ is nonzero and compact. Since the formal component is
smooth, it has no spectral obstruction directions, so $\delta_\tau$ has
nilpotent spectral singular support; this is the ind-coherent singular-support
condition, not the microlocal support of a constructible skyscraper.
Choose an equivalence \eqref{eq:unramified-equivalence}, and let $D_\tau$ be
the inverse image of $\delta_\tau$. It is compact in $\mathcal D_C$ and in the
ambient automorphic category. Compact automorphic objects are bounded
constructible on every finite-type smooth chart
\cite[Appendix~F, Section~F.2.2]{AGKRRV}, so $D_\tau$ is locally bounded
constructible. Its full eigenstructure comes from \cite[Lemma~8.1]{CT}: the
module identity
\begin{equation}\label{eq:skyscraper-evaluation}
A\otimes\delta_\tau\simeq i_{\tau,*}^{\IndCoh}(i_\tau^*A),
\qquad A\in\QCoh(\mathcal S),
\end{equation}
applied to universal evaluation over $C^I$, gives
\begin{equation}\label{eq:unramified-eigenmaps}
\Hecke_{I,(V_i)}(D_\tau)
\simeq D_\tau\boxtimes
\Bigl(\boxtimes_{i\in I}(V_i)_\tau\Bigr).
\end{equation}
The unit, tensor, permutation, and fusion maps all come from the same
universal evaluation system; spectral linearity transports them together.
\begin{lemma}\label{lem:spectral-action-transport}
Semilinear transport by $f$ identifies the automorphic spectral action with
the action for the transported curve and its restricted stack of local systems.
Consequently, if $f^*\tau\simeq\tau$, transport preserves the summand of
$\mathcal D_C$ corresponding to $\mathcal S_\tau$.
\end{lemma}
\begin{proof}
We compare the finite-set Hecke action data that characterize the spectral
action: derived representation labels, leg factors
$\operatorname{QLisse}(C)^{\otimes I}$, and their higher compatibilities
\cite[Section~8.1.1]{AGKRRV}. Exterior product realizes the leg factors on
$C^I$. Both the Hecke diagrams and this quasi-lisse factorization must be
transported.
For the Hecke diagrams: transport by $f$ is an exact $\E$-linear equivalence
preserving nilpotent singular support. It takes every Hecke correspondence over
$C^I$ to its transported correspondence and commutes with pullback, tensor
product, convolution pushforward, and their exchange maps.
Lemma~\ref{lem:satake-transport} compares the Satake kernels and their tensor
and fusion maps. Dualizing complexes are transported as well, so the
comparison also applies with dualizing pullback conventions. The result is a
comparison of actual sheaf operations and their structural maps, including the
geometric maps in the leg categories.
To pass to derived labels, use \cite[Sections~B.3.6--B.3.8]{GKRV}, recalled in
\cite[Section~14.2.2]{AGKRRV}: bounded-derived extension, restriction to
compact objects, then ind-extension. On perverse representation objects the
preceding comparison identifies the Satake diagrams with all their structural
maps; the universal property of the bounded derived category extends it
exactly and preserves the coherent finite-set diagrams. The resulting right-lax
monoidal structure maps are isomorphisms, so restriction to compact objects and
ind-extension give the continuous derived action comparison. This uses the
specified extension, without identifying the whole derived spherical sheaf
category with the derived representation category.
It remains to compare the quasi-lisse factorization: transport identifies
$\operatorname{QLisse}(C)^{\otimes I}$ with the corresponding category for the
transported curve. The exterior-product embedding used in
\cite[Section~14.2.5]{AGKRRV} is fully faithful, so the constructed comparison
also identifies the Hecke action with values in these leg categories, which is
the action of \cite[Sections~14.2.5--14.2.8 and Main Theorem~14.3.2]{AGKRRV}.
We have now compared the full finite-set action data. Transport also identifies
the universal evaluation objects on the restricted stacks. By
\cite[Theorem~8.1.4]{AGKRRV}, universal evaluation gives an equivalence between
the space of $\QCoh(\mathcal S)$-actions and these data with their higher
compatibilities. Consequently the conjugate of the spectral action under
transport is the spectral action of the transported curve.
The component $\mathcal S_\tau$ is carried to the component of $f^*\tau$, and
when these parameters are isomorphic its component idempotent is preserved.
This idempotent acts as the identity on $D_\tau$ and hence on $f^*D_\tau$,
proving the final assertion.
\end{proof}
\subsection{Recognizing the transported object}
Lemma~\ref{lem:spectral-action-transport} places $f^*D_\tau$ in the same
spectral component as $D_\tau$. We now recognize its image there. The
elementary criterion below is useful independently of the spectral setting.
\begin{lemma}\label{lem:point-object}
Let $(R,\mathfrak m)$ be a commutative local $\E$-algebra with residue field
$\E$, and let $P$ be a nonzero bounded complex of $R$-modules whose cohomology
modules have finite length. Suppose
\[
\Hom_{D(R)}(P,P)=\E,\qquad
\Hom_{D(R)}(P,P[j])=0\quad(j<0).
\]
Then $P\simeq\E[r]$ for an integer $r$.
\end{lemma}
\begin{proof}
Let $a$ and $b$ be the least and greatest degrees with nonzero cohomology. Any
two nonzero finite-length modules over $R$ admit a nonzero map from the first
to the second: take a quotient of the first onto $\E$ and an inclusion of $\E$
into the socle of the second. Thus, if $a<b$, there is a nonzero map
$h\colon H^b(P)\to H^a(P)$. The truncation maps give a composite
\[
P\longrightarrow H^b(P)[-b]
\xrightarrow{h[-b]}H^a(P)[-b]
\longrightarrow P[a-b],
\]
whose map on degree-$b$ cohomology is $h$, so it is a nonzero negative-degree
endomorphism. This contradicts the hypothesis. Hence $a=b$, and $P$ is a shift
of a single finite-length module $M$.
Every element of $\mathfrak m$ acts nilpotently on $M$. Since
$\End_R(M)=\E$, this action is a nilpotent scalar, hence zero. Thus $M$ is an
$\E$-vector space and all of its $\E$-linear endomorphisms are $R$-linear. The
equality $\End_R(M)=\E$ forces $\dim_{\E}M=1$.
\end{proof}
For the skyscraper, and therefore for $D_\tau$, we have
\begin{equation}\label{eq:unramified-endomorphisms}
H^0\RHom(D_\tau,D_\tau)=\E,\qquad
H^j\RHom(D_\tau,D_\tau)=0\quad(j<0).
\end{equation}
These are the ambient automorphic Hom groups, since $\mathcal D_C$ is a full
subcategory; transport preserves both them and compactness.
Lemma~\ref{lem:spectral-action-transport} places $f^*D_\tau$ in the
$\mathcal S_\tau$ summand, so its image under \eqref{eq:unramified-equivalence}
is a bounded coherent complex supported at the origin of the smooth formal
component. Applying Lemma~\ref{lem:point-object} to the local ring at that
origin gives
\begin{equation}\label{eq:unramified-possible-shift}
f^*D_\tau\simeq D_\tau[r]
\end{equation}
for some $r\in\Z$.
We eliminate the shift on a bounded open, so no global cohomological bound on
$\Bun_G(C)$ is needed. Use the $f$-invariant ample line bundle $\mathcal L$ of
the statement. For each sufficiently large integer $m$, let
$\mathcal U_m\subset\Bun_{\SL_n}(C)$ be the open substack on which the
associated rank-$n$ bundle $\mathcal V$ satisfies
\[
\mathcal V\otimes\mathcal L^m\text{ globally generated},
\qquad H^1(C,\mathcal V\otimes\mathcal L^m)=0.
\]
These conditions are open and the opens cover $\Bun_{\SL_n}$ by Serre
vanishing. Each $\mathcal U_m$ is of finite type: rank and trivialized
determinant fix the Hilbert polynomial, the displayed conditions fix $h^0$, and
a choice of basis of sections gives the usual finite-type Quot presentation;
adding the determinant trivialization is also a finite-type condition. Since
$f$ preserves $\mathcal L$, every $\mathcal U_m$ is invariant under transport.
Choose $m$ with $D_\tau|_{\mathcal U_m}\ne0$. On a finite-type smooth atlas of
$\mathcal U_m$, local bounded constructibility gives a finite cohomological
interval; descent gives the same boundedness on $\mathcal U_m$. Consequently
\[
S_m=\{j\in\Z:\mathcal H^j(D_\tau|_{\mathcal U_m})\ne0\}
\]
is finite and nonempty. Pullback along the automorphism of $\mathcal U_m$ is
exact and conservative, so it preserves $S_m$.
Equation~\eqref{eq:unramified-possible-shift} gives $S_m=S_m-r$, which forces
$r=0$. We have obtained an isomorphism $b\colon f^*D_\tau\simeq D_\tau$; it
remains to verify its compatibility with the Hecke eigenmaps.
