\section{Hecke symmetry and transverse test functions}
\label{fl:sec:noncharacteristic}

Theorem~\ref{fl:thm:compact-automorphic-family} extended the family
$P\mapsto\mathcal F_P$ over the valuation base $S$ as a family of
universally locally acyclic (ULA) complexes. To reach a trace we
shall take vanishing cycles of these complexes along a function $f$
on a chart of $\Bun_G$, and for that we must know how they behave
on the hypersurfaces $f=t$. Restriction to a hypersurface does not
preserve the ULA property in general. The main result of this
section, Theorem~\ref{fl:thm:noncharacteristic-hecke-comparison},
says that for the members of a Hecke family it does, at every
point where $df$ lies outside the global nilpotent
cone. This is a relative form, over a valuation ring and in mixed
characteristic, of the theorem that Hecke eigensheaves have
nilpotent singular support
\cite[Main Theorems~14.4.3 and~14.4.4]{AGKRRV2022}; this theorem
is the converse of the theorem of Nadler and Yun \cite{NY2019}
that the Hecke transforms of a sheaf with nilpotent singular
support are lisse along the curve. Its proof follows the same
mechanism: a non-nilpotent covector can be moved by a Hecke
modification into the direction of the moving point, where the
local systems attached to the family are lisse.

The rest of the section applies the theorem. We show that near the
canonical bundle the nilpotent cone has a single smooth branch
(Proposition~\ref{fl:prop:regular-nilpotent-branch}), construct a
function $f$, homogeneous of weight two and defined over
$\mathbb F_q$, whose differential meets the nilpotent cone only at
that point and transversally there
(Lemma~\ref{fl:lem:graded-morse-chart}), and identify the resulting
vanishing cycles on the generic fibre with the Whittaker
coefficient (Proposition~\ref{fl:lem:vacuum-microstalk-comparison}).

A \emph{test function} is a regular function $f$ on a smooth
chart of a stack of bundles. Its differential is the covector
against which the singular support of a complex is tested.

\subsection{The comparison theorem}

\begin{setup}\label{rw:setup:rank-one}
Let $S_1$ be the spectrum of an absolutely integrally closed
valuation ring of rank one, with generic point $\eta$ and closed
point $s$. Let $\mathcal C/S_1$ be a projective Gorenstein curve
with geometrically reduced fibres, and let $U\subset\mathcal C$ be
an open subscheme, smooth over $S_1$, that is dense in every
component of every fibre. Let $G$ be split semisimple, and assume
that the residue characteristic of $S_1$ is zero or does not divide
$|W_G|$. Put $\mathcal B=\Bun_G(\mathcal C/S_1)$.
\end{setup}

The cotangent space of $\mathcal B$ at a bundle $E$ on a fibre $C$
is the space $H^0(C,\operatorname{ad}(E)^*\otimes\Omega_C)$ of Higgs
fields, where $\Omega_C$ is the dualizing line bundle. The
\emph{global nilpotent cone} $\Nilp$ consists of the pairs
$(E,\theta)$ with $\theta$ nilpotent at every point. For a smooth
chart $\pi:Y\to\mathcal B$ we write $\pi^\circ\Nilp\subset T^*Y$
for the set of covectors $d\pi^*\theta$ with $\theta$ nilpotent;
these covectors vanish on the tangent spaces of the fibres of
$\pi$.

Let $\mathcal K$ be a complex on a separated space $Z$ of finite
presentation over $S_1$, and let $i_Z:Z_s\to Z$ and
$j_Z:Z_\eta\to Z$ be the inclusions of the fibres. The
\emph{specialization cone}, which is computed on an \'etale chart
when $Z$ is not a scheme, is
\[
 \Delta_Z(\mathcal K)=\operatorname{Cone}
 \bigl(i_Z^*\mathcal K\longrightarrow
       i_Z^*Rj_{Z,*}j_Z^*\mathcal K\bigr),
\]
a complex on $Z_s$. By Input~\ref{rw:in:hs}, a complex
$\mathcal K$ with constructible generic fibre is ULA over $S_1$ if
and only if $\mathcal K\simeq Rj_{Z,*}j_Z^*\mathcal K$, that is, if
and only if $\Delta_Z(\mathcal K)=0$. On an open subscheme of
$Z$, the complex is ULA over $S_1$ if and only if all the stalks of
$\Delta_Z(\mathcal K)$ vanish there. Three properties
follow from the base-change
theorems and \cite[Theorem~4.1]{HS}: the formation of $\Delta$
commutes with proper pushforward and with smooth pullback, and
$\Delta_Z(\mathcal K\otimes\mathcal L)=\Delta_Z(\mathcal K)\otimes
\mathcal L|_{Z_s}$ when $\mathcal L$ is lisse on $Z$.

For a dominant coweight $\lambda$ let $V_\lambda$ be the irreducible
representation of $\widehat G$ of highest weight $\lambda$, and let
$\mathsf H_{V_\lambda}$ be the corresponding Hecke functor with
one leg in $U$, that is, the functor $\mathsf H_{I,V}$ of
Section~\ref{rw:sec:geometric-route} for a set $I$ with one
element; it sends complexes on $\mathcal B$ to complexes on
$\mathcal B\times_{S_1}U$.

\begin{theorem}[Transverse Hecke comparison]
\label{fl:thm:noncharacteristic-hecke-comparison}
In Setup~\ref{rw:setup:rank-one}, let $\mathcal K$ be a locally
bounded constructible complex on $\mathcal B$ that is ULA over
$S_1$. Put $\lambda_j=|W_G|\varpi_j^\vee$, where the
$\varpi_j^\vee$ are the fundamental coweights, and suppose that for
every $j$ there is an isomorphism
\begin{equation}\label{fl:eq:total-hecke-input}
 \mathsf H_{V_{\lambda_j}}(\mathcal K)
 \simeq\bigoplus_a \mathcal K_{j,a}\boxtimes \mathcal E_{j,a},
\end{equation}
in which each $\mathcal K_{j,a}$ is a complex on $\mathcal B$ that
is ULA over $S_1$ and each $\mathcal E_{j,a}$ is lisse on $U$.

Let $\pi:Y\to\mathcal B$ be a smooth chart, separated and of finite
presentation over $S_1$, let $f\in\Gamma(Y,\mathcal O_Y)$, and let
$y$ be a point of $V(f)_s$ such that
$df_y\notin\pi^\circ\Nilp$. Then
\[
 \Delta_{V(f)}(\pi^*\mathcal K|_{V(f)})_y=0.
\]
\end{theorem}

\begin{remark}[Naturality]\label{rw:rem:transverse-naturality}
The conclusion says that the canonical map
$i^*\mathcal K'\to i^*Rj_*j^*\mathcal K'$, for
$\mathcal K'=\pi^*\mathcal K|_{V(f)}$, is an isomorphism at $y$.
Since this map is canonical, it commutes with every isomorphism
induced by an automorphism of the data
$(\mathcal C,Y,f,\mathcal K)$. In particular, when these data are
defined over a finite field and $\mathcal K$ carries a Weil
structure, the isomorphism at $y$ commutes with Frobenius.
\end{remark}

\medskip\noindent\textit{Outline of the proof.}\ If $df_y$ does not vanish on the fibre of $\pi$ through $y$, then
$V(f)$ is smooth over $\mathcal B$ near $y$ and there is nothing to
prove. Otherwise $df_y=d\pi^*\theta$ for a Higgs field $\theta$ that
is not nilpotent, and the argument has four steps, carried out in
Sections~\ref{rw:sec:detectors}--\ref{rw:sec:stabilization}.
\begin{enumerate}
\item \emph{Linear algebra}
(Lemma~\ref{fl:lem:fixed-maximal-detectors}). At a point $u\in U$
where $\theta$ is not nilpotent, $\theta$ can be conjugated into a
maximal Levi subalgebra $\mathfrak m_j$ in such a way that it acts
invertibly on $\mathfrak g/\mathfrak m_j$ and pairs nontrivially
with $\lambda_j$.
\item \emph{Geometry of the Hecke correspondence}
(Lemma~\ref{rw:lem:hecke-cotangent-projections}). Modify the bundle
at $u$ by $\lambda_j$ inside this Levi subgroup. Then $\theta$
extends to the modified bundle. Near this modification the
conormal variety of the Hecke correspondence is \'etale over the
cotangent bundles of both ends, and its covector in the direction
of the moving point $u$ is not zero.
\item \emph{Isolation}
(Lemma~\ref{rw:lem:isolated-hecke-contribution}). The Hecke
transform of $\mathcal K$ is ULA in the direction of $u$ by
\eqref{fl:eq:total-hecke-input}. Since the Hecke transform is a
proper pushforward, this gives the vanishing of the cohomology of a
specialization cone over a whole fibre of the correspondence. Step
(2) shows that near the chosen modification this cone is supported
at one point, so its stalk there is a direct summand of zero.
\item \emph{Stabilization}
(Lemmas~\ref{rw:lem:exact-hyperbolic-coordinates}
and~\ref{fl:lem:quadric-detection}). The hypersurface on which the
stalk of step (3) lives is, in suitable coordinates, the
hypersurface $f+\sum r_is_i=0$. Adding the nondegenerate quadratic
form $\sum r_is_i$ in new variables does not change the
specialization cone, up to a shift and twist.
\end{enumerate}
The steps are combined at the end of
Section~\ref{rw:sec:stabilization}.

\medskip

A vanishing theorem for a proper pushforward does not by itself
give information about the complex being pushed forward, because a
nonzero complex on a proper variety can have zero cohomology. Step
(3) is where this is addressed, and it is the reason the proof
needs the geometric information of step (2) and not only the
identity \eqref{fl:eq:total-hecke-input}.

\subsection{Placing a non-nilpotent Higgs field in a Levi subalgebra}
\label{rw:sec:detectors}

Let $\kappa$ be the normalized invariant bilinear form on
$\mathfrak g$, defined over $\mathbb Z$ on the Chevalley form of the
Lie algebra.

\begin{lemma}[Detecting non-nilpotence]
\label{fl:lem:fixed-maximal-detectors}
Work over an algebraically closed ground field $\Bbbk$, of
characteristic zero or of characteristic $p\nmid|W_G|$.
\begin{enumerate}
\item Each $\lambda_j=|W_G|\varpi_j^\vee$ is a dominant cocharacter
in the coroot lattice.
\item The form $\kappa$ is nondegenerate on $\mathfrak g$, on every
Levi subalgebra, and on the centre of every Levi subalgebra.
\item Let $a(t)\,dt\in\mathfrak g[[t]]\,dt$ with $a(0)$ not
nilpotent. After conjugation by an element of $G[[t]]$ there is an
index $j$ such that $a(t)\in\mathfrak m_j[[t]]$, where $M_j$ is the
standard maximal Levi subgroup obtained by omitting the simple root
$\alpha_j$, and
\begin{equation}\label{fl:eq:central-detector}
 \operatorname{ad}(a(0))\ \text{is invertible on }
 \mathfrak g/\mathfrak m_j,\qquad
 \kappa(a(0),d\lambda_j)\ne0.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
(1) For a coweight $\nu$ and $w\in W_G$ the difference $\nu-w\nu$
lies in the coroot lattice $Q^\vee$, and $\sum_ww\nu=0$ because
$G$ is semisimple. Hence $|W_G|\nu=\sum_w(\nu-w\nu)\in Q^\vee$.

(2) In a basis of simple coroots, the matrix of $\kappa$ on the
Cartan subalgebra is the symmetrized Cartan matrix of $G$. Its
determinant is a product of determinants of Cartan matrices of
the simple factors of $G$ and of ratios of squared lengths of
roots. For a standard Levi subgroup, the matrix of $\kappa$ on the
span of its simple coroots is obtained in the same way from the
Cartan matrix of its subdiagram, with the lengths of the roots
inherited from $G$. All the primes which divide these determinants
divide $|W_G|$. Away from these primes the pairing between two
opposite root spaces is perfect as well. A Levi subalgebra is the
sum of the whole Cartan subalgebra and of pairs of opposite root
spaces, so $\kappa$ is nondegenerate on it. The same argument,
with the span of its coroots in place of the Cartan subalgebra,
shows that $\kappa$ is nondegenerate on its derived subalgebra.
The orthogonal complement of the derived subalgebra in the Levi
subalgebra is the centre: an element orthogonal to all brackets
commutes with the Levi subalgebra, by invariance and
nondegeneracy. So $\kappa$ is nondegenerate on the centre. A
change of the lattice of cocharacters within the isogeny class
changes the indices by divisors of the Cartan determinants, which
are invertible in the characteristic considered.

(3) We first treat the constant term. Conjugate $a(0)$ into a Borel
subalgebra and write it as $h+n'$ with $h$ in the Cartan subalgebra
$\mathfrak t$ and $n'$ in the nilpotent radical. Conjugating by
root subgroups in order of increasing height removes the components
of $n'$ in the root spaces $\mathfrak g_\beta$ with $\beta(h)\ne0$.
We obtain $a(0)=h+n$ with $n$ nilpotent and $[h,n]=0$, and $h\ne0$
because $a(0)$ is not nilpotent.

