\section{Excursion operators on functions with compact support}
\label{fl:sec:compact-foundations}
\label{fl:sec:canonical-compact}

The constructions of \cite{HuII} and
\cite{HuIII} produce functions on
$G(F)\backslash G(\mathbb A_F)$, not automorphic representations,
and at the moment they are produced these functions are not known
to be cuspidal. Lafforgue's excursion operators are defined on cusp
forms. To say that a function has parameter $\sigma$, and to prove
afterwards that it is cuspidal, we need excursion operators on a
larger space. This section and the next one construct them.

The starting point is Xue's extension of the theory of shtukas from
the cuspidal part to cohomology with compact support
(Input~\ref{rw:in:xue}). It gives excursion operators on functions
with compact support, for tuples of elements of the Weil group. Two
things are missing for our purposes, and each occupies one
section.
\begin{itemize}
\item Xue's results are stated after dividing by a lattice $\Xi$ in
the centre. Our functions live on $G(F)\backslash G(\mathbb A_F)$
itself, where the connected components are indexed by all degrees.
In this section we remove the lattice
(Proposition~\ref{fl:prop:canonical-compact}), prove that the
operators are transposed by the natural pairing
(Proposition~\ref{fl:prop:compact-excursion-transpose}), and use
the transpose to let them act on arbitrary functions of finite
level (Definition~\ref{fl:def:function-excursion-action}).
\item The parameter $\sigma$ is a representation of the Galois
group, and its excursion character involves all Galois tuples. The
passage from Weil tuples to Galois tuples needs integral structures
and is the subject of Section~\ref{rw:sec:function-spaces-section}.
\end{itemize}
In this section and the two following ones the space on which the
operators act, and the tuples for which they are defined, change
several times. The table records where one is at each point.
\begin{center}
\small
\begin{tabular}{@{}p{0.47\textwidth}p{0.12\textwidth}p{0.31\textwidth}@{}}
\toprule
space & tuples in & statement\\
\midrule
functions with compact support of level $N$ & $W_F^I$ &
Proposition~\ref{fl:prop:canonical-compact}\\
all functions of level $N$, by transposition & $W_F^I$ &
Definition~\ref{fl:def:function-excursion-action}\\
finite Hecke quotients of the functions with compact support &
$\Gamma_F^I$ & Theorem~\ref{fl:thm:finite-integral-excursion}\\
$\mathcal A^{\mathrm{cont}}(G)$, made of duals of such quotients &
$\Gamma_F^I$ & Theorem~\ref{fl:thm:function-space}\\
constant terms of all functions of finite level & $W_F^I$ &
Theorem~\ref{fl:thm:normalized-ct}\\
constant terms of elements of duals of finite quotients &
$\Gamma_F^I$ & Corollary~\ref{fl:cor:finite-model-ct}\\
\bottomrule
\end{tabular}
\end{center}

\subsection{Notation}

Fix a split connected reductive group $G/F$. For a finite subscheme
$N\subset X$, the \emph{level}, write
\[
 K_N=\ker\bigl(G(\textstyle\prod_v\mathcal O_v)\to G(\mathcal O_N)\bigr),
 \qquad Y_N=G(F)\backslash G(\mathbb A)/K_N.
\]
Let $E/\mathbb Q_\ell$ be a finite extension containing the chosen
square root $q^{1/2}$, with ring of integers $\mathcal O=\mathcal O_E$.
The modules of functions with finite support on $Y_N$ are denoted
\[
 C_{N,\mathcal O}=\mathcal O^{(Y_N)},\qquad C_{N,E}=E^{(Y_N)};
\]
we call their elements \emph{functions with compact support}. For a
place $u\notin N$, let $\mathscr H_{G,u,\mathcal O}$ be the
spherical Hecke algebra with coefficients in $\mathcal O$, in which
$G(\mathcal O_u)$ has volume one. The normalized Satake isomorphism
identifies it with the representation ring of the split dual group
over $\mathcal O$; the same square root $q^{1/2}$ is used for every
Levi subgroup.

For a function $c$ with compact support and an arbitrary function
$h$ on $Y_N$, put
\[
 [c,h]_N=\sum_{y\in Y_N}w_N(y)c(y)h(y),
 \qquad
 w_N([g])=\frac{\operatorname{vol}(K_N)}
 {|G(F)\cap gK_Ng^{-1}|}.
\]
The sum is finite. We omit the subscript when the level is fixed,
and use the same notation on the quotient of $Y_N$ by a central
lattice, with the corresponding weights. Over $E$ or over
$k$ this pairing identifies the space of all functions with the
algebraic dual of the space of functions with compact support.