\subsection{Uniqueness of the full eigenstructure}
For $I$ a finite set and $L_1,L_2$ finite-rank lisse $\E$-sheaves on $C^I$,
there is a natural external-product formula
\begin{equation}\label{eq:unramified-hom-kunneth}
\RHom(D_\tau\boxtimes L_1,D_\tau\boxtimes L_2)
\simeq
\RHom(D_\tau,D_\tau)\otimes_{\E}
\RGamma(C^I,L_1^\vee\otimes L_2).
\end{equation}
On every finite-type smooth chart, the restrictions of $D_\tau$ are bounded
constructible and the usual constructible Hom--K\"unneth formula applies;
equivalently, use adjunction for the projection to the chart, proper base
change for $C^I$, and the projection formula. The second factor is a bounded
finite-dimensional $\E$-complex, hence perfect, so tensoring with it commutes
with the limits computing smooth descent and restriction over finite-type opens
of the bundle stack. Chartwise formulas therefore glue to
\eqref{eq:unramified-hom-kunneth}. On constructible chart objects these are
also the Hom complexes in the ambient ind-sheaf category.
Both factors on the right of \eqref{eq:unramified-hom-kunneth} are concentrated
in nonnegative degrees, by \eqref{eq:unramified-endomorphisms} and the fact that
the $L_i$ are ordinary lisse sheaves. Thus
\begin{align}
\Hom(D_\tau\boxtimes L_1,D_\tau\boxtimes L_2)
&\simeq\Hom(L_1,L_2),\label{eq:unramified-hom-zero}\\
H^j\RHom(D_\tau\boxtimes L_1,D_\tau\boxtimes L_2)
&=0\qquad(j<0),\label{eq:unramified-hom-negative}
\end{align}
where the first identification sends a lisse-sheaf map $a$ to
$\id_{D_\tau}\boxtimes a$.
\begin{lemma}\label{lem:unramified-eigenstructure-unique}
The complex $D_\tau$ admits a unique full coherent Hecke eigenstructure with
eigenvalue $\tau$. Consequently every isomorphism $f^*D_\tau\simeq D_\tau$ is
compatible with the eigenstructure and the given isomorphism
$u\colon f^*\tau\simeq\tau$.
\end{lemma}
\begin{proof}
Compare two full eigenstructures. For each finite $I$ and tuple
$(V_i)_{i\in I}$, their eigenmaps differ by an automorphism of
\[
D_\tau\boxtimes\Bigl(\boxtimes_{i\in I}(V_i)_\tau\Bigr).
\]
By \eqref{eq:unramified-hom-zero} this automorphism is uniquely
$\id_{D_\tau}\boxtimes a_{I,(V_i)}$, where $a_{I,(V_i)}$ is an automorphism of
the lisse factor. Naturality makes the one-leg maps $a_V$ natural in $V$.
Compatibility with successive Hecke operations on the locus of distinct legs
identifies the maps $a_{I,(V_i)}$ with exterior products of the one-leg maps;
equality on that dense open determines maps of lisse sheaves on $C^I$. Fusion
along the diagonal then gives
\[
a_{V\otimes W}=a_V\otimes a_W,
\qquad a_{\E}=\id.
\]
Thus the $a_V$ form a tensor automorphism of the local-system functor
$V\mapsto V_\tau$. Its group is the centralizer of the monodromy of $\tau$ in
$\Gd$, which is trivial by \eqref{eq:unramified-centralizer}. All $a_V$, and
hence all $a_{I,(V_i)}$, are identities, and the two systems of eigenmaps
agree.
This also proves uniqueness at the level of coherent structures.
Equation~\eqref{eq:unramified-hom-negative} says the mapping spaces between the
displayed targets have no positive homotopy groups. The eigenmaps identify the
successive Hecke transforms in the compatibility diagrams with targets of the
same form, so there is no additional choice of higher homotopies in the unit,
convolution, permutation, or fusion compatibilities. The resulting natural
transformation on ordinary representation labels has a unique exact continuous
extension by the universal properties of the stable and ind constructions
defining the spectral action.
Now transport the eigenstructure of $D_\tau$ by $f$, use $u$ on its
eigenvalue, and use any isomorphism $b\colon f^*D_\tau\simeq D_\tau$ on the
automorphic factor. Lemma~\ref{lem:satake-transport} makes this a full
eigenstructure on $D_\tau$ with eigenvalue $\tau$. The uniqueness just proved
identifies it with the original one, which is the claimed compatibility
of $b$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:equivariant-unramified}]
The inverse image of $\delta_\tau$ constructed above is nonzero, compact,
locally bounded constructible, and carries the coherent geometric
eigenstructure \eqref{eq:unramified-eigenmaps}. Its membership in $\mathcal D_C$
gives nilpotent singular support. Spectral-action transport,
Lemma~\ref{lem:point-object}, and the invariant-open argument give an
isomorphism $b\colon f^*D_\tau\simeq D_\tau$.
Lemma~\ref{lem:unramified-eigenstructure-unique} proves its compatibility with
$u$ and all eigenmaps. Iterating $b$ and its inverse defines the action of the
free cyclic group, with the corresponding iterations of $u$ on the eigenvalue;
the eigenstructure compatibilities persist under these iterations. This is the
asserted $\Z$-equivariant eigencomplex.
\end{proof}
\section{Arithmetic towers and equivariant gluing at a node}
\label{sec:parameters}
We build the parameter to which Proposition~\ref{prop:equivariant-unramified}
will be applied. Two steps. First we lift the arithmetic parameter $\rho$ to a
punctured curve in characteristic zero, retaining its Frobenius action and its
full boundary homomorphism. Second we place this lift on one component of a
separating nodal curve and extend to an unramified parameter on a smooth
generic fiber; the gluing must preserve Frobenius \emph{and} compactness of the
monodromy image. A ramified auxiliary component satisfies both at once.
\subsection{Lifting the arithmetic parameter}
With $k=\overline{\F}_q$, set
\begin{equation}\label{eq:coefficient-traits}
R_0=W(\F_q),\qquad D=\Frac(R_0),\qquad
R=W(k),\qquad K=\overline{\Frac(R)},\qquad
\Dbar=\overline D\subset K.
\end{equation}
Here $\Dbar$ is the algebraic closure of $D$ inside $K$, so $R$ is an excellent
complete strictly henselian discrete valuation ring with residue field $k$.
Extend the Witt-vector automorphism of $R$ induced by $a\mapsto a^q$ on $k$ to
an automorphism $\varphi$ of $K$; it fixes $D$ and preserves $\Dbar$. On
$\ell$-power roots of unity it acts by $\zeta\mapsto\zeta^q$, via their
Teichm\"uller lifts in $R$. We use the same letter for these compatible
actions, with the coordinate orientation fixed in Section~\ref{sec:conventions}.
Choose a smooth projective pointed lift
\[
(\mathscr X,\mathfrak x)\longrightarrow\Spec R_0
\quad\text{of }(X_0,x_0),
\qquad \mathscr U=\mathscr X\setminus\mathfrak x.
\]
Pointed deformations are unobstructed because
$H^2(X_0,T_{X_0}(-x_0))=0$; an ample line bundle also lifts, and formal
algebraization gives a projective family. This is the lifting construction of
\cite[Section~8.5]{CT}, here performed over $R_0$ so as to retain the
definition field. Write
\[
(C_1,x_1)=(\mathscr X,\mathfrak x)_D,
\qquad U_1=C_1\setminus\{x_1\}.
\]
By smoothness, a function $t$ on a neighborhood of $\mathfrak x$ over $R_0$ may
be chosen as a relative parameter, with differential trivializing the conormal
line of the section. We also regard $t$ as a rational function on $C_1$.
For an integer $d$ invertible in $R_0$, the coordinate $t$ identifies the
residual gerbe of $\mathscr X(\sqrt[d]{\mathfrak x})$ with $B\mu_d$ over $R_0$:
it trivializes the divisor line by $1/t$ and thereby specifies its trivial $d$th
root. Pulling a finite torsor back to this neutral object gives its boundary
fiber. A frame over the strictly henselian base $R$ identifies that fiber with
its finite structure group acting on itself on the right, so both inertia and
semilinear transport are recorded by left multiplication. The construction
below chooses such frames compatibly through the tower.
\begin{proposition}\label{prop:lifted-parameter}
The arithmetic representation $\rho$ determines a continuous parameter
\[
\sigma_1\colon\pi_1^{\mathrm{et}}((U_1)_{\Dbar})
\longrightarrow\PGL_n(L')
\]
for a finite coefficient extension $L'/L$, together with $\varphi$-equivariance,
such that:
\begin{enumerate}[(i)]
\item Its geometric image is Zariski dense. After compatible choices of frames
it is the same compact subgroup as $\rho(\pi_1^{\mathrm{et}}(U_{0,k}))$.
\item Its boundary homomorphism is
\begin{equation}\label{eq:lifted-boundary}
\gamma\longmapsto\exp(t_\ell(\gamma)N_1),
\end{equation}
with $N_1$ regular nilpotent. Every prime factor of characteristic-zero
inertia other than $\ell$ acts trivially.
\item In compatible frames at the coordinate-neutralized root gerbes of $x_1$,
the equivariance has a single value $A\in H_0=\rho(\pi_1^{\mathrm{et}}(U_0))$,
and
\begin{equation}\label{eq:frobenius-scales-nilpotent}
\Ad(A)N_1=qN_1.