Next we show that, after conjugation by an element of the Weyl
group, the roots vanishing on $h$ are exactly those of a standard
Levi subgroup $M_I$. This is automatic in characteristic zero; in
characteristic $p$ we argue as follows. Let
$W_h\subset W_G$ be the stabilizer of $h$. Its order is prime to
$p$, so averaging over $W_h$ is a projector on
$X_*(T)\otimes\mathbb Z_{(p)}$. Let $\Lambda_h$ be its image, the
lattice of $W_h$-invariants; then $h$, being $W_h$-invariant, lies
in $\Lambda_h\otimes\Bbbk\subset\mathfrak t$. If
$\beta(h)=0$, the reflection $s_\beta$ fixes $h$, so it belongs to
$W_h$ and $\beta$ vanishes on $\Lambda_h$. If $\beta(h)\ne0$, then
$\beta$ does not vanish on $\Lambda_h$, since otherwise it would
vanish on $\Lambda_h\otimes\Bbbk$. Choose a rational vector in
$\Lambda_h\otimes\mathbb Q$ outside the hyperplanes of the roots of
the second kind, and move it into the closed dominant chamber by
an element of $W_G$. The roots vanishing on it are those of a
standard Levi subgroup $M_I$, and after the corresponding
conjugation they are the roots vanishing on $h$.

Now $h+n\in\mathfrak m_I$, and $h$ lies in the centre of
$\mathfrak m_I$. The vectors $d\lambda_j$ with $j\notin I$ form a
basis of that centre, because $|W_G|$ is invertible. By (2) there
is $j\notin I$ with $\kappa(h,d\lambda_j)\ne0$, and
$\kappa(n,d\lambda_j)=0$ because root spaces are orthogonal to
$\mathfrak t$. Since $j\notin I$ we have
$\mathfrak m_I\subset\mathfrak m_j$, and every root $\beta$ outside
$\mathfrak m_j$ satisfies $\beta(h)\ne0$. Thus $\operatorname{ad}(h)$
is invertible on $\mathfrak g/\mathfrak m_j$, and so is
$\operatorname{ad}(h+n)$, because $\operatorname{ad}(n)$ is
nilpotent and commutes with $\operatorname{ad}(h)$. This
proves \eqref{fl:eq:central-detector} for the constant term.

Finally we treat the whole series. The conjugation map
$G\times^{M_j}\mathfrak m_j\to\mathfrak g$ has differential
$(v,m)\mapsto[v,a(0)]+m$ at the point $(1,a(0))$, which is an
isomorphism from
$\mathfrak g/\mathfrak m_j\oplus\mathfrak m_j$ to $\mathfrak g$ by
the invertibility just proved. The map is therefore \'etale there,
and Hensel's lemma lifts $a(t)$ uniquely to a $\Bbbk[[t]]$-point of
$G\times^{M_j}\mathfrak m_j$ with the given reduction. Its first
coordinate is a $\Bbbk[[t]]$-point of $G/M_j$; since $M_j$ is smooth it
lifts to $G[[t]]$, and conjugating by this element puts $a(t)$ in
$\mathfrak m_j[[t]]$.
\end{proof}

\subsection{The Hecke correspondence near the chosen modification}
\label{rw:sec:hecke-geometry}

Fix a geometric point of $U$ in a fibre $C$ of $\mathcal C$, with
local coordinate $t$, a $G$-bundle $E_0$ on $C$ with a Higgs field
$\theta$, and a trivialization of $E_0$ near the point in which
$\theta=a(t)\,dt$ satisfies \eqref{fl:eq:central-detector} for some
$j$. Let $E_1$ be the modification of $E_0$ at the point defined by
$t^{\lambda_j}$ in this trivialization. Since $\lambda_j$ is central
in $M_j$ and $a(t)\in\mathfrak m_j[[t]]$, conjugation by
$t^{\lambda_j}$ fixes $a(t)$, so $\theta$ is also a Higgs field
$\theta'$ on $E_1$. We call $h_*=(E_0,E_1)$, with its leg, the
\emph{chosen modification}. It lies in the open stratum of the
Hecke correspondence bounded by $\lambda_j$, whose fibres over
either end have dimension $a=\langle2\rho_G,\lambda_j\rangle$, where
$2\rho_G$ is the sum of the positive roots.

The Hecke correspondence is a correspondence between stacks. We
describe it near $h_*$ by schemes, in the following form.

\begin{definition}[Hecke charts]
\label{rw:def:hecke-chart}
A \emph{Hecke chart} at the chosen modification consists of
\begin{itemize}
\item two schemes $Y_0$ and $Y_1$ with smooth maps to
$\mathcal B$, both smooth of relative dimension $n$ over $S_1$;
\item an \'etale map $M\to Y_0\times Y_1\times U$ and a closed
subscheme $\Gamma\subset M$, with projections $h_0$, $h_1$ and
$\operatorname{leg}$ to $Y_0$, $Y_1$ and $U$;
\item an \'etale map over $Y_1\times U$ from $\Gamma$ to the open
stratum of the pullback to $Y_1\times U$ of the Hecke
correspondence bounded by $\lambda_j$, under which the first end
of the modification attached to a point $h$ of $\Gamma$ is the
bundle of $h_0(h)$;
\item a point of $\Gamma$ over the chosen modification, again
denoted by $h_*$.
\end{itemize}
The Higgs fields $\theta$ and $\theta'$ define covectors on $Y_0$
at $h_0(h_*)$ and on $Y_1$ at $h_1(h_*)$, by pullback from
$\mathcal B$. The following properties are required.
\begin{enumerate}
\item The scheme $\Gamma$ is smooth over $S_1$, of codimension
$c=n-a$ in $M$, and its projections to $Y_0\times U$ and to
$Y_1\times U$ are smooth of relative dimension $a$.
\item There is a covector $\xi$ at the leg such that
$(\theta,-\theta',-\xi)$ is conormal to $\Gamma$ at $h_*$. The two
projections from the conormal bundle $N^*\Gamma$ to
$T^*Y_0\times U$ and to $T^*Y_1\times U$, which forget the covector
at the other end and the covector at the leg, are \'etale at this
point.
\item The covector $\xi$ is not zero.
\end{enumerate}
\end{definition}

\begin{lemma}[Hecke charts of framed bundles]
\label{rw:lem:hecke-cotangent-projections}
There are a divisor $Q$ on $\mathcal C$, disjoint from the leg, and
a Hecke chart $(Y_0,Y_1,M,\Gamma)$ at the chosen modification in
which $Y_0$ and $Y_1$ are \'etale charts of the stack of
$G$-bundles with a trivialization along $Q$. Its covector at the
leg is
\begin{equation}\label{fl:eq:leg-derivative}
 \xi=\pm\kappa(a(0),d\lambda_j)\,dx,
\end{equation}
where $x$ is the coordinate of the leg. The charts and the divisor
$Q$ can be chosen so that properties (1)--(3) hold in a
neighbourhood of $h_*$ in the family over $S_1$.
\end{lemma}
\begin{proof}
Steps~1 and~2 construct the charts and describe their tangent
spaces at the chosen modification. The content is in step~3: the
condition that the Higgs field remains regular when the
modification moves is expressed by the pairing
\eqref{fl:eq:root-hessian}, and this pairing is nondegenerate
because of \eqref{fl:eq:central-detector}.
Steps~4 and~5 deduce the properties (2) and (3) of a Hecke chart.

\step{1}{charts} A bound on the relative position bounds the poles
of a modification, in a faithful representation $V$, by a multiple
$mx$ of the leg. Choose $Q$ disjoint from the leg and so large that
\[
 H^0\bigl(\operatorname{Hom}(V(E_0),V(E_1))(mx-Q)\bigr)=0,\qquad
 H^0\bigl(\operatorname{End}(V(E_i))(-Q)\bigr)=0\quad(i=0,1).
\]
The second vanishing says that a bundle with a trivialization along
$Q$ has no nontrivial automorphisms, so the framed bundles near
$E_i$ form an algebraic space, of which we take \'etale charts
$Y_i$. The modifications considered identify the two framings
along $Q$, so each framing determines the other. When the two
framed ends and the leg are fixed, an infinitesimal variation of
the modification is, in the faithful representation, a section of
the first vector bundle of the display. This bundle has no
sections, so the map from the Hecke correspondence to
$Y_0\times Y_1\times U$ is unramified at $h_*$. An unramified
morphism becomes a closed immersion after an \'etale base change
of its target near the point \cite[Tag~0G1D]{StacksProject}:
there are an \'etale map $M\to Y_0\times Y_1\times U$ and a closed
immersion into $M$ of a neighbourhood $\Gamma$ of $h_*$ which
lifts the given map. By semicontinuity the two vanishings hold in
a neighbourhood. The
projections of the open stratum to either end and the leg are
smooth with fibres the open orbit of dimension $a$ in the affine
Grassmannian. So $\Gamma$ is smooth of dimension $n+1+a$ and of
codimension $(2n+1)-(n+1+a)=n-a$. Finally, a framing of the second
end of a modification determines a framing of the first. So the
pullback to $Y_1\times U$ of the Hecke correspondence is the
pullback of the correspondence between framed bundles, and the
scheme of which $\Gamma$ is an \'etale neighbourhood is its base
change along the \'etale chart $Y_0$. Hence $\Gamma$ is \'etale
over this pullback, with first end the bundle of $h_0$. These are
the data of a Hecke chart, with property (1).

\step{2}{the local model} All tangent spaces are taken over the
residue field of the leg. Put
$\mathfrak g_{\rm in}=\mathfrak g[[t]]$,
$\mathfrak g_{\rm out}=\operatorname{Ad}(t^{\lambda_j})\mathfrak g_{\rm in}$
and
$\mathfrak i=\mathfrak g_{\rm in}\cap\mathfrak g_{\rm out}$,
and let $\mathfrak g_b$ be the subspace of $\mathfrak g$ on which
$\lambda_j$ has weight $b$. Then
\[
 \mathfrak g_{\rm in}/\mathfrak i
   =\bigoplus_{b>0}\mathfrak g_b[t]/t^b\mathfrak g_b[t],
 \qquad
 \mathfrak g_{\rm out}/\mathfrak i
   =\bigoplus_{b<0}t^b\mathfrak g_b[t]/\mathfrak g_b[t].
\]
The first space is the tangent space at $h_*$ to the modifications
of $E_0$ at the fixed leg, and the second is the tangent space to
the modifications of $E_1$. Both have dimension $a$, and the
pairing $(v_+,v_-)\mapsto\operatorname{Res}\kappa(v_+,v_-)\,dt$
between them is perfect.

\step{3}{the Hessian} Fix the end $E_0$, its Higgs field $\theta$
and the leg, and move the modification by $v_+$. To first order the
modified bundle is changed by conjugation by $1+v_+$, and $\theta$
remains regular on it if and only if $[v_+,a(t)]\,dt$ has no polar
part with respect to $\mathfrak g_{\rm out}$. By the duality of step
2 this is the vanishing, for all $v_-$, of
\begin{equation}\label{fl:eq:root-hessian}
 (v_+,v_-)\longmapsto
 \operatorname{Res}\kappa(a(t),[v_+,v_-])\,dt
 =\operatorname{Res}\kappa(\operatorname{ad}(a(t))v_+,v_-)\,dt.
\end{equation}
We claim that this pairing is nondegenerate. Since
$a(t)\in\mathfrak m_j[[t]]$ has weight zero,
$\operatorname{ad}(a(t))$ preserves each $\mathfrak g_b[t]/t^b$. It
is triangular for the filtration by powers of $t$, with diagonal
blocks $\operatorname{ad}(a(0))|_{\mathfrak g_b}$, and these are
invertible for $b\ne0$ by \eqref{fl:eq:central-detector}. Hence
$\operatorname{ad}(a(t))$ is an automorphism of
$\mathfrak g_{\rm in}/\mathfrak i$, and the claim follows from the
perfect pairing of step 2. The same holds with the two ends
exchanged.

To relate this to the global geometry, let $\mathcal A$ be the
sheaf of infinitesimal automorphisms of the modification $h_*$; it
is the adjoint bundle away from the leg and has stalk
$\mathfrak i$ there. At $h_*$, the framed Hecke correspondence
with fixed leg has tangent complex
$R\Gamma(\mathcal C,\mathcal A(-Q))[1]$, and under Serre duality on
the Gorenstein curve the maps induced on cotangent spaces by the
two projections are given by the local spaces of step 2.