A \emph{central lattice} is a subgroup $\Xi\subset C_{T_G}$ as in
Definition~\ref{fl:def:central-period-lattice}, and the
\emph{$\Xi$-coinvariants} of a module of functions, or of a
cohomology group, are its coinvariants under translation by $\Xi$;
geometrically, they are the corresponding object for the quotient
of the stack of shtukas by $\Xi$.

Let $I$ be a finite set, $V$ a representation of $\widehat G^I$
and $V_{\mathcal O}$ a stable $\mathcal O$-lattice in $V$. The
cohomology of shtukas with legs indexed by $I$ is
\[
 \mathcal H^j_{N,I,V_{\mathcal O}}
 =\bigoplus_{\delta\in\Lambda_G}\varinjlim_\mu
 R^j(p_{I,\mu,\delta})_!\mathcal F_{I,V_{\mathcal O}},
 \qquad
 H^j_{N,I,V_{\mathcal O}}
 =(\mathcal H^j_{N,I,V_{\mathcal O}})_{\bar\eta_I}.
\]
Here the degree of the first bundle of a shtuka is an element of
$\Lambda_G=X_*(G/G_{\mathrm{der}})$; $p_{I,\mu,\delta}$ is the map to
$(X-N)^I$
from the part of the stack of shtukas of level $N$ where that
degree is $\delta$ and the Harder--Narasimhan polygon is bounded by
$\mu$, the sheaf $\mathcal F_{I,V_{\mathcal O}}$ is the Satake
sheaf in the normalization relative to $X^I$ of
\cite[\S1.1.9]{XueIntegral2023}, and $\bar\eta_I$ is a geometric
generic point of $X^I$.

With coefficients in $\mathcal O$ the level $N$ is assumed to be
nonempty. The reason is the following property of the automorphism
groups $A_y=G(F)\cap gK_Ng^{-1}$ of the points $y=[g]$ of $Y_N$.

\begin{lemma}[Automorphism groups at nonempty level]
\label{rw:rem:nonempty-level}
If $N\ne\varnothing$, every group $A_y$ is a finite $p$-group.
\end{lemma}
\begin{proof}
The group $A_y$ is discrete and compact, hence finite. Let
$a\in A_y$, regarded as an automorphism of a bundle with a
trivialization over $N$, and take a faithful representation of $G$
of dimension $n$. The coefficients of the characteristic polynomial
of $a$ on the associated vector bundle are regular functions on the
proper, geometrically connected curve $X$, hence constants. At a
point of $N$ the automorphism is the identity, so the polynomial is
$(T-1)^n$. By the Cayley--Hamilton theorem $(a-1)^n=0$, and
$a^{p^k}=1$ as soon as $p^k\ge n$.
\end{proof}

So $|A_y|$ is a unit of $\mathcal O$, and the cohomology with
compact supports of the classifying groupoid of $A_y$ with
coefficients in $\mathcal O$ is $\mathcal O$ in degree zero and
zero in the other degrees; if $|A_y|$ were not a unit, the torsion
in the cohomology or the homology of $A_y$ could enter. It follows
that for $I=\varnothing$, $V=\mathbf 1$ and $j=0$ the module above
is $C_{N,\mathcal O}$: the constant section on the component of $y$
corresponds to the characteristic function of $y$, and the factor
$|A_y|^{-1}$ is put in the pairing by integration, through the
weights $w_N$.

Functions of empty level are included in this setting. They are
the functions of level $N$ invariant under the finite group
$G(\mathcal O_N)$, which commutes with the Hecke algebra at a place
outside $N$ and with the excursion operators. Since a submodule of
a finitely generated module over a noetherian algebra is finitely
generated, the finiteness statements below pass to these invariants
with coefficients in $\mathcal O$. The projection onto the
invariants by averaging divides by $|G(\mathcal O_N)|$; we use it
only with coefficients in $E$.

\subsection{Removing a central lattice}

Xue fixes a discrete lattice in the centre
\cite[Convention~0.0.6]{XueIntegral2023}, and his finiteness
theorem \cite[Theorem~5.0.4]{XueIntegral2023} concerns the quotient
of the stack of shtukas by that lattice. We need the same statement
for the direct sum over all degrees. The passage rests on two
facts: translation by a central element identifies the components
of different degrees, and such translations are invertible elements
of a spherical Hecke algebra.