\end{equation}
\item For every algebraic representation of $\PGL_n$, the associated local
system over $(U_1)_K$ admits an invariant lattice whose finite reductions
extend lisse over $\mathscr U_R$, $\varphi$-equivariantly. Their
specializations recover the tensor local system of $\rho$, including its
given arithmetic action.
\end{enumerate}
\end{proposition}
The proof uses a finite-cover lemma, reused in the nodal smoothing, that
compares monodromy orbits on finite torsor fibers and allows disconnected
torsors (needed when we enlarge the compact group recording monodromy).
\begin{lemma}\label{lem:component-orbits}
Let $S$ be the spectrum of a complete strictly henselian discrete valuation
ring with algebraically closed residue field, and let $\mathcal Y\to S$ be a
proper flat tame Deligne--Mumford curve with finite diagonal, projective coarse
space, and geometrically reduced fibers. Let $\mathcal T\to\mathcal Y$ be a
finite \'etale right $Q$-torsor, $Q$ finite, and $y\colon S\to\mathcal Y$ a
section in its smooth scheme locus. Identify the geometric special and generic
fibers of $y^*\mathcal T$ by a trivialization over $S$. The partitions of this
finite set by connected components of $\mathcal T_{\bar s}$ and
$\mathcal T_{\bar\eta}$ agree.
\end{lemma}
\begin{proof}
The stack $\mathcal T$ is again proper, flat, and tame with geometrically
reduced fibers. Proper henselian invariance lifts the open-and-closed
decomposition of $\mathcal T_{\bar s}$ to one of $\mathcal T$
\cite[Theorem~4.5]{Rydh}. Consider a lifted summand $\mathcal Z$. Its special
fiber is connected, reduced, and proper over an algebraically closed field, so
$H^0(\mathcal Z_{\bar s},\mathcal O)=k(\bar s)$. Coarse-space pushforward for a
tame stack is exact and commutes with base change
\cite[Lemmas~2.3.3--2.3.4]{AbramovichVistoli}. The coarse space is flat over
$S$; locally, taking invariants under a linearly reductive finite group
preserves flatness. Semicontinuity on this proper coarse space gives
$h^0(\mathcal Z_{\bar\eta},\mathcal O)\le1$; flatness makes the generic fiber
nonempty, so the dimension is exactly one. In particular $\mathcal Z_{\bar\eta}$
is connected. Every lifted special component therefore has exactly one
geometric generic component. Intersecting with the trivialized section fiber
proves the assertion.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:lifted-parameter}]
The use of the arithmetic image $H_0$ retains the action needed at the end.
This image is compact. Choose a faithful linear representation with
$H_0$-invariant lattice; let $H_{0,r}$ be a decreasing sequence of open normal
congruence subgroups cofinal at the identity, and put
$\Gamma_r=H_0/H_{0,r}$. The finite quotients of $\rho$ give a compatible tower
of right $\Gamma_r$-torsors on $U_0$, with the torsor convention chosen so that
associated local systems have monodromy $\rho$.
At stage $r$ the inertia image has $\ell$-power order. Choose indices $d_r$,
each a power of $\ell$, with $d_r\mid d_{r+1}$ and with $d_r$ divisible by that
order. The stage-$r$ torsor extends uniquely to a finite \'etale torsor on the
root stack $X_0(\sqrt[d_r]{x_0})$: a Kummer root of the local parameter kills
the finite tame inertia, after which the cover is \'etale across the root
divisor. This local description also proves the assertion over the original
finite field, or one descends the geometric extension by its uniqueness. The
tame-cover interpretation is \cite[Proposition~A.10]{LieblichOlsson}.
The relative root stack $\mathscr X(\sqrt[d_r]{\mathfrak x})$ is smooth,
proper, and tame over $R_0$, with finite diagonal, since $d_r$ is invertible in
$R_0$. Proper henselian invariance therefore lifts the torsor to this relative
root stack \cite[Theorem~4.5]{Rydh}; see also
\cite[Theorem~A.11 and Corollaries~A.12--A.13]{LieblichOlsson} and
\cite[Sections~8.3 and~8.5]{CT}. Full faithfulness lifts its group action and
the transition maps, comparing on a divisible root index where necessary, and
preserves all relations among those maps. Only the root indices are required to
be prime to $p$; the finite groups $\Gamma_r$ may have order divisible by $p$.
Restriction to $\mathscr U$ and then to $(U_1)_{\Dbar}$ gives the parameter
$\sigma_1$ with values in $\varprojlim_r\Gamma_r=H_0$, after framing. All the
lifted data are over $R_0$. Their base changes to $R$ and $\Dbar$ therefore
carry the descended semilinear $\varphi$-actions and their compatible
transition maps.
\emph{Geometric image.}\quad Choose a section of $\mathscr U_R$ through a
geometric special point and compatible frames of the tower along it, which
exist because this section is strictly henselian. Apply
Lemma~\ref{lem:component-orbits} at each stage after base change to $R$. The
geometric fibers of the root stacks and their finite \'etale covers are smooth
curves as stacks, so a connected component is irreducible, and removing the
marked gerbes and their inverse images preserves its connectedness. The
component partitions on the punctured geometric special and generic fibers
agree. In a torsor fiber these partitions are the monodromy orbits, and the
orbit of the chosen frame is precisely the image subgroup of $\Gamma_r$. Thus
the geometric images agree in every $\Gamma_r$. Both are compact, hence closed
in $H_0$, so they are equal. Connected components are unchanged by extension
from $\Dbar$ to $K$, giving the assertion for $\sigma_1$ itself. Density follows
from the hypothesis on $\rho$. This is (i).
\emph{Boundary action.}\quad Use $t$ to neutralize the residual gerbe of each
root stack as $B\mu_{d_r}$, compatibly under power maps. Pullback to its
neutral object over $R$ gives a finite \'etale torsor on a strictly henselian
trait. The resulting nonempty finite sets of frames have surjective transition
maps, so a compatible frame can be chosen throughout the tower. Its
$\mu_{d_r}$-action is the same on special and generic fibers under the
identification of prime-to-$p$ roots of unity. These actions record the entire
peripheral homomorphism, not just its conjugacy class. Passing to the limit
gives \eqref{eq:lifted-boundary}, with $N_1$ conjugate to the original regular
nilpotent after the fixed generator convention. Since only $\ell$-power root
indices are used, each $\Z_a(1)$-factor of characteristic-zero inertia with
$a\ne\ell$ acts trivially, including the factor $a=p$. This is (ii).
\emph{Frobenius scaling.}\quad The neutral objects defined by $t$ are themselves
defined over $R_0$. In their compatible frames the semilinear action is left
multiplication by elements of $\Gamma_r$ compatible in $r$, hence by one element
$A\in H_0$. Its compatibility with the gerbe action reads
\[
A\,\exp(t_\ell(\gamma)N_1)\,A^{-1}
=\exp(q\,t_\ell(\gamma)N_1).
\]
Taking the logarithm of a nontrivial boundary element proves
\eqref{eq:frobenius-scales-nilpotent}. All variants use the chosen coordinate
orientation; replacing the Frobenius generator $\varphi$ by $\varphi^{-1}$
replaces $A$ by $A^{-1}$ and $q$ by $q^{-1}$. This is (iii).
\emph{Lattice specialization.}\quad An algebraic representation over a finite
coefficient extension has a lattice invariant under the compact group $H_0$.
Each finite reduction factors through some $\Gamma_r$, so the lifted torsor
extends that reduction over $\mathscr U_R$. These are equivariant extensions,
and their special fibers are the reductions of the original arithmetic local
system. Tensor products and morphisms compare through the same tower; for
morphisms one may clear denominators. This gives the tensor specialization data
in (iv).
\end{proof}
\subsection{A compact conjugator and an auxiliary component}
The first parameter is lifted with its arithmetic action. To extend it across
a separating node, the two boundary actions must agree when expressed with the
opposite orientations of the two branches. The following observation supplies
an identification that also commutes with $A$ and belongs to a fixed compact
subgroup.
\begin{lemma}\label{lem:compact-conjugator}
Suppose $N_1\in\mathfrak{pgl}_n(L')$ is regular nilpotent,
$A\in\PGL_n(L')$ satisfies \eqref{eq:frobenius-scales-nilpotent}, and
$H_0\subset\PGL_n(L')$ is compact with $A\in H_0$. After a finite coefficient
extension there exist a cocharacter $c\colon\mathbb G_m\to\PGL_n$, a compact
open subgroup $J$ containing $H_0$, and an integer $e>1$ prime to $\ell$, such
that
\begin{equation}\label{eq:compact-conjugator}
c(z)A=Ac(z),\qquad \Ad(c(z))N_1=zN_1,
\qquad P:=c(-1/e)\in J.
\end{equation}
Consequently $P$ commutes with $A$ and
\begin{equation}\label{eq:inverse-boundary-conjugation}
P\exp(-e aN_1)P^{-1}=\exp(aN_1)
\qquad(a\in L').