\step{4}{the projections are \'etale} Consider the projection
$N^*\Gamma\to T^*Y_0\times U$. Its source and target are smooth of
the same dimension $2n+1$, so it is enough to show that its tangent
map at the chosen point is injective. A tangent vector in the
kernel fixes $E_0$, the covector $\theta$ and the leg. What remains
are the $a$ tangent directions of the modification and the $c$
directions in the fibre of the conormal bundle. Because the
projection $\Gamma\to Y_1\times U$ is smooth, the differentials in
the directions of $Y_0$ of the $c$ equations of $\Gamma$ are
linearly independent; equivalently, the map from a fibre of the
conormal bundle to the covectors at the first end is injective. Now
let a tangent vector in the kernel move the modification by $v_+$.
Along it the covector $\theta$ at $E_0$ stays in the image of the
conormal bundle, which means that $\theta$ continues to extend to
the moved second end, to first order. By step 3 this forces
$v_+=0$. The vector then lies in a fibre of the conormal bundle and
has zero image in the covectors at the first end, so it is zero by
the injectivity just noted. The argument for the other projection is the same with
the ends exchanged. This proves (2), given that
$(\theta,-\theta')$ extends to a conormal vector, which is the
statement that $\theta$ and $\theta'$ agree away from the leg.

\step{5}{the covector at the leg} Moving the leg from $0$ to $x$
replaces $t^{\lambda_j}$ by $(t-x)^{\lambda_j}$, whose logarithmic
derivative in $x$ at $x=0$ is $-d\lambda_j/t$. Pairing with
$\theta=a(t)\,dt$ gives
$\mp\operatorname{Res}\kappa(a(t),d\lambda_j)\,dt/t
=\mp\kappa(a(0),d\lambda_j)$, which is not zero by
\eqref{fl:eq:central-detector}. This proves (3) and
\eqref{fl:eq:leg-derivative}.
\end{proof}

Theorem~\ref{fl:thm:noncharacteristic-hecke-comparison} concerns an
arbitrary smooth chart of $\mathcal B$. After pullback to the stack
of framed bundles such a chart is not \'etale but smooth, of some
relative dimension $d$. A smooth morphism need not be a product in
any \'etale neighbourhood, but it admits, locally, an \'etale map
to the product of its target with $\mathbb A^d$. The next lemma
shows that this is enough: a Hecke chart can carry $d$ additional
parameters, which a modification leaves unchanged.

\begin{lemma}[Hecke charts with parameters]
\label{rw:lem:hecke-chart-parameters}
Let $(Y_0,Y_1,M,\Gamma)$ be a Hecke chart at the chosen
modification, and let
$(\varpi,w):Y_0^+\to Y_0\times\mathbb A^d$ be an \'etale map with a
point $y_0^+$ over $h_0(h_*)$. Put $Y_1^+=Y_1\times\mathbb A^d$,
let $M^+$ be the pullback of $M$ to $Y_0^+\times Y_1^+\times U$,
and let $\Gamma^+\subset M^+$ be the part of the inverse image of
$\Gamma$ on which the coordinate $w'$ of the factor $\mathbb A^d$
of $Y_1^+$ is equal to $w(y)$, where $y$ is the first end. Then
$(Y_0^+,Y_1^+,M^+,\Gamma^+)$, with the point $h_*^+=(h_*,y_0^+)$,
is a Hecke chart at the chosen modification, of relative dimension
$n+d$ and with the same covector at the leg.
\end{lemma}
\begin{proof}
The maps $Y_0^+\to Y_0\to\mathcal B$ and
$Y_1^+\to Y_1\to\mathcal B$ are smooth. The map
$M^+\to M\times\mathbb A^d\times\mathbb A^d$ defined by $w$ and
$w'$ is \'etale, because $(\varpi,w)$ is, and $\Gamma^+$ is the
pullback along it of the product of $\Gamma$ with the diagonal of
$\mathbb A^d$. Hence $\Gamma^+$ is closed in $M^+$ and \'etale over
$\Gamma\times\mathbb A^d$, which is \'etale over the pullback to
$Y_1^+\times U$ of the Hecke correspondence; the first end of a
modification is the bundle of $\varpi(y)$, as required. The points
of $\Gamma^+$ are the pairs of a point $h$ of $\Gamma$ and a point
of $Y_0^+$ over $h_0(h)$, and $h_*^+$ is one of them.

Consequently $\Gamma^+$ is smooth of codimension
$c+d=(n+d)-a$ in $M^+$. Its projection to $Y_0^+\times U$ is a base
change of the projection of $\Gamma$ to $Y_0\times U$, and its
projection to $Y_1^+\times U$ is the composite of the \'etale map
to $\Gamma\times\mathbb A^d$ with the product of the projection of
$\Gamma$ to $Y_1\times U$ and the identity of $\mathbb A^d$. Both
are smooth of relative dimension $a$. This is (1).

The conormal bundle of $\Gamma^+$ is the pullback of the product of
$N^*\Gamma$ with the conormal bundle of the diagonal, and the
latter maps isomorphically to the cotangent bundle of either factor
$\mathbb A^d$. The pullbacks of $\theta$ and $\theta'$ to $Y_0^+$
and $Y_1^+$ vanish on the directions of $\mathbb A^d$, so
$(\theta,-\theta',-\xi)$ is conormal to $\Gamma^+$ at $h_*^+$, with
the same $\xi$. The two projections from $N^*\Gamma^+$ to
$T^*Y_0^+\times U$ and to $T^*Y_1^+\times U$ are \'etale at every
point lying over a point at which the projections from $N^*\Gamma$
are \'etale. This gives (2) and (3).
\end{proof}

\subsection{Isolating one stalk in a Hecke fibre}
\label{rw:sec:isolation}

The hypothesis \eqref{fl:eq:total-hecke-input} is about the Hecke
transform of $\mathcal K$, that is, about an integral of
$\mathcal K$ over all modifications with a given second end. The
theorem is about $\mathcal K$ itself, near one point and in the
codirection of one test function $f$. This subsection passes from
the integral to one of its terms. We introduce a function $F$ of
the two ends of a modification and of $c$ auxiliary variables. On
$\Gamma$ it is $f$ of the first end, and at the base point its
differential in the direction of the second end and the leg is
$(\theta',\xi)$. Because $\xi\ne0$, the integral over the
modifications which end at the base point has zero specialization
cone. Among these modifications the chosen one is isolated in the
locus where the integrand can have a nonzero cone, because the
projection from the conormal bundle of $\Gamma$ to
$T^*Y_1\times U$ is \'etale there. So the term of the chosen
modification has zero cone by itself
(Lemma~\ref{rw:lem:isolated-hecke-contribution}). The next
subsection identifies this term with the specialization cone of
$\mathcal K$ on the hypersurface $f=0$.

In this subsection and the next, $(Y_0,Y_1,M,\Gamma)$ is a Hecke
chart at the chosen modification, of relative dimension $n$, and
$c=n-a$. Let $f$ be a function on $Y_0$, and let $y_0=h_0(h_*)$,
a point of the special fibre of $Y_0$ lying over $E_0$. Assume that
\[
 f(y_0)=0,\qquad df(y_0)=\theta ,
\]
where $\theta$ is the covector on $Y_0$ defined by the Higgs
field. Write $B=Y_1\times U$, and let
$x:M\to Y_0$ and $b:M\to B$ be the two components of the \'etale
map $M\to Y_0\times B$. Choose equations $g_1,\ldots,g_c$ for
$\Gamma$ in $M$ near $h_*$. On $M\times\mathbb A^c$ put
\[
 F(m,\zeta)=f(x(m))-\sum_{j=1}^c\zeta_jg_j(m).
\]
Here $x(m)$ is the image of $m$ in $Y_0$. It is the first end of
a modification ending at $b(m)$ exactly when $m$ lies in $\Gamma$.
Since $M$ is
\'etale over $Y_0\times B$, a tangent vector at $m$ is a pair of
tangent vectors at $x(m)$ and at $b(m)$. By
Definition~\ref{rw:def:hecke-chart}(2) the covector
$(\theta,-\theta',-\xi)$ is conormal to $\Gamma$ at $h_*$, so there
is a unique $\zeta_*$ with
$\sum_j\zeta_{*,j}\,dg_j=(\theta,-\theta',-\xi)$ at $h_*$. We write
$m_*$ for the image of $h_*$ in $M$, $b_*$ for $b(m_*)$ and $b'$
for $(m_*,\zeta_*)$. Then
$F(b')=0$ and
\begin{equation}\label{rw:eq:dF-at-base-point}
 d_xF=0,\qquad d_\zeta F=0,\qquad
 d_bF=(\theta',\xi)\qquad\text{at }b'.
\end{equation}

Let $q:\overline{\mathsf H}\to B$ be the pullback to $B$ of the
Hecke correspondence bounded by $\lambda_j$, with its proper
projection to the second end and the leg: a point of
$\overline{\mathsf H}$ over $(y_1,u)$ is a modification at $u$,
bounded by $\lambda_j$, whose second end is the bundle of $y_1$.
Let $\bar h_0:\overline{\mathsf H}\to\mathcal B$ be the first end,
and let $A$ be the Satake kernel of $V_{\lambda_j}$ on
$\overline{\mathsf H}$. The first end of a modification is a point
of the stack $\mathcal B$ and not of the chart $Y_0$; but $\Gamma$
is an \'etale neighbourhood of $h_*$ in $\overline{\mathsf H}$, and
on it $\bar h_0$ is the composite of $h_0:\Gamma\to Y_0$ with the
chart $Y_0\to\mathcal B$. Pulling back along $V(F)\to B$ gives a
proper map
\[
 q':\overline{\mathsf H}\times_BV(F)\longrightarrow V(F),
\]
and we call the pullback of $\bar h_0^*\mathcal K\otimes A$ to its
source the \emph{integrand}. The pushforward of the integrand along
$q'$ is the pullback of $\mathsf H_{V_{\lambda_j}}(\mathcal K)$
to $V(F)$.

\begin{lemma}[The isolated contribution]
\label{rw:lem:isolated-hecke-contribution}
Assume \eqref{fl:eq:total-hecke-input}, with $\mathcal K$ and all
$\mathcal K_{j,a}$ ULA over $S_1$. Then the specialization cone of
the integrand has zero stalk at the point $(h_*,b')$.
\end{lemma}
\begin{proof}
Let $\mathcal D$ be the specialization cone of the integrand.

\step{1}{the cohomology of $\mathcal D$ on the fibre of $q'$ over
$b'$ vanishes} By \eqref{rw:eq:dF-at-base-point} and
$\xi\ne0$, the differential of $F$ at $b'$ is not zero in the
direction of the leg. Hence the projection $V(F)\to Y_1$ is smooth
near $b'$. The pullback of $\mathcal K_{j,a}$ along a smooth map is
ULA over $S_1$, and it stays so after tensoring with the pullback
of the lisse sheaf $\mathcal E_{j,a}$. By \eqref{fl:eq:total-hecke-input},
the pullback of the Hecke transform to $V(F)$ therefore has zero
specialization cone at $b'$. Since $\Delta$ commutes with the
proper pushforward $q'_*$,
\begin{equation}\label{rw:eq:fibre-cohomology-vanishes}
 R\Gamma\bigl(q'^{-1}(b'),\mathcal D\bigr)=0.
\end{equation}
This statement concerns the whole fibre, which is the closed
Schubert variety of modifications ending at $b_*$; it has no
analogue for an open part of the fibre.

\step{2}{near $h_*$, the restriction of $\mathcal D$ to the fibre
is supported at $h_*$} Since $\Delta$ commutes with \'etale
pullback, we may compute near $h_*$ on the \'etale neighbourhood
$\Gamma$ of $h_*$ in $\overline{\mathsf H}$. The kernel $A$ is
lisse of rank one there. Up to this factor the integrand is the
pullback of $\mathcal K$ along
\[
 \Gamma\times_BV(F)\longrightarrow Y_0,\qquad
 (h,m,\zeta)\longmapsto h_0(h).
\]
Since the pullback of $\mathcal K$ to the smooth chart $Y_0$ is
ULA, the cone $\mathcal D$ vanishes at every point where this map
is smooth. Its source is the hypersurface $F(m,\zeta)=0$ in
$(\Gamma\times_B M)\times\mathbb A^c$, a scheme which is \'etale over
$\Gamma\times Y_0\times\mathbb A^c$; so the map is smooth at a
point unless $dF$ vanishes there on all tangent vectors with
$dh_0=0$. At a point $(h,b')$ of the fibre over $b'$, the
derivatives of $F$ in $x$ and $\zeta$ are those of
\eqref{rw:eq:dF-at-base-point} and vanish. The remaining tangent
vectors are those of $\Gamma$ at $h$ with $dh_0=0$, on which $dF$
is the covector $(\theta',\xi)$. Hence, if the cone is not
zero at $(h,b')$, then $(\theta',\xi)$ vanishes on
$\ker(dh_0)\subset T_h\Gamma$; in other words,
$(\theta_0,-\theta',-\xi)$ is conormal to $\Gamma$ at $h$ for some
covector $\theta_0$ at $h_0(h)$.