\begin{lemma}[Central translations]
\label{fl:lem:central-hecke-units}
At a fixed principal level, every translation by $T_G(\mathbb A)$
is a product of invertible central elements of spherical Hecke
algebras at places outside the level. Translation by
$\lambda(\varpi_v)$ corresponds under the Satake isomorphism to the
character $\alpha_G^*\lambda$ of $\widehat G$. The same holds for
the action on the modules $H^j_{N,I,V_{\mathcal O}}$ and on their
coinvariants under central lattices.
\end{lemma}

\begin{proof}
Let $t\in T_G(\mathbb A)$. By weak approximation in the split torus
$T_G$ there is $t_0\in T_G(F)$ such that $t_0^{-1}t$ lies in $K_N$
at the places of $N$. Translation by a rational central element is
the identity on functions on $G(F)\backslash G(\mathbb A)$, and on
shtukas it is an isomorphism of all the data (bundles, level
structures, modifications and Frobenius); so we may replace $t$ by
$t_0^{-1}t$. Translation by an element of
$T_G(\prod_v\mathcal O_v)\cap K_N$ is trivial at level $N$. What
remains is a finite product of elements $\lambda(\varpi_v)$ with
$v\notin N$ and $\lambda$ a cocharacter of $T_G$.

Translation by $\lambda(\varpi_v)$ is the Hecke operator of the
double coset of $\lambda(\varpi_v)$, which is a single coset
because the element is central. The corresponding Schubert variety
is a point and the modulus character is trivial on it, so under
the normalized Satake isomorphism the operator corresponds to the
one-dimensional representation $\alpha_G^*\lambda$ of $\widehat G$.
It is invertible, its inverse being the operator of $-\lambda$. On
stacks of shtukas the Hecke correspondence of this double coset is
the graph of the translation. The operator is the pullback along
the translation by $\lambda(\varpi_v)$, so it moves the supports
in the direction opposite to that in which the translation moves
points. This gives the assertion for the modules
$H^j_{N,I,V_{\mathcal O}}$ and for their coinvariants.
\end{proof}

\begin{lemma}[An exact degree in a central quotient]
\label{fl:lem:central-degree-stacks}
Put $\Lambda_G=X_*(Q_G)$, where $Q_G=G/G_{\mathrm{der}}$,
and grade the shtuka stack by the degree $\delta\in\Lambda_G$
of its first bundle. Fix a place $u\notin N$. The central ideles
$\lambda(\varpi_u)$, for $\lambda\in X_*(T_G)$, generate a
central lattice $\Xi_u$. Their degree shifts form
\[
 \Lambda_u=-\deg(u)(r_G)_*X_*(T_G)\subset\Lambda_G,
 \qquad r_G:T_G\longrightarrow Q_G.
\]
For every adjoint Harder--Narasimhan bound $\mu$ and every
$\delta$, the quotient map induces an equivalence
\begin{equation}\label{fl:eq:exact-degree-central-quotient}
 \operatorname{Cht}^{\le\mu}_{G,N,I,V,\delta}
 \ \xrightarrow{\ \sim\ }
 \bigl(\operatorname{Cht}^{\le\mu}_{G,N,I,V}/\Xi_u\bigr)_{
                                           \delta\bmod\Lambda_u}.
\end{equation}
The equivalence is over the leg base, preserves the Satake
coefficient, and is compatible with increasing $\mu$.
It also holds with $m\Xi_u$ and $m\Lambda_u$ for every $m\ge1$.
\end{lemma}

\begin{proof}
Since $r_G$ is an isogeny, $\lambda\mapsto-\deg(u)(r_G)_*\lambda$
is injective and $\Lambda_u$ has finite index in $\Lambda_G$.
Twisting a $G$-bundle by the $T_G$-bundle attached to
$\lambda(\varpi_u)$ changes its degree in $\Lambda_G$ by
$-\deg(u)(r_G)_*\lambda$ and does not change the associated bundle
for the adjoint group, so it preserves the bound $\mu$. Applying
the same twist to all the bundles of a shtuka is compatible with
the modifications, the level structure and the Frobenius
isomorphism. Thus $\Xi_u$ acts on the stack of shtukas, and
$\lambda(\varpi_u)$ maps the part of degree $\delta$ isomorphically
onto the part of degree $\delta-\deg(u)(r_G)_*\lambda$.

Consider the union of the parts whose degree lies in
$\delta+\Lambda_u$. By the injectivity just noted, each of them is
the translate of the part of degree $\delta$ by exactly one element
of $\Xi_u$. So every object of the quotient of this union by
$\Xi_u$ is isomorphic to an object of degree $\delta$, and a
morphism in the quotient between two objects of degree $\delta$
involves the translation by an element of $\Xi_u$ which does not
change the degree, that is, by the identity. Hence the functor
\eqref{fl:eq:exact-degree-central-quotient} is essentially
surjective and fully faithful; in particular it preserves
automorphism groups. The part of degree $\delta$ may
have several connected components, indexed by a subset of
$\pi_1(G)$; all of them are kept on both sides.