\end{equation}
\end{lemma}
\begin{proof}
Identify $N_1$ with its traceless nilpotent matrix and choose a matrix lift
$\widetilde A$ of $A$. The relation $\widetilde A N_1=qN_1\widetilde A$ preserves
the regular Jordan filtration. On the successive one-dimensional quotients the
eigenvalues of $\widetilde A$ form a geometric progression with ratio $q$, and
are distinct: $q>1$ is an integer and the coefficient field has characteristic
zero. Thus $\widetilde A$ is diagonalizable after a finite extension. Choose an
eigenvector $v_0$ generating a Jordan chain, and put $v_i=N_1^iv_0$ for
$0\le i<n$. For some $a_0\ne0$,
\[
\widetilde A v_i=a_0q^i v_i,
\qquad N_1v_i=v_{i+1}\ (i<n-1),\qquad N_1v_{n-1}=0.
\]
The projective class of the diagonal cocharacter $c(z)v_i=z^iv_i$ commutes with
$A$ and scales $N_1$ by $z$.
Choose a closed faithful linear representation of $\PGL_n$ and an $H_0$-stable
lattice $\Lambda$. The inverse image of $\GL(\Lambda)$ is a compact open
subgroup $J\subset\PGL_n(L')$ containing $H_0$; it is open by continuity and
compact because the embedding is closed. Since $c(1)=1$ and $J$ is a
neighborhood of $1$, choose $e>1$ with $e\equiv-1$ modulo a sufficiently high
power of $\ell$; then $-1/e$ is close enough to $1$ that $c(-1/e)\in J$, and
this $e$ is prime to $\ell$. The final identity follows by applying
$\Ad(P)N_1=-N_1/e$ to the exponential.
\end{proof}
Fix $c,J,e,P$ as in the lemma, enlarging the coefficient field once and again
denoting it by $L'$. Retain the notation for all parameters. Define a cover
\begin{equation}\label{eq:auxiliary-cover}
h\colon C_2\longrightarrow C_1
\quad\text{by normalizing }C_1\text{ in }D(C_1)(w),
\qquad w^e=t.
\end{equation}
The valuation of $t$ at $x_1$ is one, so the polynomial $w^e-t$ is Eisenstein
there, even after extension to $\Dbar$; this is a degree-$e$ geometrically
connected cover. The curve $C_2$ is smooth and projective over $D$, with a
unique point $x_2$ over $x_1$. It is $D$-rational and totally ramified, with
local parameter $w$; this is visible in the completed local extension
$D[[t]]\subset D[[w]]$. Riemann--Hurwitz gives $g(C_2)\ge2$.
Put $U_2=C_2\setminus\{x_2\}$ and let $\sigma_2=h^*\sigma_1$ on
$(U_2)_{\Dbar}$. Any other branch point of $h$ lies over $U_1$ and causes no
ramification in this pullback local system. In the coordinate $w$, its boundary
homomorphism is $\exp(e\,t_\ell(\gamma)N_1)$.
For the rest of the construction choose open normal subgroups $J_r\subset J$,
decreasing and cofinal at $1$, and set
\begin{equation}\label{eq:nodal-finite-quotients}
Q_r=J/J_r.
\end{equation}
The map $H_0\to Q_r$ factors through one of the earlier $\Gamma_j$, since
$H_0\cap J_r$ is open in $H_0$. Extension of structure group therefore gives a
right $Q_r$-torsor on the first component with its original equivariance;
pullback by $h$ gives the one on the second. These torsors need not be
connected. For any common root index $b$ divisible by their stage-$r$ inertia
orders they extend over $C_i(\sqrt[b]{x_i})_{\Dbar}$. The cover $h$ induces a
map of these root stacks, since $h^*x_1=e x_2$. In coordinates it sends a $b$th
root of $t$ to the $e$th power of a $b$th root of $w$; on the residual gerbes
its map is
\begin{equation}\label{eq:gerbe-power-map}
\mu_b\longrightarrow\mu_b,
\qquad \zeta\longmapsto\zeta^e.
\end{equation}
The neutral objects fixed by $t$ and $w$ correspond under this map: in their
divisor-line trivializations, $h^*(1/t)=(1/w)^e$, with no scalar factor. The
second torsor and its frame are pulled back from the first, and this fiber
identification is defined over $D$. Thus the induced frame on the second gerbe
has exactly the same equivariance value $A$ as the first. This equality, on
top of the factor $e$ in the inertia, is the arithmetic information supplied by
the algebraic cover.
\subsection{The balanced smoothing and its parameter}
Identify $x_1$ with $x_2$ and write $C_0=C_1\cup_{x_1=x_2}C_2$. Choose a
projective smoothing $\mathscr C\to\Spec D[[s]]$ of this separating node, with a
relatively ample line bundle. The curve has unobstructed deformations and an
independent smoothing parameter at the split node; lifting an ample line bundle
algebraizes the deformation. As in \cite[Section~8.2]{CT}, the completed local
equation can be chosen as
\begin{equation}\label{eq:ordinary-node}
a b'=s,
\end{equation}
where $a$ restricts to a parameter on the first branch and $b'$ to one on the
second. Rescaling their tangent coordinates over $D$, and rescaling $s$
accordingly, ensures
\[
(a/t)(x_1)=1,\qquad (b'/w)(x_2)=1.
\]
These equalities identify the coordinate neutralizations of the branch gerbes
without a constant unit twist.
Choose compatible roots $s_b$ of $s$ and put
\begin{equation}\label{eq:nodal-traits}
S_b=\Spec\Dbar[[s_b]],\qquad s_b^b=s,
\qquad \Omega=\overline{\Dbar((s))}.
\end{equation}
Extend $\varphi|_{\Dbar}$ coefficientwise to $\Dbar((s))$, then to the union
obtained by adjoining the compatible roots $s_b$ while fixing those roots, and
finally to $\Omega$. We again denote this automorphism by $\varphi$. The smooth
projective geometric generic fiber of $\mathscr C$ will be denoted $C/\Omega$.
It is connected of genus $g(C_1)+g(C_2)$, and its polarization is
$\varphi$-invariant.
For each $b$, let $\mathcal C(b)\to S_b$ be the balanced twisted model with node
chart
\begin{equation}\label{eq:balanced-node}
\left[\Spec\bigl(\Dbar[[s_b]][u,v]/(uv-s_b)\bigr)/\mu_b\right],
\quad
\begin{cases}
a=u^b,\quad b'=v^b,\\
\zeta(u,v)=(\zeta u,\zeta^{-1}v).
\end{cases}
\end{equation}
Away from the closed node this is the ordinary family after base change. This
is the balanced twisted-curve construction of \cite[Theorems~1.9--1.10,
Remark~1.11, and Proposition~2.2(ii)]{OlssonTwisted}; the chart and its power
maps are described in \cite[Section~8.2]{CT}. Each model is proper, flat, and
tame, with projective coarse space and geometrically reduced fibers. Its
special-fiber normalization is the disjoint union of the two root stacks
$C_i(\sqrt[b]{x_i})_{\Dbar}$. For $b\mid m$ the power maps on $u,v$ give the
corresponding maps between models over $S_m\to S_b$.
The construction may be made over $D[[s_b]]$ with the group scheme $\mu_b$, so
the models and these maps carry the chosen semilinear actions. The ratios $a/t$
and $b'/w$ have residue one. Their formal roots with constant term one are
unique and compatible under power maps in characteristic zero; they are also
preserved by $\varphi$. Consequently the node chart identifies the branch
neutral objects with those already used for $t,w$. For general $b$, only the
$\ell$-primary quotient of the gerbe acts on our torsors. On that quotient
$\varphi$ acts by the $q$th power; the other prime factors act trivially on the
torsors.
We fix at this point the index that produces Borel level on the special-fiber
bundle stack. Choose a strictly dominant cocharacter $\lambda\in X_*(T)$ for the
split torus $T\subset B\subset G$ and an integer $b_*$ satisfying
\begin{equation}\label{eq:alcove-index}
0<\langle\alpha,\lambda\rangle<b_*
\qquad\text{for every positive root }\alpha.
\end{equation}
Choose the parameter indices $b_r$ so that $b_*\mid b_r\mid b_{r+1}$ and $b_r$
kills the two stage-$r$ inertia actions. The sheaf construction uses the fixed
model $\mathcal C(b_*)$ and inertia type $\lambda|_{\mu_{b_*}}$; the varying
indices $b_r$ supply only finite-stage extensions of the parameter. Let
$\mathcal V\subset\mathcal C(b_*)$ be its smooth scheme locus. Its geometric
generic fiber is $C$ and its special fiber is $(U_1\sqcup U_2)_{\Dbar}$.
\begin{proposition}\label{prop:nodal-parameter}
There exists a continuous Zariski-dense parameter
\[
\tau\colon\pi_1^{\mathrm{et}}(C)\longrightarrow J
\subset\PGL_n(L')
\]
equivariant under $\varphi$. For every algebraic representation of $\PGL_n$, an
invariant lattice in its evaluation local system has finite reductions
extending lisse over $\mathcal V\times_{S_{b_*}}S_{b_r}$ for sufficiently large
$r$. These extensions are equivariant and specialize on the two punctured
components to $\sigma_1$ and $\sigma_2$ with their given actions, compatibly
with the entire tensor system.