Let $\mathcal Z$ be the set of these points $h$; it is closed in
the fibre. Since the projection of $\Gamma$ to $Y_0\times U$ is
smooth, $dh_0$ is surjective, so $\theta_0$ is determined by $h$,
and it depends continuously on $h\in\mathcal Z$. Thus
$h\mapsto(h;\theta_0,-\theta',-\xi)$ is a continuous section of
$N^*\Gamma$ over $\mathcal Z$. Its values lie in the fibre of the
projection $N^*\Gamma\to T^*Y_1\times U$ over the fixed point
defined by $(b_*,\theta')$ and the leg, and its value at $h_*$ is
the point $(h_*;\theta,-\theta',-\xi)$. By
Definition~\ref{rw:def:hecke-chart}(2) this projection
is \'etale at that point, which is therefore isolated in its
fibre. Hence $h_*$ is isolated in $\mathcal Z$. As
$\Gamma\to\overline{\mathsf H}$
is \'etale, hence open, there is a neighbourhood
$\mathcal N$ of $h_*$ in $q'^{-1}(b')$ such that
$\mathcal D|_{\mathcal N}$ is supported at $h_*$.

\step{3}{conclusion} Put $F_*=q'^{-1}(b')$. By step~2 the complex
$\mathcal D|_{F_*}$ vanishes on $\mathcal N-\{h_*\}$. So it is the
direct image of its restriction to the union of the two disjoint
closed subsets $\{h_*\}$ and $F_*-\mathcal N$, and it is the
direct sum of a complex supported at $h_*$, with stalk
$\mathcal D_{(h_*,b')}$, and of a complex supported away from
$h_*$. Thus $\mathcal D_{(h_*,b')}$ is a
direct summand of \eqref{rw:eq:fibre-cohomology-vanishes} and
vanishes.
\end{proof}

\subsection{Removing the auxiliary variables}
\label{rw:sec:stabilization}

Lemma~\ref{rw:lem:isolated-hecke-contribution} is a statement about
the hypersurface $F=0$ in
$(\Gamma\times_B M)\times\mathbb A^c$. We now show that this
hypersurface differs from $V(f)$ by a nondegenerate quadratic form
in additional variables, and that such a form does not affect the
specialization cone.

\begin{lemma}[Hyperbolic coordinates]
\label{rw:lem:exact-hyperbolic-coordinates}
Near $(h_*,b')$ there are functions $r_1,\ldots,r_n$ and
$s_1,\ldots,s_n$ on $(\Gamma\times_B M)\times\mathbb A^c$, vanishing
at $(h_*,b')$, such that the map
\[
 (\Gamma\times_B M)\times\mathbb A^c\longrightarrow
 (Y_0\times U)\times\mathbb A^{2n},\qquad
 (h,m,\zeta)\longmapsto
 \bigl(h_0(h),\operatorname{leg}(h),r,s\bigr),
\]
is \'etale at $(h_*,b')$ and
\begin{equation}\label{fl:eq:exact-quadratic}
 F=f(h_0(h))+\sum_{i=1}^nr_is_i.
\end{equation}
\end{lemma}
\begin{proof}
Choose \'etale coordinates $x_1,\ldots,x_n$ on $Y_0$ near $y_0$ and
put $s_i=x_i(x(m))-x_i(h_0(h))$. Because the $x_i$ are \'etale
coordinates, the locus $s=0$ coincides near the point with the
locus $x(m)=h_0(h)$, which inside $\Gamma\times_BM$ is
$\Gamma\times_{Y_0\times B}M$. This scheme is \'etale over
$\Gamma$, and the closed immersion $\iota:\Gamma\to M$ defines a
section of it, which is an open immersion. So near $(h_*,b')$ the
functions $s_i$ generate the ideal of the graph $m=\iota(h)$, the
variable $\zeta$ being free. A function that vanishes on the graph is therefore a
combination of the $s_i$: there are functions $F_i$ and $G_{ji}$
with
\[
 f(x(m))-f(h_0(h))=\sum_is_iF_i,\qquad
 g_j(m)=\sum_is_iG_{ji};
\]
the second function vanishes on the graph because
$g_j(\iota(h))=0$ for $h\in\Gamma$. With
$r_i=F_i-\sum_j\zeta_jG_{ji}$, substitution into the definition of
$F$ gives \eqref{fl:eq:exact-quadratic} as an identity of
functions. We differentiate \eqref{fl:eq:exact-quadratic} with
respect to the \'etale coordinate $x_i$, with $h$ fixed: on the
locus $s=0$ the function $r_i$ is the partial
derivative of $F$ in $x_i$; in particular $r$ vanishes at
$(h_*,b')$ by \eqref{rw:eq:dF-at-base-point}.

It remains to show that the map is \'etale. Its source and target
are smooth over $S_1$, of relative dimensions
\[
 \dim\bigl((\Gamma\times_B M)\times\mathbb A^c\bigr)
 =(n+1+a)+n+(n-a)\qquad\text{and}\qquad (n+1)+2n,
\]
which are equal, so it suffices to show that a tangent vector at
$(h_*,b')$ killed by the differential of the map is zero. Such a
vector has $dh_0=0$ and $d\operatorname{leg}=0$, and then $ds=0$
gives $dx=0$. What remains are the $a$ directions of $\Gamma$ in
the fibre of $(h_0,\operatorname{leg})$ and the $c$ directions of
$\zeta$. On the locus $s=0$, to which the vector is tangent, $r_i$
is the partial derivative of $F$ in $x_i$, that is, the $i$-th
component of the covector $df-\sum_j\zeta_jd_xg_j$ at the point
$h_0(h)$ of $Y_0$. The first term does not change along the
remaining directions, since they fix $h_0(h)$. The derivative of
$r$ along them is therefore, up to sign, the tangent map of the
projection from $N^*\Gamma$ to the covector at the first end, with
the first end and the leg fixed. That map is injective by
Definition~\ref{rw:def:hecke-chart}(2), so the vector is
zero.
\end{proof}

\begin{lemma}[Quadratic stabilization]
\label{fl:lem:quadric-detection}
Let $Z$ be a scheme smooth over $S_1$, let $f$ be a function on
$Z$, and let $\mathcal K$ be a complex on $Z$ that is ULA over
$S_1$. Let $\mathcal K_m$ be the pullback of $\mathcal K$ to the
hypersurface
\[
 \mathcal Q_m=\Bigl\{f+\sum_{i=1}^mr_is_i=0\Bigr\}
 \subset Z\times\mathbb A^{2m},
\]
and identify $V(f)$ with the locus $r=s=0$ in $\mathcal Q_m$. Then
\[
 \Delta_{Q_m}(\mathcal K_m)|_{V(f)_s}
 \simeq\Delta_{V(f)}(\mathcal K|_{V(f)})(-m)[-2m].
\]
In particular, if $\Delta_{Q_m}(\mathcal K_m)$ has zero stalk at a
point $z\in V(f)_s$, then so does
$\Delta_{V(f)}(\mathcal K|_{V(f)})$.
\end{lemma}
\begin{proof}
By induction it is enough to treat $m=1$: the function
$f+\sum_{i<m}r_is_i$ on $Z\times\mathbb A^{2m-2}$ and the pullback
of $\mathcal K$, which is again ULA, reduce the case of $m$ to that
of $m-1$.

For $m=1$, compactify the fibres of $\mathcal Q_1\to Z$ to the family of
conics
\[
 \bar q:\overline{\mathcal Q}=\{rs+fw^2=0\}\subset\mathbb P^2_Z
 \longrightarrow Z,
\]
so that $\mathcal Q_1$ is the open subscheme $w=1$. This family is the
pullback along $f:Z\to\mathbb A^1$ of the universal family
$\{rs+tw^2=0\}\subset\mathbb P^2\times\mathbb A^1$. The total space
of the universal family is smooth, and the map
$[r:s:w]\mapsto[r:w]=[-tw:s]$ identifies it with the blowup of
$\mathbb P^1\times\mathbb A^1$ at the point $([0:1],0)$. The
formulas for the cohomology of a blowup and of a projective bundle
\cite[Theorem~1.2 and Proposition~2.2.2.1]{Riou2014}, proper base
change along $f$ and the projection formula give
\[
 R\bar q_*\bar q^*\mathcal K
 \simeq\mathcal K\oplus\mathcal K(-1)[-2]
       \oplus i_*(\mathcal K|_{V(f)})(-1)[-2],
\]
where $i:V(f)\to Z$ is the inclusion. The decomposition is
constructed on the universal family, so it requires no regularity
of $V(f)$.

Apply $\Delta_Z$, which commutes with $R\bar q_*$ and with $i_*$.
On the right the first two summands vanish because $\mathcal K$ is
ULA, and the third becomes
$i_*\Delta_{V(f)}(\mathcal K|_{V(f)})(-1)[-2]$. On the left we get
$R\bar q_*\Delta_{\overline{\mathcal Q}}(\bar q^*\mathcal K)$. The map
$\bar q$ is smooth outside the locus $r=s=0$ over $V(f)$, which
lies in the chart $w=1$ and is the copy of $V(f)$ in the statement.
So $\Delta_{\overline{\mathcal Q}}(\bar q^*\mathcal K)$ is supported on that
locus, on which $\bar q$ restricts to the isomorphism onto $V(f)$.
Comparing the two sides gives the assertion.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{fl:thm:noncharacteristic-hecke-comparison}]
\step{1}{the smooth case} If $df_y$ does not vanish on the tangent
space of the fibre of $\pi$ through $y$, then $V(f)\to\mathcal B$ is
smooth near $y$, and $\pi^*\mathcal K|_{V(f)}$ is ULA near $y$ as a
smooth pullback of $\mathcal K$.

\step{2}{reduction to a chart with parameters} Otherwise
$df_y=d\pi^*\theta$ for a Higgs field $\theta$ on the bundle
$E_0=\pi(y)$, and $\theta$ is not nilpotent by hypothesis. Some
invariant polynomial of $\theta$ is then a nonzero section of a
line bundle on the reduced curve, so it is nonzero at a point $u$
of the dense open $U$. Apply
Lemma~\ref{fl:lem:fixed-maximal-detectors}(3) to the germ of
$\theta$ at $u$; this gives $j$ and the chosen modification $h_*$,
and Lemma~\ref{rw:lem:hecke-cotangent-projections} gives $Q$ and
a Hecke chart $(Y_0',Y_1',M',\Gamma')$ of framed bundles.

Let $\mathcal B_Q\to\mathcal B$ be the stack of bundles with a
trivialization along $Q$; it is smooth over $\mathcal B$, and
$Y_0'\to\mathcal B_Q$ is \'etale. The fibre product
$Y\times_{\mathcal B}Y_0'$ is therefore smooth over $Y$, and it is
smooth over $Y_0'$, of some relative dimension $d$, because $\pi$
is smooth. Choose a point $z$ of it over $y$ and over the first end
of the chosen point of $\Gamma'$. By the local structure of smooth
morphisms, a neighbourhood $Y_0$ of $z$ admits an \'etale map to
$Y_0'\times\mathbb A^d$ over $Y_0'$. By
Lemma~\ref{rw:lem:hecke-chart-parameters}, $Y_0$ is the first
scheme of a Hecke chart $(Y_0,Y_1,M,\Gamma)$ at the chosen
modification, in which $h_0(h_*)=z$. Since
$\Delta$ commutes with smooth pullback, it suffices to prove the
theorem for the chart $Y_0$, the pullback of $f$ and the point
$y_0=z$. At that point the pullback of $f$ vanishes, and its
differential is the pullback of $\theta$ from $\mathcal B$, because
$df_y$ is. These are the hypotheses of
Section~\ref{rw:sec:isolation} for this Hecke chart.

\step{3}{conclusion} By
Lemma~\ref{rw:lem:isolated-hecke-contribution}, the specialization
cone of the integrand vanishes at $(h_*,b')$. Near that point the
integrand is, up to a lisse factor of rank one, the pullback of
$\mathcal K$ along $h_0$ to the hypersurface $F=0$ in
$(\Gamma\times_B M)\times\mathbb A^c$. By
Lemma~\ref{rw:lem:exact-hyperbolic-coordinates} this hypersurface
is \'etale, near $(h_*,b')$, over the hypersurface
$\{f+\sum r_is_i=0\}\subset(Y_0\times U)\times\mathbb A^{2n}$, and
on the latter the complex is pulled back along the projection to
$Y_0$. The point $(h_*,b')$ maps to the point with $r=s=0$ over
$(y_0,u)$, where $u$ is the leg of $h_*$. Since $\Delta$ commutes
with \'etale pullback, Lemma~\ref{fl:lem:quadric-detection},
applied with $Z=Y_0\times U$ and the pullback of $\mathcal K$,
which is ULA, shows that the specialization cone of the pullback of
$\mathcal K$ to $V(f)\times U$ vanishes at $(y_0,u)$. As
$V(f)\times U\to V(f)$ is smooth, the same holds on $V(f)$ at
$y_0$.
\end{proof}

\begin{corollary}[Local acyclicity along a transverse test function]
\label{fl:cor:phase-universal-local-acyclicity}
For the family of Theorem~\ref{fl:thm:compact-automorphic-family}
over the base $S$, let $Y\to\Bun_G(\mathcal C_S)$ be a smooth chart
and let $f:Y\to\mathbb A^1_S$ be a smooth function whose
differential lies outside the pullback of the nilpotent cone
except along a section $x_*$. Then, for every perfect coefficient
$P$, the pullback of $\mathcal F_P$ to $Y$ is ULA over
$\mathbb A^1_S$ away from $x_*$.
\end{corollary}
\begin{proof}
By the valuative criterion \cite[Corollary~3.10]{HS} it suffices to
test universal local acyclicity after base change along maps
$\operatorname{Spec}V\to\mathbb A^1_S$ from absolutely integrally
closed valuation rings of rank one. If the coordinate of
$\mathbb A^1$ pulls back to $t\in V$, the base change of $Y$ is the
hypersurface $f-t=0$ in $Y_V$. The Hecke identities
\eqref{fl:eq:total-hecke-input} hold for $\mathcal F_P$ over $S$
with the ULA complexes $\mathcal F_{P\otimes W_a}$ and lisse
factors on $U^\circ$ (Lemma~\ref{fl:lem:coarse-covariance} and
Theorem~\ref{fl:thm:compact-automorphic-family}), and they survive
base change to $V$, as does the density of the open set of legs in
every component. Theorem~\ref{fl:thm:noncharacteristic-hecke-comparison},
applied to the function $f-t$ on $Y_V$, gives the vanishing of the
specialization cone at every point outside $x_*$. The maps
$\operatorname{Spec}V\to\mathbb A^1_S$ include those that change
$t$ and the characteristic at once, and those that lie in a single
fibre of $S$.
\end{proof}

\subsection{The nilpotent cone at the canonical bundle}
\label{fl:sec:boundary-whittaker-geometry-section}

Let $C$ be a fibre of our family of curves, with dualizing line
bundle $\Omega_C$, and let $(L,L^2\simeq\Omega_C)$ be the theta
pair that is part of the data of the family
(Section~\ref{fl:sec:punctured-spectral-fibre}). Fix a pinning of $G$ and put
\[
 E_*=\lambda_{\rm pr}(L)\times^TG ,
\]
where $\lambda_{\rm pr}$ is the sum of the positive coroots.
A root of height $h$ contributes the line bundle $\Omega_C^h$ to
$\operatorname{ad}E_*$, so the pinned generators of the negative
simple root spaces define sections of
$\operatorname{ad}E_*\otimes\Omega_C$ that vanish nowhere. Their sum
is a Higgs field $\theta_*$ on $E_*$ that is regular nilpotent at
every point. We call $(E_*,\theta_*)$ the \emph{canonical point} of
the nilpotent cone.

\begin{proposition}[The nilpotent cone at the canonical point]
\label{fl:prop:regular-nilpotent-branch}
Suppose that $\Omega_C$ has positive degree on every component of
$C$. Near $(E_*,\theta_*)$ the global nilpotent cone is smooth and
Lagrangian, relative to the base. If $m_1,\ldots,m_r$ are the
exponents of $G$, its tangent space at $(E_*,\theta_*)$ is
\[
 \bigoplus_iH^1(C,\Omega_C^{-m_i}),
 \qquad\text{of rank}\quad
 \sum_i\operatorname{rk}H^1(C,\Omega_C^{-m_i})
 =\dim G\,(g(C)-1).
\]
The pair $(E_*,\theta_*)$ has no infinitesimal automorphisms, and
these statements commute with base change.
\end{proposition}
\begin{proof}
\step{1}{deformations} The centralizer of a regular nilpotent
element is a smooth commutative group scheme
\cite[Proposition~4.4.1]{Riche}, and the principal grading acts on
its Lie algebra with weights $-2m_i$. In characteristic zero the
second assertion is classical: for the element $f$ of a principal
$\mathfrak{sl}_2$-triple, the Lie algebra is a sum of irreducible
representations with highest weights $2m_i$, and the centralizer of
$f$ is spanned by their lowest weight vectors. In characteristic
$p$ we deduce it as follows. Let $A=\mathbb Z[1/|W_G|]$; the primes
which are not very good divide $|W_G|$, so the results of
\cite[\S4]{Riche} hold over $A$. Let $f$ be the sum of the root
vectors of the pinning for the negative simple roots. Its
centralizer is smooth over $A$, by
\cite[Proposition~4.4.1]{Riche} applied with the opposite Borel
subgroup, so the Lie algebra of the centralizer is a finite
locally free $A$-module. The principal
cocharacter acts on $f$ by $t^{-2}$, hence normalizes the
centralizer and acts on this module. The weight spaces are direct
summands, and their ranks are constant because
$\operatorname{Spec}A$ is connected. So the weights are the same in
characteristic $p$ as in characteristic zero. If the group
is semisimple but not simply connected, these results are applied
to its simply connected cover: under our hypothesis on the
characteristic the cover is a central \'etale isogeny, it induces
an isomorphism of Lie algebras, and the inverse image of the
centralizer of an element of the Lie algebra is the centralizer of
the same element. Twisting by
$E_*$ gives a smooth commutative group scheme $J$ over $C$ with
$\operatorname{Lie}J=\bigoplus_i\Omega_C^{-m_i}$. Because the orbit
map of the regular nilpotent orbit is smooth, every regular
nilpotent Higgs field near $\theta_*$ is locally conjugate to
$\theta_*$, and the pairs $(E,\theta)$ with $\theta$ regular
nilpotent near $(E_*,\theta_*)$ are the $J$-torsors near the
trivial one. Their automorphisms, deformations and obstructions are
$H^0$, $H^1$ and $H^2$ of $\operatorname{Lie}J$. Here $H^0=0$ by
the positivity of $\Omega_C$ and $H^2=0$ because $C$ is a curve.
Hence $H^1$ is locally free and commutes with base change, and the
space of such pairs is smooth with the stated tangent space. Since
being regular is an open condition and $C$ is proper, every
nilpotent Higgs field near $\theta_*$ is regular nilpotent, so this
space is a neighbourhood of $(E_*,\theta_*)$ in the nilpotent cone.
The rank is computed by Riemann--Roch and $\sum_i(2m_i+1)=\dim G$.

\step{2}{the Lagrangian property} Let $(E,\theta)$ be a point of
the germ. By step 1 it is locally isomorphic to $(E_*,\theta_*)$,
and a tangent vector to the germ at this point is represented by a
cocycle $\gamma$ with values in the centralizer of $\theta$, which
is locally $\operatorname{Lie}J$. The Liouville form, evaluated on
this tangent vector, is the Serre duality pairing of $\gamma$ with
$\theta$, that is, the class of the cocycle $\kappa(\theta,\gamma)$
with values in $\Omega_C$. Locally $\theta$ has principal weight
$-2$ and $\gamma$ has weights $-2m_i<0$, while $\kappa$ pairs only
opposite weights; so $\kappa(\theta,\gamma)=0$. Thus the Liouville
form vanishes at every point of the germ, and therefore so does the
symplectic form.

The rank computed in step~1 is $\dim G\,(g-1)$, the dimension of
the stack of bundles. Now let $h:Y\to\Bun_G$ be a smooth chart of
relative dimension $r$, for instance a chart of framed bundles. The
pullback of the germ to $T^*Y$ consists of the covectors
$dh^*\theta$ at the points of $Y$ over the germ. It is smooth of
dimension $\dim G\,(g-1)+r=\dim Y$, half the dimension of $T^*Y$,
the $r$ directions of the fibres of $h$ being included. The
Liouville form of $T^*Y$ vanishes on it: evaluated on a tangent
vector it is the pairing of $\theta$ with the image of that vector
in the tangent space of $\Bun_G$, which is the Liouville form of
$T^*\Bun_G$ on a vector tangent to the germ, and that is zero by
what was just shown. So the pullback of the germ to every smooth
chart is Lagrangian; this is the form in which the statement is
used.
\end{proof}

\subsection{A homogeneous test function defined over $\mathbb F_q$}
\label{fl:sec:graded-morse-chart}

We now construct a chart at the canonical point and a test
function on it. Two requirements shape the construction. The
function must be homogeneous for a $\mathbb G_m$-action, because
Section~\ref{fl:sec:conservation} uses the action to compare
vanishing cycles at different points of $S$. And the chart, its
distinguished point and the function must be defined over
$\mathbb F_q$, because the final trace is taken over that field.

Work first over the complete local base
$S_0=\operatorname{Spec}R_0[[\varepsilon]]$, whose closed point
$s_0$ has residue field $\mathbb F_q$; the objects over $S$ are
obtained by pullback. Choose a divisor $Q$ in the smooth locus of
the curve over $S_0$, so large that
$H^0(C,\operatorname{End}(V(E_*))(-Q))=0$ for a faithful
representation $V$; by the theorem on cohomology and base change
this holds on every fibre. Bundles near $E_*$ with a trivialization
along $Q$ then have no nontrivial automorphisms, and since
deformations of bundles on a curve are unobstructed they form a
smooth algebraic space $\mathscr X_{\rm fr}$ over $S_0$, which is
quasi-separated with affine diagonal. The divisor $Q$ is finite
over the local base $S_0$, so $L$ is trivial along $Q$; a
trivialization of $L$ along $Q$ gives one of $E_*$, through its
torus reduction, and hence a section
$c:S_0\to\mathscr X_{\rm fr}$.

Changing the trivialization of a bundle along $Q$ by the constant
element $\lambda_{\rm pr}(t)$ defines an action of $\mathbb G_m$ on
$\mathscr X_{\rm fr}$. It does not change the underlying bundle,
and it fixes $c$, because $\lambda_{\rm pr}(t)$ is an automorphism of
$E_*$. This automorphism multiplies $\theta_*$ by $t^{-2}$. So
$\theta_*$, as a covector at $c$, is an eigenvector of the action,
of weight $2$ or $-2$ according to the conventions for induced
actions. We fix the direction of the action, replacing $t$ by
$t^{-1}$ if necessary, so that the weight is two in the following
sense: on a chart with an action, the differential at a fixed point
of a function of weight two has the same weight as $\theta_*$.
Finally, let $\mathbb G_m$ act on the cotangent bundle by the
induced action composed with that scaling of the fibres, by $t^2$
or $t^{-2}$, which fixes $\theta_*$. This action preserves the
graph of the differential of every function of weight two. It
preserves the pullback of the nilpotent cone, which is stable under
the induced action, since that action does not change the
underlying bundle, and under scaling. The symplectic form is
homogeneous for it, and we normalize the grading it induces on
tangent spaces at fixed points so that the symplectic form has
weight two.

\begin{lemma}[An equivariant chart through the canonical point]
\label{rw:lem:equivariant-chart-section}
There are a $\mathbb G_m$-equivariant \'etale morphism
$u:Y\to\mathscr X_{\rm fr}$ with $Y$ affine over $S_0$, and a
$\mathbb G_m$-fixed section $x_*:S_0\to Y$ with $u\circ x_*=c$,
such that $x_*(s_0)$ is an $\mathbb F_q$-rational point. After
replacing $Y$ by a $\mathbb G_m$-stable affine open subscheme
containing $x_*$, there are homogeneous \'etale coordinates on $Y$
that vanish along $x_*$.
\end{lemma}
\begin{proof}
\step{1}{the chart} By \cite[Corollary~10.2]{AHR}, applied to the
fixed point $c(s_0)$ of the action of the group $\mathbb G_m$ of
multiplicative type on $\mathscr X_{\rm fr}$, which is locally of
finite presentation, there is a $\mathbb G_m$-equivariant \'etale
map $u:(Y,y_0)\to(\mathscr X_{\rm fr},c(s_0))$ with $Y$ affine,
inducing an isomorphism of residue fields and of stabilizers at
$y_0$. So $y_0$ is an $\mathbb F_q$-rational fixed point.

\step{2}{the section} Form the fibre product
\[
\begin{tikzcd}[column sep=large,row sep=large]
 V=S_0\times_{\mathscr X_{\rm fr}}Y
   \arrow[r,"v"]\arrow[d,"p"'] & Y\arrow[d,"u"]\\
 S_0\arrow[r,"c"'] &\mathscr X_{\rm fr}.
\end{tikzcd}
\]
The map $p$ is \'etale, $V$ is affine because the diagonal of
$\mathscr X_{\rm fr}$ is affine, and $y_0$ defines a rational point
of the closed fibre of $p$. Since $S_0$ is henselian, there is a
unique section $h$ of $p$ through this point
\cite[Tag~04GR]{StacksProject}. Put $x_*=v\circ h$.

\step{3}{the section is fixed} Let $\mathbb G_m$ act trivially on
$S_0$. Then $V$ inherits an action over $S_0$. The geometric fibres
of $p$ are discrete and $\mathbb G_m$ is connected, so the action
is trivial on every geometric fibre. The locus in
$\mathbb G_m\times_{S_0}V$ where the action map agrees with the
projection is open, because $V/S_0$ is \'etale, and it contains
every geometric point. So the action on $V$ is trivial, and $h$,
hence $x_*$, is fixed.