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:nodal-parameter}]
We first glue the finite torsors on the special fiber of $\mathcal C(b_r)$.
Express both actions using the common node generator $\zeta\in\mu_{b_r}$. Write
$u_{1,r}(\zeta)\in Q_r$ for the first action in its fixed frame. On the second
branch the positive coordinate generator is $\zeta^{-1}$; combining this
orientation reversal with \eqref{eq:gerbe-power-map}, its action in the induced
frame is $u_{1,r}(\zeta)^{-e}$. If $P_r$ is the image of $P$ in $Q_r$, then
\eqref{eq:inverse-boundary-conjugation} gives
\begin{equation}\label{eq:finite-stage-gluing}
P_r\,u_{1,r}(\zeta)^{-e}P_r^{-1}=u_{1,r}(\zeta).
\end{equation}
To read this identity directly at a finite root index, lift $\zeta$ to
characteristic-zero tame inertia and use \eqref{eq:lifted-boundary} before
reducing to $Q_r$; different lifts have the same reduction, and the prime
factors other than $\ell$ contribute the identity on both sides.
Identify the second right-torsor fiber with the first by left multiplication by
$P_r$. Equation~\eqref{eq:finite-stage-gluing} says exactly that this
identification intertwines their gerbe actions. Both semilinear actions have
value $A_r$, the reduction of $A$. Since $P_rA_r=A_rP_r$, the same
identification intertwines Frobenius. In coordinates on the two fibers the
required square is the elementary equality
\[
P_r(A_rg)=A_r(P_rg)\qquad(g\in Q_r).
\]
Figure~\ref{fig:node-gluing} records the two signs and this gluing map. All the
identifications are reductions of the single element $P\in J$, so they are
compatible throughout the tower.
The normalization torsors now glue to a finite \'etale right $Q_r$-torsor
$\mathcal T_{r,0}$ on $\mathcal C(b_r)_0$. This is ordinary nodal gluing in the
equivariant chart: \'etale locally on each branch cover, the torsor is constant
across the origin, and its two constant fibers have been identified with the
same $\mu_{b_r}$-action. Glue them across $uv=0$ and take the equivariant
quotient; \'etaleness holds at the node as well as away from it, and the
$Q_r$-action and semilinear action are retained. This is the finite-cover gluing
of \cite[Section~8.3]{CT}, with the identification chosen to commute with $A$.
Proper henselian invariance lifts $\mathcal T_{r,0}$ to a finite \'etale torsor
$\mathcal T_r$ on $\mathcal C(b_r)$ \cite[Theorem~4.5]{Rydh}. Full faithfulness
lifts the group actions, the equivariance isomorphisms, and the transition maps
after pullback to divisible indices; their relations lift as well. Restriction
to the common geometric generic curve $C$ yields a tower of finite
$Q_r$-torsors and hence, after compatible framing, a continuous parameter with
image in $\varprojlim_rQ_r=J$. The lifted equivariance gives the asserted
action on this parameter.
We verify density using the first component, without imposing connectedness.
Choose a section of $\mathcal V\to S_{b_*}$ through a point of
$(U_1)_{\Dbar}$, base-change it to every $S_{b_r}$, and choose compatible
frames of the lifted tower along these sections. Apply
Lemma~\ref{lem:component-orbits} over each $S_{b_r}$. The orbit of the frame
under the monodromy of the punctured first component lies in its connected
component in the whole special torsor; the lemma identifies that component's
section fiber with the geometric generic component's section fiber, which is
the generic monodromy orbit. Thus, if $H_1$ is the image of $\sigma_1$ and
$H_\tau$ the image of $\tau$ in these compatible frames, then
\[
\operatorname{im}(H_1\to Q_r)
\subset\operatorname{im}(H_\tau\to Q_r)
\qquad\text{for every }r.
\]
Both subgroups are compact and hence closed in $J$; since the $J_r$ are
cofinal, these inclusions imply $H_1\subset H_\tau$.
Proposition~\ref{prop:lifted-parameter} makes $H_1$ Zariski dense, hence $\tau$
is Zariski dense.
Finally let $V$ be an algebraic representation over a finite coefficient
extension and choose a $J$-stable lattice in $V$. Every finite reduction
factors through some $Q_r$. Away from the node the models $\mathcal C(b_r)$
agree with the corresponding base changes of $\mathcal C(b_*)$. The torsor
$\mathcal T_r$ therefore supplies the required lisse extension on
$\mathcal V\times_{S_{b_*}}S_{b_r}$, and its special restrictions are exactly
the two component torsors with their semilinear actions. Using this one tower
for all representations supplies the tensor and morphism comparisons, as in
Proposition~\ref{prop:lifted-parameter}. These are the equivariant
lattice-specialization data needed for nearby cycles on the fixed model
$\mathcal C(b_*)$.
\end{proof}
\section{From the separating node to an equivariant flag eigensheaf}
\label{sec:nodal}
The parameter $\tau$ of Proposition~\ref{prop:nodal-parameter} lives on the
smooth projective generic curve $C/\Omega$ of the twisted degeneration.
Proposition~\ref{prop:equivariant-unramified} therefore supplies an equivariant
unramified eigencomplex. We specialize to the node, retain the first
normalization component, and remove the enhancement of its flag. Two issues are
nonvanishing on the prescribed node type and preservation of the semilinear
action when passing to a perverse eigensheaf.
\begin{proposition}\label{prop:charzero-equivariant}
Let $(C_1,x_1)/D$, $\sigma_1$, and $\varphi$ be the pointed curve, dense lifted
parameter, and semilinear automorphism of Proposition~\ref{prop:lifted-parameter}.
There exist an integer $m\ge1$ and a nonzero locally constructible perverse
geometric $\E$-adic sheaf
\[
M_{\Dbar}\quad\text{on}\quad
\mathcal A_1=\Bun_{G,B,x_1}((C_1)_{\Dbar})
\]
with a $\varphi^m$-structure and the full coherent Hecke eigenstructure for
$\sigma_1$. Every eigenisomorphism is compatible with this structure and with
the given $\varphi^m$-structure on $\sigma_1$. Moreover,
$\SS(M_{\Dbar})\subset\Lambda_{\mathrm{par}}$.
\end{proposition}
\subsection{The prescribed node type and its two flags}
Retain the index $b_*$ and strictly dominant cocharacter $\lambda$ of
Section~\ref{sec:parameters}, so
\[
0<\langle\alpha,\lambda\rangle<b_*
\qquad\text{for every positive root }\alpha.
\]
Put $S=S_{b_*}=\Spec\Dbar[[s_{b_*}]]$. In the relative bundle stack of
$\mathcal C(b_*)/S$, remove from the closed fiber every inertia type except the
conjugacy class of $\lambda|_{\mu_{b_*}}$. Denote the resulting open stack by
$\mathcal B$. The condition is open because the finitely many conjugacy classes
of homomorphisms $\mu_{b_*}\to G$ are locally constant on the residual gerbe.
The centralizer of the chosen type is $T$: the inequalities ensure that no root
is trivial on $\lambda(\mu_{b_*})$. The Hom-stack theorem gives algebraicity and
local finite presentation \cite[Theorem~1.2]{HallRydh}. Smoothness follows from
the vanishing of $H^2$ of the adjoint bundle: the curve is tame, coarse
pushforward is exact, and its coarse space has dimension one. Thus $\mathcal B$
has smooth finite-type charts over $S$, and its geometric generic fiber is
$\Bun_G(C)$.
Let $B^-$ be the Borel opposite to $B$ with the same maximal torus, and define
\[
\mathcal A_1^+=\Bun_{G,R_u(B),x_1}((C_1)_{\Dbar}),
\qquad
\mathcal A_2^+=\Bun_{G,R_u(B^-),x_2}((C_2)_{\Dbar}).
\]
An object of either stack includes an enhanced flag, i.e.\ a reduction to the
indicated unipotent radical. The calculation of \cite[Lemma~8.2]{CT} gives
\begin{equation}\label{eq:nodal-type-quotient}
\mathcal B_s\simeq
[\mathcal A_1^+\times\mathcal A_2^+/T],
\end{equation}
where $T$ changes the two frames on the node gerbe simultaneously. We recall the
calculation to verify compatibility with our semilinear action and with the two
different component curves.
On the first branch of the balanced chart, set $a=u^{b_*}$. In a frame in
which inertia is $\lambda$, a change of frame satisfies
\[
h(\zeta u)=\lambda(\zeta)h(u)\lambda(\zeta)^{-1}.
\]
Conjugating to the invariant punctured-disc frame gives
$g(a)=\lambda(u)^{-1}h(u)\lambda(u)$. If $j=\langle\alpha,\lambda\rangle$ for a
positive root, the positive and negative root coordinates respectively
transform as
\begin{equation}\label{eq:nodal-root-coordinate}
u^j f(u^{b_*})\longmapsto f(a),
\qquad
u^{b_*-j}g(u^{b_*})\longmapsto a g(a).