\step{4}{homogeneous coordinates} Let $\mathcal I$ be the ideal of
$x_*(S_0)$ in $Y$. The conormal module $\mathcal I/\mathcal I^2$ is
free of finite rank over the local ring of $S_0$ and is graded by
the weights of $\mathbb G_m$. Lift a homogeneous basis to
homogeneous elements of $\mathcal I$; this is possible because
taking a homogeneous component commutes with the surjection
$\mathcal I\to\mathcal I/\mathcal I^2$. These functions define an
equivariant map from $Y$ to an affine space with a linear action,
which is \'etale along $x_*$. The locus where it is not \'etale is
closed and $\mathbb G_m$-stable. Take a function vanishing on that
locus and nonzero at $y_0$; its component of weight zero has the
same value at $y_0$ and still vanishes on the locus, since the
ideal of a stable closed subscheme is graded. Inverting it gives a
stable affine open subscheme containing $x_*$ on which the map is
\'etale.
\end{proof}

The symplectic vector space in the next proof is the tangent space
$\mathsf E$ of the cotangent bundle of $Y$ at $(x_*,\theta_*)$,
with its grading by the weights of $\mathbb G_m$. It contains two
graded Lagrangian subspaces: the tangent space
$\Lambda_{\rm vert}$ of the cotangent fibre, and the tangent space
$\Lambda_{\rm nilp}$ of the nilpotent cone, which is Lagrangian by
Proposition~\ref{fl:prop:regular-nilpotent-branch}. The graph of
the differential of a function $f$ with $df(x_*)=\theta_*$ is a
Lagrangian complement of $\Lambda_{\rm vert}$, and it is graded if
$f$ is homogeneous of weight two. We need it to be a complement of
$\Lambda_{\rm nilp}$ as well.

\begin{lemma}[A transverse homogeneous test function]
\label{fl:lem:graded-morse-chart}
After replacing $Y$ by its product with a graded vector space and
by a $\mathbb G_m$-stable affine open neighbourhood of $x_*$, there
is a function $f$ on $Y$, homogeneous of weight two and smooth over
the base, such that the graph of $df$ meets the pullback of the
nilpotent cone exactly along $x_*$, and transversally:
\begin{equation}\label{fl:eq:homogeneous-morse-graph}
 \operatorname{graph}(df)\cap\Nilp_Y=x_*(S_0).
\end{equation}
\end{lemma}
\begin{proof}
\step{1}{linear algebra} Put
$I=\Lambda_{\rm vert}\cap\Lambda_{\rm nilp}$ and choose a graded
decomposition
\[
 \mathsf E=(I\oplus I^\vee(2))\oplus(U_+\oplus U_-),
 \qquad \Lambda_{\rm nilp}=I\oplus U_+,
 \qquad \Lambda_{\rm vert}=I\oplus U_-,
\]
in which the two summands in parentheses are orthogonal, $U_+$ and
$U_-$ are Lagrangian in the second, and $I^\vee(2)$ is an isotropic
graded complement of $I$ in the first. Such a decomposition exists:
graded complements of $I$ in $\Lambda_{\rm nilp}$ and
$\Lambda_{\rm vert}$ pair perfectly, and a graded complement of $I$
in the orthogonal of $U_+\oplus U_-$ is made isotropic by
subtracting half of its alternating self-pairing.

A common graded Lagrangian complement of $U_+$ and $U_-$ need not
exist; see Remark~\ref{rw:rem:grading-obstruction}. We therefore
enlarge the chart. Replace $Y$ by $Y\times U_-^{\rm new}$, where
$U_-^{\rm new}$ is a copy of $U_-$ as a graded vector space. This
adds to $\mathsf E$ the symplectic space
$U_-^{\rm new}\oplus U_+^{\rm new}$, where $U_+^{\rm new}$ consists
of the new cotangent directions; we identify it with a copy of
$U_+$ in such a way that the new symplectic form is the negative
of the old one under the two identifications. The vertical subspace
becomes $\Lambda_{\rm vert}\oplus U_+^{\rm new}$, and the tangent
space of the pulled-back nilpotent cone becomes
$\Lambda_{\rm nilp}\oplus U_-^{\rm new}$. Consider
\[
 \Lambda=I^\vee(2)\oplus
 \{u_++u_+^{\rm new}\}\oplus
 \{u_-+u_-^{\rm new}\},
\]
where $u_\pm$ runs through $U_\pm$ and $u_\pm^{\rm new}$ is its
copy. It is graded, and it is isotropic because the two symplectic
forms cancel on the last two summands; it has half the dimension,
so it is Lagrangian. Comparing components shows that it meets both
$\Lambda_{\rm nilp}\oplus U_-^{\rm new}$ and
$\Lambda_{\rm vert}\oplus U_+^{\rm new}$ in zero.

\step{2}{the function} As a Lagrangian complement of the vertical
subspace, $\Lambda$ is the graph of a symmetric bilinear form on
the tangent space of the enlarged chart at $x_*$, homogeneous of
weight two. This holds on the closed fibre; lift the form to a
homogeneous symmetric form in the homogeneous coordinates of
Lemma~\ref{rw:lem:equivariant-chart-section} over $S_0$, and let
$q$ be half the associated quadratic form. Let $e_0$ be the linear
function of weight two in these coordinates with
$de_0=\theta_*$ along $x_*$, and put $f=e_0+q$. Then
$df=\theta_*$ along $x_*$ and the tangent space of
$\operatorname{graph}(df)$ at $x_*(s_0)$ is $\Lambda$. The
determinant expressing transversality to the nilpotent cone is a
unit at $s_0$, hence a unit over the local base $S_0$.

\step{3}{shrinking} Let $Z_f\subset Y$ be the closed subscheme
where $\operatorname{graph}(df)$ meets $\Nilp_Y$. By
transversality it is \'etale over $S_0$ along $x_*$, and $f$ is
smooth there because $\theta_*\ne0$. We shrink $Y$ twice. First we
remove the closed subset where $Z_f$ is not \'etale over $S_0$ or
$f$ is not smooth; it is stable under the action and does not
meet $x_*(S_0)$. After this $Z_f$ is \'etale and separated over
$S_0$, so the section $x_*$ is open and closed in it, and we
remove $Z_f-x_*(S_0)$, which is again closed and stable. Each
time the result is an affine open subset, stable under the action,
which contains the whole section: if $J$ is the homogeneous ideal
of the subset to be removed and $\mathcal I$ the ideal of the
section, then $J+\mathcal I=\mathcal O_Y$, so $J$ contains an
element equal to $1$ along the section; its component of degree
zero has the same property, and we invert it.
\end{proof}

\begin{remark}[Why the chart is enlarged]
\label{rw:rem:grading-obstruction}
In a symplectic plane with weights $0$ and $2$, the two weight
lines are the only graded Lagrangian subspaces, so they have no
common graded complement. After one coordinate is added, the
quadratic form $xy$ of weight two has Hessian determinant $-1$ and
supplies one. The enlargement in
Lemma~\ref{fl:lem:graded-morse-chart} does this for all weights at
once. It depends only on the chart, so the same function $f$
serves for every coefficient $P$.
\end{remark}

Pulling back along $S\to S_0$ gives the chart, the section $x_*$
and the test function $f$ used in
Section~\ref{fl:sec:conservation}. By construction their special
fibres and the point $x_*(s_0)$ are defined over $\mathbb F_q$.

\subsection{The microstalk on the generic fibre}
\label{rw:sec:generic-microstalk}

On the generic fibre the curve is smooth and proper over a field of
characteristic zero, and the vanishing cycles of $f$ at $x_*$ can
be computed.

\begin{proposition}[The microstalk is the Whittaker coefficient]
\label{fl:lem:vacuum-microstalk-comparison}
On the geometric generic fibre there is an isomorphism of functors
on $\operatorname{Perf}(\mathfrak F_\rho)$,
\begin{equation}\label{fl:eq:generic-microstalk}
 (R\Phi_f\mathcal F_P)_{x_*}[-1]
 \simeq\Gamma(\mathfrak F_\rho,P)\otimes L_{\rm shift},
\end{equation}
where $L_{\rm shift}$ is a graded line that does not depend on $P$.
\end{proposition}

The proof is given at the end of the section. It has four
ingredients, of different kinds, which we state as separate lemmas:
a passage from \'etale sheaves to sheaves in the classical topology
(Lemma~\ref{rw:lem:microstalk-classical-reduction}); the theorem of
Nadler and Taylor (Input~\ref{rw:in:microstalk}); a computation of
vanishing cycles of a homogeneous function
(Lemmas~\ref{rw:lem:weight-two-vanishing-cycles},
\ref{rw:lem:twisted-unipotent-chart}
and~\ref{rw:lem:homogeneous-vanishing-cycles}); and the comparison
of two kernels by a calculation with $D$-modules
(Lemma~\ref{rw:lem:poincare-kernel}).

Both sides of \eqref{fl:eq:generic-microstalk} are composites of
$P\mapsto\mathcal F_P$ with a functor on constructible sheaves with
nilpotent singular support on $\Bun_G(C)$, where $C$ is the generic
fibre, a smooth proper curve over the algebraically closed field
$K_{\rm geom}$ of characteristic zero: on the left the functor
$\mu=(R\Phi_f(-))_{x_*}[-1]$, on the right, by
\eqref{fl:eq:puncture-vacuum}, the functor
$\operatorname{Wh}=\operatorname{RHom}(\operatorname{Poinc}^{\rm Vac}_!,-)$.
So the proposition says that
$\mu\simeq\operatorname{Wh}\otimes L_{\rm shift}$. Vanishing cycles
are normalized by the triangle $i^*\to R\Psi\to R\Phi$.

The functors $\mu$ and $\operatorname{Wh}$ make sense in three
settings: for \'etale sheaves over $K_{\rm geom}$, for \'etale
sheaves over $\mathbb C$, and for sheaves of $k$-vector spaces in
the classical topology. We write $\mu^{\rm cl}$ and
$\operatorname{Wh}^{\rm cl}$ for the functors in the third setting.

\begin{lemma}[Reduction to the classical topology]
\label{rw:lem:microstalk-classical-reduction}
Choose an embedding of $K_{\rm geom}$ in $\mathbb C$. If there is
an isomorphism
$\mu^{\rm cl}\simeq\operatorname{Wh}^{\rm cl}\otimes L_{\rm shift}$
of functors on sheaves of $k$-vector spaces with nilpotent singular
support, in the classical topology, on the stack of bundles of the
complex curve $C\otimes_{K_{\rm geom}}\mathbb C$, then there is an
isomorphism $\mu\simeq\operatorname{Wh}\otimes L_{\rm shift}$ of
functors on \'etale sheaves with nilpotent singular support on
$\Bun_G(C)$.
\end{lemma}
\begin{proof}
An embedding exists, because $K_{\rm geom}$ has the cardinality of
the continuum.

\emph{The form of the two functors.} Let $\pi:Y\to\Bun_G(C)$ be the
chart, a smooth map from a scheme of finite type. Then
$\mu(\mathcal F)$ is, up to a shift, the stalk at the rational
point $x_*$ of the vanishing cycles of $\pi^*\mathcal F$ for the
function $f$ on $Y$. The object $\operatorname{Poinc}^{\rm Vac}_!$
is the direct image with compact supports, along a map
$p_{\rm W}$ to $\Bun_G(C)$ from a stack of finite type, of a
complex $\mathcal K_{\rm W}$ obtained by the six operations from
constant sheaves \cite[\S1.1.1]{GR2025}\cite[\S3.3]{GLCI2024}; so
$\operatorname{Wh}(\mathcal F)=\operatorname{RHom}(\mathcal K_{\rm W},
p_{\rm W}^!\mathcal F)$. Thus each of the two functors is a
pullback to a stack of finite type, followed by operations on that
stack: the functor $\operatorname{RHom}$ out of a fixed complex in
one case, vanishing cycles at a point of a scheme in the other.

\emph{From $K_{\rm geom}$ to $\mathbb C$.} For stacks of finite
type over the field $K_{\rm geom}$, the six operations on
constructible complexes commute with the base change to
$\mathbb C$: for direct images this is the generic base change
theorem over the base $\operatorname{Spec}K_{\rm geom}$
\cite[Th.~finitude, Th\'eor\`eme~1.9]{SGA4half}; inverse images,
direct images with compact supports and tensor products commute
with every base change; and the extraordinary inverse image and the
internal $\operatorname{Hom}$ are reduced, by a decomposition into
strata on which the complexes are lisse, to direct images along
immersions and to smooth pullbacks. Moreover the pullback
functor is fully faithful on constructible complexes, the
cohomology of such a complex being unchanged by an extension of
algebraically closed fields. Vanishing cycles commute with the
change of trait from the henselization of
$\mathbb A^1_{K_{\rm geom}}$ at the origin to that of
$\mathbb A^1_{\mathbb C}$
\cite[Th.~finitude, Proposition~3.7]{SGA4half}. Singular supports
are compatible with extension of the base field: for torsion
coefficients this is \cite[Theorem~1.4(iii)]{Beilinson2016}, and for
coefficients in an algebraic extension of $\mathbb Q_\ell$ it is
\cite[Theorem~1.5(xi)]{Barrett2024}. We apply this on smooth charts
of finite type; the nilpotent cone is also compatible with the
extension of the base field, so nilpotence is preserved.