\end{equation}
The toral coordinates are series in $a$. These formulas identify the full
frame-change group with $\{g(a)\in G(\Dbar[[a]]):g(0)\in B\}$; the assertion
over arbitrary test schemes follows by the same root-group factorization in the
formal big cell. Evaluation $h\mapsto h(0)$ becomes the torus projection of this
Iwahori group. Fixing the gerbe frame therefore gives an enhanced $B$-flag.
On the second branch the node generator acts by $v\mapsto\zeta^{-1}v$. Its
positively oriented coordinate generator thus sees $-\lambda$, so the identical
calculation gives $B^-$. On both branches the fixed gerbe torsor has
automorphism group $T$, expressed using the common node generator. Gluing the
normalization bundles, with an identification on their gerbes, gives
\eqref{eq:nodal-type-quotient}. The computation is branchwise and requires no
isomorphism between $C_1$ and $C_2$.
All these prescriptions use the split group, the group-scheme cocharacter
$\lambda$, and the balanced coordinates chosen over $D$; they consequently
commute with $\varphi$, including its action on $\mu_{b_*}$: no individual root
of unity has been declared fixed. Thus \eqref{eq:nodal-type-quotient} is an
equivariant identification. A modification away from the node preserves its
gerbe frames and acts on the corresponding factor. This also identifies every
successive and multi-leg Hecke correspondence, after pullback to
$\mathcal A_1^+\times\mathcal A_2^+$, with its componentwise counterpart.
\subsection{A criterion for nonzero nearby cycles}
Nearby cycles can vanish for a nonzero generic complex. The input that excludes
this possibility here is the specialization-detection theorem of
\cite[Theorem~4.2]{CT}, in the following scope.
\begin{lemma}[Specialization detection]\label{lem:specialization-detection}
Let $S=\Spec A$ be an excellent complete strictly henselian trait with
algebraically closed residue field and $\ell$ invertible. Let
$\mathcal Y\to S$ be smooth and locally of finite type. Its special fiber is to
be an unlevelled, ordinary Borel-level, or enhanced Borel-level bundle stack on
a smooth projective curve, or, in characteristic zero, the simultaneous-torus
quotient of two enhanced-level stacks. If the residue characteristic is
positive, assume the residue field is an algebraic closure of a finite field and
impose the Lie-theoretic hypotheses of Lemma~\ref{lem:lie-hypotheses}.
Assume a smooth scheme $L\to S$ of relative dimension one provides spherical
Hecke legs, that $L_s$ contains a nonempty open of each special-fiber curve on
which modifications are allowed, and that bounded output maps to
$\mathcal Y\times_S L$ are representable and proper; exact-relative-position
strata must be smooth over both input-and-leg and output-and-leg, and their
special fibers must have the transverse-modification geometry of
\cite[Proposition~3.3]{CT}. The Satake kernel on its open stratum is constant
with the relative Satake shift and twist.
Let $F$ be a locally bounded constructible complex on
$\mathcal Y_{\bar\eta}$ with lisse spherical Hecke eigenvalues on
$L_{\bar\eta}$. Suppose each eigenvalue has a lisse lattice whose finite
reductions extend lisse after finite trait extensions, compatibly with the
special-fiber value, as in Proposition~\ref{prop:equivariant-specialization}.
If, on some smooth finite-type chart $Z\to\mathcal Y$, the closure of the
nonzero-stalk locus of $F|_{Z_{\bar\eta}}$ meets $Z_s$, then $\Psi F\ne0$.
\end{lemma}
Here and below $\Psi$ is geometric nearby cycles without inertia invariants. The
closure in the lemma may be computed after descending its finitely many
geometric generic components to a finite extension of the trait; neither the
lemma nor this interpretation requires $F$ itself to descend. These are part of
the statement and conventions of \cite[Theorem~4.2]{CT}.
We produce the required meeting by extending a bundle carrying a nonzero
generic stalk. For $G=\SL_n$ the uniformization needed here has an elementary
description. If $Y$ is a smooth projective connected curve over an
algebraically closed field and $y\in Y$, the complement $Y\setminus\{y\}$ is a
smooth affine curve whose coordinate ring is a Dedekind domain. A rank-$n$
projective module over that ring is isomorphic to the sum of a free module of
rank $n-1$ and its determinant. Consequently every $\SL_n$-bundle is trivial on
this complement, and a free basis can be adjusted by a unit so that its
determinant agrees with the given trivialization. This is the $\SL_n$ case of
the affine-curve triviality used in Drinfeld--Simpson uniformization and in
\cite[Theorem~1.1]{BelkaleFakhruddin}; see also \cite{DrinfeldSimpson}.
In particular any two $\SL_n$-bundles on $Y$ differ by a modification at $y$,
and each such modification lies in some finite Schubert bound. In a family, the
bounded Hecke space with fixed reference input and fixed smooth section $y$ is
proper over the base trait. A geometric generic modification therefore extends
after a finite trait extension: its point first descends to a finite generic
extension, and the valuative criterion then applies over the corresponding DVR
normalization of the complete trait. The output remains unchanged near every
marking or node disjoint from $y$.
\subsection{Specializing on the chosen component}
Apply Proposition~\ref{prop:equivariant-unramified} to $(C,\tau)$. The generic
curve is smooth, projective, and connected of genus at least two, with
polarization from the projective smoothing, and
Proposition~\ref{prop:nodal-parameter} supplies the continuous dense parameter
over a finite coefficient field with its equivariance. We obtain a nonzero
locally bounded constructible eigencomplex $D_\tau$ on
$\mathcal B_{\bar\eta}=\Bun_G(C)$, with its $\varphi$-structure and coherent
equivariant eigenmaps.
The trait $S_{b_*}$ is excellent, complete, and strictly henselian, with
algebraically closed characteristic-zero residue field, in particular $\ell$
invertible. Use $L=\mathcal V$, the smooth scheme locus of $\mathcal C(b_*)$, a
relative curve with $L_{\bar\eta}=C$ and
$L_s=U_{1,\Dbar}\sqcup U_{2,\Dbar}$. At these legs the bounded correspondences
have the split affine Grassmannian models in smooth frame charts; output maps
are representable and proper, and exact-position maps are smooth over both
bundle-and-leg projections. These local models have the normalized constant
kernels on their open cells. Modifications there preserve the node type. On the
special fiber \eqref{eq:nodal-type-quotient}, the cotangent description, cone
dimension bound, and transverse modifications are precisely the torus-quotient
case of \cite[Section~3 and Section~8.2]{CT}. The required Lie-theoretic
properties hold in characteristic zero. Thus the geometric hypotheses of
Proposition~\ref{prop:equivariant-specialization} and
Lemma~\ref{lem:specialization-detection} hold for this family.
For their eigenvalue hypothesis use the equivariant finite-torsor tower of
Proposition~\ref{prop:nodal-parameter}. For each representation, an invariant
lattice has reductions extending lisse over the smooth scheme locus after the
extensions $S_{b_r}\to S_{b_*}$. Their specializations are the component
parameter systems, and all tensor maps and semilinear actions are carried by the
same tower. Hence
\begin{equation}\label{eq:nodal-nearby}
F_s:=\Psi D_\tau
\end{equation}
is locally bounded constructible and carries the coherent component
eigenstructure and its $\varphi$-action.
To prove $F_s\ne0$ on this type, first choose a point of
$\mathcal A_1^+\times\mathcal A_2^+$. A smooth chart of $\mathcal B$ through its
image, followed by henselian lifting, gives a reference bundle $P_0$ over $S$
with the prescribed node type. Lift also a point of $L_s$ to a section $y$ of
$L/S$. Choose a bundle $P$ on $C/\Omega$ where $D_\tau$ has nonzero stalk; such
a point exists on a finite-type chart because $D_\tau$ is nonzero and locally
bounded constructible.
The Dedekind-domain argument identifies $P$ and $(P_0)_\Omega$ off
$y_\Omega$. Extend the resulting bounded modification after a finite trait
extension. Its output lies in $\mathcal B$, since its node restriction still
equals that of $P_0$. Take a smooth finite-type chart $Z\to\mathcal B$ through
its special value and a residue-field point of the chart above it. If $S'$ is
the extended trait, the morphism $Z\times_{\mathcal B}S'\to S'$ is smooth;
henselian lifting of the above point gives a section, after a further finite
extension if necessary. Its geometric generic point has the chosen nonzero
stalk. This section witnesses a meeting of the closure in
Lemma~\ref{lem:specialization-detection} with the special fiber. Thus
\begin{equation}\label{eq:nodal-nonzero}
F_s\ne0\quad\text{on }[\mathcal A_1^+\times\mathcal A_2^+/T].
\end{equation}
Finite extensions in this argument only locate the closure inside the fixed
geometric generic fiber; they impose no new descent condition on $D_\tau$ or its
semilinear structure, and need not carry $\varphi$: geometric nearby cycles are
unchanged by this finite-trait replacement, and the equivariant object
\eqref{eq:nodal-nearby} was already constructed over $S$.