\emph{From \'etale sheaves to the classical topology.} Over
$\mathbb C$ there is a fully faithful functor from constructible
\'etale $k$-sheaves to constructible sheaves of $k$-vector spaces
in the classical topology, which commutes with the direct and
inverse images of both kinds
\cite[\S2.2.1]{GR2025}\cite[\S6.1.2]{BBD1982} and with vanishing
cycles \cite[Expos\'e~XIV, Th\'eor\`eme~2.8]{SGA7II}, and which
preserves nilpotence of the singular support on $\Bun_G(C)$
\cite[\S2.2.2]{GR2025}. Being fully faithful, it preserves
$\operatorname{RHom}$. It sends the complex $\mathcal K_{\rm W}$ to
the complex defined by the same recipe.

\emph{Conclusion.} Let $\operatorname{an}$ denote the composite of
the two functors of comparison, from nilpotent \'etale sheaves on
$\Bun_G(C)$ to nilpotent sheaves in the classical topology on the
stack of bundles of the complex curve. By the two paragraphs above
there are isomorphisms of functors
$\mu\simeq\mu^{\rm cl}\circ\operatorname{an}$ and
$\operatorname{Wh}\simeq\operatorname{Wh}^{\rm cl}\circ\operatorname{an}$.
They are isomorphisms of functors on the homotopy categories which
commute with the shift, and this is all we need (Remark
\ref{rw:rem:homotopy-naturality}). Composing the given isomorphism
with $\operatorname{an}$ gives the assertion.
\end{proof}

The comparison with the classical topology is used only to
produce this isomorphism of functors. The action of Frobenius is
brought in afterwards, on the \'etale side
(Proposition~\ref{fl:prop:composite-trace-nonzero}, step~3).

\emph{From here to the proof of
Proposition~\ref{fl:lem:vacuum-microstalk-comparison}, $C$ is a
smooth proper complex curve, and all sheaves are sheaves of
$k$-vector spaces in the classical topology.} Let
$i:\Bun_N^\omega\to\Bun_G$ be the stack of $N$-bundles twisted by
the torus bundle of $E_*$, and let
$f_N:\Bun_N^\omega\to\mathbb A^1$ be the sum of the coordinates on
the simple root spaces. The one-parameter group $\lambda_{\rm pr}$ acts
on $\Bun_N^\omega$, fixing $E_*$ and contracting to it, and $f_N$
has weight two. So $f_N$ descends to
\[
 \bar f:\Bun_N^\omega/\mathbb G_m\longrightarrow
 [\mathbb A^1/\mathbb G_m],
\]
where $\mathbb G_m$ acts on $\mathbb A^1$ with weight two. Let
$i_{\mathbb G_m}:\Bun_N^\omega/\mathbb G_m\to\Bun_G$ be the map
induced by $i$, let $j:B\mu_2\to[\mathbb A^1/\mathbb G_m]$ be the
open point, and let $k[\mu_2]$ be the regular representation.

Two steps below identify the cohomology of a space which an
action contracts to a point with the stalk at that point. We state
this once, in the form in which it is used.

\begin{lemma}[Contraction by an action with positive weights]
\label{rw:lem:positive-weight-contraction}
Let $\mathbb R_{>0}$ act on $E=\mathbb C^n$ by
\[
 r\cdot(z_1,\ldots,z_n)=(r^{w_1}z_1,\ldots,r^{w_n}z_n)
\]
with integers $w_i>0$, and let $\mathcal F$ be a bounded
constructible complex on $E$ whose cohomology sheaves are locally
constant along the orbits. Then the restriction map
\[
 R\Gamma(E,\mathcal F)\longrightarrow\mathcal F_0
\]
is an isomorphism. More generally, let $f:E\to\mathbb C$ be a
polynomial which is homogeneous of weight $d>0$ for this action,
let $\epsilon>0$, and let $U_\epsilon\subset E$ be the open subset
where $|f|<\epsilon$. Then the restriction map
$R\Gamma(U_\epsilon,\mathcal F)\to\mathcal F_0$ is an isomorphism.
\end{lemma}

These are restriction maps. So they are natural in $\mathcal F$,
and compatible with the restriction from $U_\epsilon$ to
$U_{\epsilon'}$ for $\epsilon'<\epsilon$.

\begin{proof}
Write $z_i=x_i+\sqrt{-1}\,y_i$, and let
$v=\sum_iw_i(x_i\partial_{x_i}+y_i\partial_{y_i})$ be the vector
field of the action. Away from the origin the action is free, and
locally $E$ is the product of an interval of an orbit and a
transversal. The cohomology sheaves of $\mathcal F$ are locally
constant along the orbits, so over $E\setminus\{0\}$ the
microsupport of $\mathcal F$ consists of covectors which vanish on
$v$ \cite[Proposition~4.1.2(ii)]{KS1985}. At the origin $v=0$.

Choose an integer $D>0$ divisible by all the $w_i$, and put
\[
 \rho(z)=\sum_i|z_i|^{2D/w_i}.
\]
This is a polynomial in the $x_i$ and $y_i$. It is proper, it
vanishes only at the origin, and $d\rho(v)=2D\rho$. So away from
the origin $d\rho$ does not vanish on $v$, and the conormal
directions of the level sets of $\rho$ do not meet the microsupport
of $\mathcal F$. The open sets $E_s=\{\rho<s\}$, $s>0$, are
relatively compact, increase with $s$ and exhaust $E$. By the
theorem on noncharacteristic deformations
\cite[Theorem~4.4.1(1)]{KS1985} the restriction maps
\[
 R\Gamma(E,\mathcal F)\longrightarrow R\Gamma(E_s,\mathcal F)
\]
are isomorphisms. The sets $E_s$ form a basis of neighbourhoods of
the origin, so the colimit of the right-hand side as $s\to0$ is
$\mathcal F_0$. This proves the first assertion.

For the second, replace $\rho$ by the function
\[
 \rho_\epsilon=\rho+\frac{|f|^2}{\epsilon^2-|f|^2}
\]
on $U_\epsilon$. It is smooth and vanishes only at the origin. The
set where $\rho_\epsilon\le s$ is compact: on it $\rho\le s$ and
$|f|^2\le s\epsilon^2/(1+s)<\epsilon^2$. Since $|f|^2$ is
homogeneous of weight $2d$,
\[
 d\rho_\epsilon(v)=2D\rho
   +\frac{2d\,\epsilon^2|f|^2}{(\epsilon^2-|f|^2)^2},
\]
which is positive away from the origin. The set $U_\epsilon$ is not
stable under the whole action, but the cohomology sheaves of
$\mathcal F$ are locally constant along the parts of the orbits
which lie in it, and the statement about the microsupport uses
nothing more. The argument then continues as before, with the sets
$\{\rho_\epsilon<s\}$.
\end{proof}

\begin{lemma}[Vanishing cycles on the line with the action of weight two]
\label{rw:lem:weight-two-vanishing-cycles}
For a constructible complex $\mathcal G$ on
$[\mathbb A^1/\mathbb G_m]$, with pullback $\mathcal G'$ to
$\mathbb A^1$, there is an isomorphism
\begin{equation}\label{rw:eq:kirillov-kernel}
 (R\Phi_t\mathcal G')_0[1]\simeq
 \operatorname{RHom}\bigl(j_*k[\mu_2][-1],\mathcal G\bigr),
\end{equation}
natural in $\mathcal G$.
\end{lemma}
\begin{proof}
Complexes on this stack split according to the two characters of
the central subgroup $\mu_2$.

For the trivial character we are on the quotient $\mathcal Q_1$ of
$\mathbb A^1$ by the action of weight one, with open point
$j_1:\operatorname{pt}\to\mathcal Q_1$, closed point
$i_1:B\mathbb G_m\to\mathcal Q_1$ and atlas
$a:\mathbb A^1\to\mathcal Q_1$, and we must show that
$(R\Phi_t\mathcal G')_0\simeq\operatorname{RHom}(j_{1,*}k,\mathcal G)$.
With our normalization, the stalk of the vanishing cycles is the
cone of the specialization map from the stalk $\mathcal G'_0$ to
the stalk $\mathcal G'_1$ at the point $1$; the latter is the stalk
of the nearby cycles, because $\mathcal G'$ is equivariant. Both
stalks are corepresentable functors of $\mathcal G$. Indeed
$\mathcal G'_1=\operatorname{RHom}(j_{1,!}k,\mathcal G)$; and
$\mathcal G'_0=R\Gamma(\mathbb A^1,\mathcal G')$ by
Lemma~\ref{rw:lem:positive-weight-contraction}, and this is
$\operatorname{RHom}(a_!k[2],\mathcal G)$ because $a$ is smooth of
relative dimension one. By the Yoneda lemma the specialization map
is induced by a morphism
$\operatorname{sp}:j_{1,!}k\to a_!k[2]$, and the stalk of the
vanishing cycles is $\operatorname{RHom}(\mathcal T,\mathcal G)$,
where $\mathcal T$ is the fibre of $\operatorname{sp}$. It remains
to see that $\mathcal T\simeq j_{1,*}k$. For
$\mathcal G=j_{1,*}k$ the vanishing cycles are a line in degree
zero, and
$\operatorname{RHom}(\mathcal T,j_{1,*}k)=
\operatorname{RHom}(j_1^*\mathcal T,k)$; so
$j_1^*\mathcal T\simeq k$. The restriction $i_1^*\mathcal T$ is
$i_1^*a_!k[1]=p_!k[1]$, where
$p:\operatorname{pt}\to B\mathbb G_m$, and $i_1^*j_{1,*}k=p_*k$.
These two objects are isomorphic: for a torsor under
$\mathbb G_m$, both are the cone of the Euler class $k[-2]\to k$.
The endomorphisms of degree zero of $p_*k$ are the scalars, since
$p^*p_*k=k\oplus k[-1]$. So the map
$i_1^*\mathcal T\to i_1^*j_{1,*}j_1^*\mathcal T$ that glues
$\mathcal T$ from its two restrictions is either an isomorphism, in
which case $\mathcal T\simeq j_{1,*}k$, or zero, in which case
$\mathcal T\simeq j_{1,!}k\oplus i_{1,*}i_1^*\mathcal T$. The
second case is excluded by taking $\mathcal G=j_{1,!}k$: its
vanishing cycles are a line, whereas $\operatorname{RHom}$ from the
direct sum to $j_{1,!}k$ has total dimension three.

For the nontrivial character, the stalk and the costalk at the
origin vanish, because the central $\mu_2$ acts trivially on every
complex on $B\mathbb G_m$; so $\mathcal G$ is the direct image of
its restriction $V$ to the open point, and both sides of
\eqref{rw:eq:kirillov-kernel} are $V[1]$.
\end{proof}

\begin{lemma}[A linear chart of the stack of twisted unipotent bundles]
\label{rw:lem:twisted-unipotent-chart}
There are an affine space with a linear action of $\mathbb G_m$,
all of whose weights are positive, and a map from it to
$\Bun_N^\omega$, equivariant for the action of $\lambda_{\rm pr}$, which
is a torsor under a unipotent group and maps the origin to $E_*$.
\end{lemma}
\begin{proof}
Let $N^\omega$ be the group scheme over the curve obtained by
twisting $N$ by the fixed torus bundle, so that $\Bun_N^\omega$
classifies its torsors. Filter $N$ by the height of the roots. The
successive quotients of $N^\omega$ are central, and they are the
direct sums of the line bundles $\mathcal L_\alpha$ attached to the
roots $\alpha$ of a given height. Choose an effective divisor $Q$
so large that $H^0(\mathcal L_\alpha(-Q))=0$ for all positive roots
$\alpha$, and let $\Bun_{N,Q}^\omega$ classify the torsors with a
trivialization along $Q$. Lifting a torsor with its trivialization
from one quotient of the filtration to the next meets no
obstruction, since $H^2$ vanishes on a curve; the lifts have no
automorphisms; and they form a torsor under the vector group
$\bigoplus_\alpha H^1(\mathcal L_\alpha(-Q))$, the sum being over
the roots of the given height. So $\Bun_{N,Q}^\omega$ is obtained
from a point by successive torsors under vector groups. The group
$\lambda_{\rm pr}$ acts on all these data, and on the vector group of
the roots of height $h$ with weight $2h$. A torsor under a vector
group over an affine scheme is trivial, and it has a
trivialization invariant under $\lambda_{\rm pr}$: its sections form an
affine space under a rational representation of $\mathbb G_m$, and
$\mathbb G_m$ is linearly reductive. By induction
$\Bun_{N,Q}^\omega$ is an affine space with a linear action of
positive weights, whose origin is the trivial torsor. The map
$\Bun_{N,Q}^\omega\to\Bun_N^\omega$ which forgets the
trivialization is a torsor under the group of sections of
$N^\omega$ over $Q$, a unipotent group. It commutes with the
action of $\lambda_{\rm pr}$, and it maps the origin to the point of
$\Bun_N^\omega$ defined by the trivial torsor, whose image in
$\Bun_G$ is $E_*$.
\end{proof}

\begin{lemma}[Vanishing cycles of the homogeneous function]
\label{rw:lem:homogeneous-vanishing-cycles}
For a constructible complex $\mathcal F$ on $\Bun_G$ there is an
isomorphism
\[
 (R\Phi_{f_N}i^!\mathcal F)_{E_*}[-1]\simeq
 \operatorname{RHom}(\mathcal P_1,\mathcal F),\qquad
 \mathcal P_1=i_{\mathbb G_m,!}\,\bar f^*\bigl(j_*k[\mu_2]\bigr)[-1],
\]
natural in $\mathcal F$.
\end{lemma}
\begin{proof}
Put $\mathcal J=i_{\mathbb G_m}^!\mathcal F$, a complex on the
quotient. Its pullback $\mathcal J'$ to $\Bun_N^\omega$ is
$i^!\mathcal F[-2]$, because the quotient map is smooth of relative
dimension one. By adjunction and
Lemma~\ref{rw:lem:weight-two-vanishing-cycles},
\[
 \operatorname{RHom}(\bar f^*j_*k[\mu_2][-1],\mathcal J)
 \simeq\bigl(R\Phi_t(f_{N,*}\mathcal J')\bigr)_0[1],
\]
the stalk at the origin of the vanishing cycles of the direct image
of $\mathcal J'$ under $f_N$; here smooth base change identifies
the pullback of $\bar f_*\mathcal J$ with $f_{N,*}\mathcal J'$.