Ordinary pullback of $F_s$ to $\mathcal A_1^+\times\mathcal A_2^+$ is nonzero,
because this map is smooth and surjective. Choose a $\Dbar$-point $(a_1,a_2)$
with nonzero stalk. The bundle and enhanced flag defining $a_2$ descend to a
finite extension $D'/D$: they are of finite presentation and $\Dbar/D$ is
algebraic. Pass to a finite normal extension containing $D'$. Since $\varphi$
fixes $D$, its action on this normal extension has finite order, so some power
$\varphi^m$ fixes it pointwise, and the descended model supplies an
identification $\varphi^{m*}a_2\simeq a_2$. We henceforth use this power as
generator.
Restricting along $\mathcal A_1^+\times\{a_2\}$ gives a nonzero locally bounded
constructible complex $F_1^+$ on $\mathcal A_1^+$ with a $\varphi^m$-structure.
The quotient description of the Hecke correspondences and proper base change for
each bound give
\begin{equation}\label{eq:nodal-enhanced-eigen}
\Hecke_{I,(V_i)}(F_1^+)
\simeq F_1^+\boxtimes
\mathop{\boxtimes}_{i\in I}(V_i)_{\sigma_1}
\end{equation}
for every finite leg set $I$. The same comparisons apply on the
successive-modification spaces and on every fusion diagonal; they are natural
under semilinear transport, so all these eigenmaps and their coherence
constraints are equivariant. Ordinary restriction here is sufficient, and no
perversity assertion is made for $F_1^+$.
\subsection{Perverse cohomology and removal of the enhancement}
To pass from $F_1^+$ to an ordinary-level perverse eigensheaf, we take perverse
cohomology, push forward along the torsor that forgets the enhancement, and take
perverse cohomology once more. The first truncation supplies nilpotent singular
support, which controls monodromy along the enhancement torus; unipotence of
that monodromy will make the direct image nonzero; the second truncation then
gives the required perverse object.
The geometric inputs are \cite[Propositions~6.1 and 7.5]{CT}. Their scope is a
smooth projective complex curve with one marked point and a split simple simply
connected group. Proposition~6.1 applies at ordinary or enhanced Borel level:
for a locally bounded constructible geometric adic complex with a coherent
eigenstructure for a Zariski-dense parameter, every perverse cohomology inherits
the full eigenstructure and nilpotent singular support, and some perverse
cohomology is nonzero if the complex is. Proposition~7.5 removes the enhancement
when the boundary monodromy is unipotent. Our group $\SL_n$ meets the group
assumptions, and Proposition~\ref{prop:lifted-parameter} proves both density and
the required unipotent boundary monodromy for $\sigma_1$.
Choose an abstract field isomorphism $\Dbar\simeq\C$ to apply these geometric
statements. Such an isomorphism exists: both fields are algebraically closed of
characteristic zero and have transcendence degree $2^{\aleph_0}$ over $\Q$. We
use their Betti arguments to establish geometric properties while making all
sheaf operations and induced maps in the geometric adic category over $\Dbar$.
First choose $j$ with $Q={}^pH^j(F_1^+)\ne0$. Its eigenstructure retains the
semilinear action as follows. For a finite set $I$, put $d=|I|$. The proof of
\cite[Proposition~6.1, Section~6.3]{CT} constructs, on the objects under
consideration, the functorial adic comparison
\begin{equation}\label{eq:nodal-truncation}
{}^pH^j\bigl(\Hecke_{I,(V_i)}(F_1^+)[d]\bigr)
\simeq \Hecke_{I,(V_i)}({}^pH^jF_1^+)[d].
\end{equation}
The Betti calculation gives the needed perverse bounds; the comparison follows
from the universal property of the adic truncation triangles. On the eigenvalue
side, put $\mathcal L_I=\boxtimes_{i\in I}(V_i)_{\sigma_1}$. Since this sheaf is
lisse on the smooth $d$-dimensional leg space, exterior product with
$\mathcal L_I[d]$ is perverse exact, so
\[
{}^pH^j(F_1^+\boxtimes\mathcal L_I[d])
\simeq Q\boxtimes\mathcal L_I[d].
\]
Applying these comparisons to \eqref{eq:nodal-enhanced-eigen} and canceling the
common shift $[d]$ gives the relative-normalized eigenmaps for $Q$.
For a collision $\Delta\colon U_1^J\to U_1^I$, the corresponding truncation
comparison uses $(\id\times\Delta)^*[|J|-|I|]$ on these locally constant leg
families; comparisons for successive Hecke operations use the total dimension of
their retained leg space, counting a shared leg only once. Thus
\eqref{eq:nodal-truncation} carries the entire unit, convolution, permutation,
and fusion diagrams. Pullback by $\varphi^m$ preserves the perverse structure
and its truncation maps, so these adic comparisons commute with it; no
compatibility between a chosen complex realization and $\varphi$ is required.
Now let
\[
q\colon\mathcal A_1^+\longrightarrow\mathcal A_1
\]
forget the enhancement. This is a $T$-torsor of relative dimension $r=n-1$. Set
$F=q_!Q$. We need the intermediate assertion in the proof of
\cite[Proposition~7.5]{CT}: this particular direct image is nonzero. We recall
the mechanism, since a general torus direct image does not preserve
nonvanishing.
At enhanced level a cotangent residue lies in $\operatorname{Lie}B$. In an
adapted local frame write a generically nilpotent Higgs field as
$\theta=b(t)\,dt/t$, with $b(t)$ regular. The positive-degree invariant
polynomials vanish on $b(t)$ and hence on the residue $b(0)$; the toral
projection of this residue is therefore zero. Thus the enhanced nilpotent cone
is the smooth cotangent pullback of the ordinary-level cone and annihilates
tangent directions to the $T$-fibers \cite[Section~3.1]{CT}. The microlocal
criterion of \cite[Lemma~7.1]{CT} now makes the Betti realization of $Q$
monodromic along this torsor: locally on the base it is constructible for a
product stratification with the whole torus as second factor, so its cohomology
sheaves on each fiber are local systems.
For a representation $V$, let $\chi_V$ be its character on the dual torus
$\widehat T$. The central-sheaf calculation, based on Gaitsgory's construction
\cite{GaitsgoryCentral} and proved in the needed form in
\cite[Lemma~7.4 and proof of Proposition~7.5]{CT}, gives, for every
$V\in\Rep(\Gd)$,
\begin{equation}\label{eq:nodal-character}
\chi_V(\text{enhancement monodromy})
=\operatorname{Tr}(\text{boundary monodromy on }V_{\sigma_1})
\,\id_Q
= (\dim V)\,\id_Q.
\end{equation}
The last equality uses unipotence at $x_1$. On any stalk cohomology of a torsor
fiber the commuting monodromies have joint generalized eigencharacters, viewed
as points of the dual torus $\widehat T(\E)$. Character functions of
representations span the Weyl-invariant regular functions on $\widehat T$ and
separate its Weyl orbits. Equation~\eqref{eq:nodal-character} therefore puts
each eigencharacter in the orbit of $1$, which consists of $1$ alone. All
enhancement monodromies on these finite-dimensional spaces are unipotent.
Choose a fiber $T=q^{-1}(a)$ where $Q_T:=Q|_T$ is nonzero, and let $j_0$ be the
largest index with $\mathcal H^{j_0}(Q_T)\ne0$. Its fiber $W$ is a nonzero
finite-dimensional representation of $\Gamma=X_*(T)$ with commuting unipotent
operators. The augmentation ideal of $\E[\Gamma]$ acts nilpotently on this
space, so the coinvariants $W_\Gamma$ are nonzero. Poincar\'e duality gives
\[
H_c^{2r}(T,\mathcal H^{j_0}(Q_T))
\simeq W_\Gamma(-r)\ne0.
\]
In the compact-support hypercohomology spectral sequence this term has no
incoming differential, by maximality of $j_0$, and no outgoing differential,
because $H_c^i(T,-)=0$ for $i>2r$. Consequently $H_c^{2r+j_0}(T,Q_T)\ne0$.
Comparison and base change for the adic functor $q_!$ imply $F_a\ne0$. This
argument uses a nilpotence bound only on the chosen finite-dimensional space.
The morphism $q$ has finite type and fixed relative dimension, so $F$ is locally
bounded constructible. Away from $x_1$, Hecke modifications transport an
enhanced flag as well as its ordinary flag. Proper base change on a bounded
Hecke correspondence and the projection formula therefore give the natural
comparison
\begin{equation}\label{eq:nodal-torus-hecke}
\Hecke_{I,(V_i)}(q_!Q)
\simeq (q\times\id_{U_1^I})_!\Hecke_{I,(V_i)}(Q)
\simeq q_!Q\boxtimes
\mathop{\boxtimes}_{i\in I}(V_i)_{\sigma_1}.
\end{equation}
These comparisons also hold on the successive and collision correspondences, so
they preserve every coherence constraint. Since $q$ is defined over $D$, its
direct image and these natural comparison maps carry the
$\varphi^m$-structure.
Finally choose a nonzero perverse cohomology $M_{\Dbar}={}^pH^a(F)$.