The map $f_N$ is not proper, and vanishing cycles do not commute
with a direct image which is not proper. What makes the comparison
possible here is the contraction: we claim that
\[
 \bigl(R\Phi_t(f_{N,*}\mathcal J')\bigr)_0
 \simeq(R\Phi_{f_N}\mathcal J')_{E_*},
\]
naturally in $\mathcal J'$. Both sides are cones of specialization
maps. On the left the stalk of $f_{N,*}\mathcal J'$ at $0$ is the
cohomology of the inverse image of a small disc, which the action
contracts to $E_*$; so it is $\mathcal J'_{E_*}$, by
Lemma~\ref{rw:lem:positive-weight-contraction} applied on the chart
of Lemma~\ref{rw:lem:twisted-unipotent-chart}. The nearby stalk is the
cohomology of the whole fibre $f_N^{-1}(1)$, because over the
punctured line $f_N$ is a product, after a covering of degree two,
by homogeneity and equivariance. On the right the stalk is again
$\mathcal J'_{E_*}$, and the nearby stalk is the cohomology of a
Milnor fibre, which the action identifies with the part of
$f_N^{-1}(1)$ where a norm $\nu$ is less than a radius that tends
to infinity as the value of $f_N$ tends to $0$. So the claim
amounts to this: for $R$ large the restriction
\[
 R\Gamma(f_N^{-1}(1),\mathcal J')\longrightarrow
 R\Gamma(f_N^{-1}(1)\cap\{\nu<R\},\mathcal J')
\]
is an isomorphism.

We prove this on the chart of
Lemma~\ref{rw:lem:twisted-unipotent-chart}. The map from the chart
to $\Bun_N^\omega$ is smooth, so nearby cycles commute with
pullback along it, and its fibres are contractible, so for a
complex pulled back from $\Bun_N^\omega$ it preserves the
cohomology of the fibre of $f_N$ and of the inverse image of the
disc. On the affine space, $\nu$ is the maximum of the functions
$|x_i|^{1/w_i}$, where $w_i>0$ is the weight of the coordinate
$x_i$. It is a proper semialgebraic function, and $\mathcal J'$ is
constructible for a finite algebraic stratification. We claim that
the complex $R\nu_*\mathcal J'$ on the half-line is locally
constant on an interval $(R_0,\infty)$. Put $u=\nu/(1+\nu)$, a
function with values in $[0,1)$, and let $\Gamma$ be the graph of
$u$ on the fibre $f_N^{-1}(1)$, as a subset of
$\mathbb P\times\mathbb R$, where $\mathbb P$ is the projective
space which contains the affine space. It is a semialgebraic set
with compact closure. So the extension by zero of $\mathcal J'$
from $\Gamma$ to $\mathbb P\times\mathbb R$ is an
$\mathbb R$-constructible complex with compact support, and its
direct image on $\mathbb R$ is $\mathbb R$-constructible
\cite[Proposition~8.3.1]{KS1985}. In particular this direct image
is locally constant on an interval $(1-\delta,1)$. Since $\nu$ is
proper, the points of the closure of $\Gamma$ which do not belong
to $\Gamma$ lie over $u=1$; so over $[0,1)$ the direct image is
$Ru_*\mathcal J'$, and the claim follows. Covering the half-line by
$[0,R)$ and $(R_0,\infty)$ now shows that the restriction to
$\nu<R$ is an isomorphism for $R>R_0$.

The isomorphism so obtained is natural in $\mathcal J'$. For each
$R$ the restriction map displayed above is a natural transformation
between functors of $\mathcal J'$, and these transformations are
compatible when $R$ increases. The action identifies the nearby
stalk with the right-hand side for all large $R$, compatibly with
the restriction maps between different radii. So there is one
natural transformation from the cohomology of the fibre
$f_N^{-1}(1)$ to the nearby stalk; what depends on $\mathcal J'$ is
only the radius beyond which the individual restriction maps are
isomorphisms. Together, and with
$\mathcal J'=i^!\mathcal F[-2]$, this gives the statement. It says
that the Whittaker functional is corepresented by the Whittaker
sheaf \cite[Remark~1.1.1]{NT2026}.
\end{proof}

\begin{lemma}[The kernel of the Poincar\'e object]
\label{rw:lem:poincare-kernel}
The object $\mathcal P_1$ of
Lemma~\ref{rw:lem:homogeneous-vanishing-cycles} is isomorphic to
$\operatorname{Poinc}^{\rm Vac}_!$.
\end{lemma}
\begin{proof}
The object $\operatorname{Poinc}^{\rm Vac}_!$ is defined in the
same way as $\mathcal P_1$, with the torus $T$ in place of
$\lambda_{\rm pr}$, the map $\chi:\Bun_N^\omega\to\mathbb A^r$ to the
simple root spaces in place of
$f_N=\operatorname{sum}\circ\chi$, and the kernel
$\mathcal K_T=j_{{\rm reg},*}R_{Z(G)}[-r]$ on $[\mathbb A^r/T]$,
where $r$ is the rank of $G$,
$j_{\rm reg}:BZ(G)\to[\mathbb A^r/T]$ is the open point and
$R_{Z(G)}$ is the direct image with compact supports of $k$ under
$\operatorname{pt}\to BZ(G)$ \cite[\S\S3.3.3--3.3.7]{GLCI2024}. By
base change along $\chi$ it suffices to prove that
\begin{equation}\label{rw:eq:kernel-comparison}
 q_!\operatorname{sum}^*\bigl(j_*k[\mu_2][-1]\bigr)\simeq
 \mathcal K_T,\qquad
 q:[\mathbb A^r/\mathbb G_m]\to[\mathbb A^r/T].
\end{equation}
Both sides are obtained by the six operations from constant
sheaves, so they are defined with coefficients in $\mathbb Q$, on
the same analytic stack. For two bounded constructible complexes
with coefficients in $\mathbb Q$ on $[\mathbb A^r/T]$, which has
finitely many orbits, the space $\operatorname{Hom}^0$ has finite
dimension and commutes with the extension of the coefficients to
$\mathbb C$. The isomorphisms form a Zariski open subset of this
space, defined over $\mathbb Q$, because a morphism is an
isomorphism if it is one on the stalks at a point of each orbit;
and such an open subset has a rational point if it has a complex
point. So we may take $k=\mathbb C$ and use $D$-modules. Under the
Riemann--Hilbert correspondence $\mathcal K_T$ corresponds to the
direct image with compact supports, under
$\mathbb A^r\to[\mathbb A^r/T]$, of the pullback of the exponential
$D$-module under the sum \cite[Remark~3.3.4]{GLCI2024}. By base
change and transitivity of direct images,
\eqref{rw:eq:kernel-comparison} follows once we know that
$j_*k[\mu_2][-1]$ corresponds to the direct image $\mathcal E_2$
with compact supports of the exponential $D$-module under
$\mathbb A^1\to[\mathbb A^1/\mathbb G_m]$, for the action of weight
two. This direct image is regular holonomic: on the open orbit
its pullback by the orbit map $z\mapsto z^2$ is a trivial
connection, so it is a sum of regular Kummer connections, and the
only other stratum is the origin.
For the trivial character of $\mu_2$ we are in the case of
weight one, treated in \cite[Remark~3.3.2]{GLCI2024}. The part of
$\mathcal E_2$ with nontrivial character is the direct image of its
restriction to the open point, as in the proof of
Lemma~\ref{rw:lem:weight-two-vanishing-cycles}. That restriction is
the cohomology with compact supports of $\mathbb G_m$ with
coefficients in the product of the exponential $D$-module and the
local system of order two. It is dual to the cohomology of the
complex $h\mapsto h'-(1+1/(2x))h$ on Laurent polynomials. The
kernel of this operator is zero, as one sees from the term of
highest degree. Its cokernel is one-dimensional: the relations
$[x^n]=(n-\tfrac12)[x^{n-1}]$, for all integers $n$, show that it
is spanned by the class of $1$, and that class is not zero, because
the linear form $x^n\mapsto c_n$ with $c_0=1$ and
$c_n=(n-\tfrac12)c_{n-1}$ vanishes on the image. So
the restriction is the nontrivial character, in the same degree as
for the trivial character. This proves
\eqref{rw:eq:kernel-comparison}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{fl:lem:vacuum-microstalk-comparison}]
By Lemma~\ref{rw:lem:microstalk-classical-reduction} it suffices
to construct an isomorphism
$\mu^{\rm cl}\simeq\operatorname{Wh}^{\rm cl}\otimes L_{\rm shift}$
for a complex curve; we drop the superscripts.

\step{1}{the theorem of Nadler and Taylor} By
Lemma~\ref{fl:lem:graded-morse-chart} the graph of $df$ meets the
nilpotent cone transversally at the smooth point
$(E_*,\theta_*)$. For such a function the functor of vanishing
cycles factors through the category of sheaves localized at this
point of the cotangent bundle, which is equivalent to the category
of complexes of vector spaces
\cite[Proposition~2.3.1 and its proof]{NT2026}; so $\mu$ is the
microstalk at $(E_*,\theta_*)$, up to a shift. We need the
comparison of the functors attached to two transverse functions as
an isomorphism of functors, natural in the sheaf, and not only for
each sheaf separately. It is. Choose once a contact transformation
near the point, with its kernel. The two functors defined by the
kernel are inverse to each other
\cite[Theorems~6.3.4 and~6.3.5]{KS1985}, and the kernel is
determined up to tensor product with an invertible complex of
coefficients \cite[Remark~6.3.6]{KS1985}, which over the field $k$
is a graded line; the coefficients in that section are a ring of
finite weak global dimension, so $k$ is allowed. After the
transformation the functor attached to a transverse function is the
tensor product with a graded line which depends only on the
function \cite[Lemmas~7.2.3 and~7.2.4]{KS1985}. The comparison of
two transverse functions is then the choice, made once, of an
isomorphism between two graded lines. By
Input~\ref{rw:in:microstalk}, which is
\cite[Theorem~3.3.4]{NT2026}, the microstalk at the same point is
also computed, up to a shift, by
$\mathcal F\mapsto(R\Phi_{f_N}i^!\mathcal F)_{E_*}$.

\step{2}{conclusion} By
Lemma~\ref{rw:lem:homogeneous-vanishing-cycles} the last functor
is, up to a shift, $\operatorname{RHom}(\mathcal P_1,-)$, and by
Lemma~\ref{rw:lem:poincare-kernel} this is $\operatorname{Wh}$. So
$\mu\simeq\operatorname{Wh}\otimes L_{\rm shift}$, where the graded
line $L_{\rm shift}$ collects the shifts. It does not depend on the
sheaf, and the isomorphism is one of functors: the isomorphism of
step~1 is one of functors, that of
Lemma~\ref{rw:lem:homogeneous-vanishing-cycles} is natural, and
Lemma~\ref{rw:lem:poincare-kernel} is an isomorphism between two
objects which corepresent functors, so it induces a natural
isomorphism between them whatever isomorphism is chosen. The
remaining ambiguity is a scalar, by
Lemma~\ref{fl:lem:scalar-coefficient-covariance}.
\end{proof}