Proposition~6.1 of \cite{CT} applies now at ordinary Borel level, and the same
adic truncation argument \eqref{eq:nodal-truncation} proves equivariance of all
the induced eigenmaps. It also gives nilpotent singular support; at ordinary
flag level this is $\Lambda_{\mathrm{par}}$. This completes the proof of
Proposition~\ref{prop:charzero-equivariant}.
\begin{figure}[!b]
\centering
\begin{tikzpicture}[>=Stealth,thick,font=\small]
\coordinate (n) at (0,0);
\draw[linkblue] (-4.6,0.2) .. controls (-2.7,0.8) and (-1.3,0.5) .. (n);
\draw[linkblue] (n) .. controls (1.3,-0.5) and (2.7,-0.8) .. (4.6,-0.2);
\fill (n) circle (2pt);
\node[above=5pt] at (-3.2,0.45) {$(C_1,x_1)$};
\node[below=5pt] at (3.2,-0.45) {$(C_2,x_2)$};
\node[above=7pt] at (n) {$B\mu_{b_r}$};
\node[align=center] at (-2.8,-0.75)
{$u\mapsto\zeta u$\\$u_{1,r}(\zeta)$};
\node[align=center] at (2.8,0.75)
{$v\mapsto\zeta^{-1}v$\\$u_{1,r}(\zeta)^{-e}$};
\node[draw,rounded corners,inner sep=6pt] (f1) at (-2.8,-2.05)
{first fiber: $Q_r$};
\node[draw,rounded corners,inner sep=6pt] (f2) at (2.8,-2.05)
{second fiber: $Q_r$};
\draw[->] (f2.west) -- node[above] {$g\mapsto P_rg$}
node[below] {$P_rA_r=A_rP_r$} (f1.east);
\end{tikzpicture}
\caption{The two branches of the balanced node and the torsor fibers on
its normalization, shown schematically. The cover $w^e=t$ multiplies
boundary monodromy by $e$, and the second branch reverses its
orientation. Multiplication by $P_r$ identifies the resulting action
$u_{1,r}^{-e}$ with $u_{1,r}$ while commuting with the common Frobenius
value $A_r$.}
\label{fig:node-gluing}
\end{figure}
\section{Specialization to the finite field}
\label{sec:final}
We now have a perverse eigensheaf with an action of $\varphi^m$ on the lifted
pointed curve over $\Dbar$. The arithmetic torsor tower of
Proposition~\ref{prop:lifted-parameter} identifies its eigenvalue after
mixed-characteristic specialization with the original arithmetic parameter,
including its Frobenius structure.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Take $m$ and $M_{\Dbar}$ from Proposition~\ref{prop:charzero-equivariant}. Recall
that $R=W(k)$, $K=\overline{\Frac(R)}$, and $\Dbar\subset K$ is preserved by
$\varphi$. Pullback under this algebraically closed field extension gives
\[
M_K\quad\text{on}\quad
\Bun_{G,B,\mathfrak x}(\mathscr X_K).
\]
It remains nonzero, locally constructible, and perverse, and inherits the
$\varphi^m$-structure. These assertions are checked on finite-type smooth
charts: extension of algebraically closed fields preserves nonzero constructible
stalks, support dimensions, and Verdier duality, hence perversity. Base change
for the bounded Hecke correspondences, with the tensor Satake comparison of
Section~\ref{sec:conventions}, transports the entire eigenstructure. Its
eigenvalue is the base change of $\sigma_1$, with the action provided by the
arithmetic tower.
Consider the relative ordinary-level bundle stack
\begin{equation}\label{eq:final-relative-stack}
\mathscr A=
\Bun_{G,B,\mathfrak x}(\mathscr X_R/R),
\qquad L=\mathscr U_R,
\qquad M=\Psi M_K\quad\text{on }\mathscr A_k.
\end{equation}
We verify the hypotheses of Proposition~\ref{prop:equivariant-specialization}
and Lemma~\ref{lem:specialization-detection} in this second family. The ring
$R$ is an excellent complete strictly henselian DVR with algebraically closed
residue field $k$, and $\ell$ is invertible because $\ell\ne p$. The stack
$\mathscr A$ is smooth and locally of finite type over $R$: $H^2$ of the
adjoint bundle vanishes on the fibers of the smooth relative curve, and adding
a flag is a smooth $G/B$-fibration. The leg space $L$ is smooth of relative
dimension one, with special fiber $U$.
Since all legs avoid $\mathfrak x$, their spherical Hecke correspondences have
the ordinary split affine Grassmannian models in frames. Bounded output maps
are representable and proper, both exact-position maps over bundle-and-leg are
smooth, and the open-stratum kernels have the fixed relative Satake
normalization. The special-fiber cotangent and transverse-modification geometry
is the ordinary Borel-level case of \cite[Section~3]{CT}. Its Lie-theoretic
hypotheses (H1)--(H4) hold by Lemma~\ref{lem:lie-hypotheses}, using $p>n$.
Finally, the invariant lattices supplied by the arithmetic tower have lisse
finite reductions on $\mathscr U_R$, with their $\varphi$-actions; their
specializations, including all tensor maps, are exactly the local systems
associated to $\rho$. Thus all geometric and eigenvalue hypotheses of both
specialization statements hold.
It follows that $M$ is locally constructible and perverse, and has the natural
multi-leg eigenisomorphisms
\begin{equation}\label{eq:final-eigen}
\Hecke_{I,(V_i)}(M)
\simeq M\boxtimes
\mathop{\boxtimes}_{i\in I}(V_i)_{\rho|_{\pi_1(U)}}.
\end{equation}
The same proposition gives a $\varphi^m$-structure on $M$ and equivariance of
these maps. The special-fiber action of $\varphi^m$ is $q^m$-Frobenius. On the
eigenvalue system its action is precisely the one obtained by restricting
$\rho$ to $\pi_1(U_{0,\F_{q^m}})$, because the finite torsors used for the lift
descended from that arithmetic representation. This identifies the arithmetic
eigenvalue as well as its geometric restriction.
We still have to prove $M\ne0$. Choose a $K$-point $(P,\beta)$ with nonzero
stalk of $M_K$, where $\beta$ is its Borel reduction at $\mathfrak x_K$. Use the
trivial $G$-bundle on $\mathscr X_R$ as reference, and lift a point of $U(k)$
to a section $y\in\mathscr U_R(R)$ by smoothness and the henselian property. The
affine-curve argument of Section~\ref{sec:nodal} trivializes $P$ off $y_K$,
compatibly with its determinant. The resulting modification of the reference
bundle lies in a finite Schubert bound, whose Hecke space is proper over the
reference input and $y$. After a finite trait extension it extends to a bundle
$\mathscr P$ on the whole relative curve.
The remaining datum is the flag. Its generic value $\beta$ is a point of the
associated flag bundle $\mathscr P|_{\mathfrak x}\times^G G/B$ over the extended
trait. This bundle is proper, so the valuative criterion extends $\beta$ as
well. We have thereby extended $(P,\beta)$ to a section of $\mathscr A$. Lift
this section to a smooth finite-type chart through its special value, using
henselian lifting as in the nodal argument. The section has a nonzero generic
stalk, so the closure of the nonzero-stalk locus on that chart meets its special
fiber. All hypotheses having been verified above,
Lemma~\ref{lem:specialization-detection} proves $M\ne0$.
Apply now the field-level nilpotent-detection theorem \cite[Theorem~4.1]{CT}.
For the level stacks over $k=\overline{\F}_q$ under its Lie-theoretic
hypotheses, it states that a locally bounded constructible complex with
spherical eigenisomorphisms and lisse eigenvalues has singular support in the
cone of generically nilpotent Higgs fields on every smooth finite-type chart.
Here $\mathscr A_k$ is the ordinary Borel-level stack, the characteristic
assumptions hold, and \eqref{eq:final-eigen} supplies those lisse eigenvalues.
At an ordinary flag $\beta$, the cotangent residue condition is
\[
\operatorname{Res}_{x}\theta
\in\operatorname{Lie}R_u(B_\beta).
\]
The cone in the detection theorem is therefore exactly
$\Lambda_{\mathrm{par}}$ of Theorem~\ref{thm:main}. This proves
$\SS(M)\subset\Lambda_{\mathrm{par}}$ in the stated chartwise sense.
Equip $M$ with its $\varphi^m$-structure and denote the resulting Weil sheaf
over $\F_{q^m}$ by $M_0$. The split Weil Satake comparison of
Section~\ref{sec:conventions} identifies the transported Hecke functors with the
relative-Satake-normalized Weil functors in the theorem. Thus
\eqref{eq:final-eigen} is an isomorphism of Weil sheaves for every finite leg
set and every collection of representations.
Every operation above transports the full eigenstructure by comparisons
natural for the semilinear generator. These comparisons were constructed on
the successive-modification correspondences and on every collision diagonal,
with the tensor unit and permutations respected throughout. The resulting unit,
convolution, permutation, and fusion diagrams therefore commute as Weil
diagrams, including all iterated collisions. A structure for the group generated
by Frobenius is all that is required for a Weil sheaf; no profinite descent
assertion is used. The resulting $M_0$ has all the claimed properties.
\end{proof}
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