The Satake sheaf is defined in terms of the relative positions of
the bundles of a shtuka at the legs, which are not changed by a
simultaneous central twist; so it corresponds to itself under the
equivalence. Everything commutes with restriction to open subsets
of $X^I$ and with enlarging $\mu$, and the argument applies without
change to the subgroup $m\Xi_u$.
\end{proof}

On cohomology this equivalence takes the following algebraic form.
The lemma has two uses: part (2) transfers finite generation from the
coinvariants to the whole module, and part (3) allows identities
between operators to be checked on coinvariants.

\begin{lemma}[Graded modules and central translations]
\label{fl:lem:graded-central-period}
Let $\mathscr R$ be an algebra graded by a free abelian group
$\Gamma$, and let $\mathcal V=\bigoplus_{\delta\in\Gamma}
\mathcal V_\delta$ be a graded $\mathscr R$-module. Suppose a
finite-index sublattice $\Lambda\subset\Gamma$ acts by central
homogeneous units $z_\lambda\in\mathscr R_\lambda$, with
$z_{\lambda+\lambda'}=z_\lambda z_{\lambda'}$.
Write $\mathcal V_{\Lambda}=\mathcal V/
\sum_{\lambda\in\Lambda}(z_\lambda-1)\mathcal V$.
Then the following statements hold.
\begin{enumerate}
\item The map $\mathcal V_\delta\to
(\mathcal V_{\Lambda})_{\bar\delta}$ is an isomorphism.
\item If $\mathcal V_{\Lambda}$ is finite over $\mathscr R$,
then $\mathcal V$ is finite over $\mathscr R$.
\item A vector of $\mathcal V$ that maps to zero in
$\mathcal V_{m\Lambda}$ for every $m\ge1$ is zero.
Consequently compatible homomorphisms or operator identities
can be tested on all these modules of coinvariants.
\end{enumerate}
\end{lemma}

\begin{proof}
(1) The quotient $\mathcal V_\Lambda$ is graded by $\Gamma/\Lambda$,
and its part of degree $\bar\delta$ is a quotient of
$\bigoplus_{\lambda\in\Lambda}\mathcal V_{\delta+\lambda}$. On this
sum define
\begin{equation}\label{fl:eq:degree-recentering}
 (v_{\delta+\lambda})_{\lambda\in\Lambda}
       \longmapsto\sum_{\lambda\in\Lambda}z_{-\lambda}
                                      v_{\delta+\lambda}
 \in\mathcal V_\delta .
\end{equation}
The sum is finite. The map vanishes on the elements
$(z_\lambda-1)v$, so it factors through
$(\mathcal V_\Lambda)_{\bar\delta}$, and it is inverse to the map
of the statement in both directions.

(2) Choose finitely many generators of $\mathcal V_\Lambda$,
homogeneous for the grading by $\Gamma/\Lambda$, and lift them to
homogeneous elements $v_1,\ldots,v_s$ of $\mathcal V$, of degrees
$\delta_j$. Let $v\in\mathcal V_\delta$, and write its image as
$\sum_ja_j\bar v_j$ with $a_j\in\mathscr R$. Decomposing the $a_j$
into homogeneous components $a_{j,\eta}$ and keeping only the
terms of degree $\bar\delta$, we may assume
$\eta+\delta_j-\delta\in\Lambda$ for all terms. The element
\[
 \sum_{j,\eta}z_{\delta-\eta-\delta_j}a_{j,\eta}v_j
\]
of $\mathcal V$ then has degree exactly $\delta$ and the same
image as $v$ in $\mathcal V_\Lambda$, so it equals $v$ by (1). Thus
$v_1,\dots,v_s$ generate $\mathcal V$.

(3) A vector has components in finitely many degrees. For $m$
large these degrees are distinct modulo $m\Lambda$, and by (1),
applied to $m\Lambda$, each component maps injectively to
$\mathcal V_{m\Lambda}$. So a vector whose image in every
$\mathcal V_{m\Lambda}$ is zero is zero. For the last statement
apply this to the value of the difference of two operators on a
given vector.
\end{proof}

\begin{proposition}[Finiteness over one spherical Hecke algebra]
\label{fl:prop:classical-one-place-finiteness}
For every place $u\notin N$, every finite set $I$, every
representation $V$ with lattice $V_{\mathcal O}$ and every $j$, the
module $H^j_{N,I,V_{\mathcal O}}$ is finitely generated over
$\mathscr H_{G,u,\mathcal O}$. So are its coinvariants under
any central lattice. In particular $C_{N,\mathcal O}$ is
finitely generated over $\mathscr H_{G,u,\mathcal O}$.
\end{proposition}

\begin{proof}
For the $\Xi_u$-coinvariants the finite
generation is \cite[Theorem~5.0.4]{XueIntegral2023}; for
$I=\varnothing$ the module is the space of functions with compact
support, as in Example~2.3.13 of that paper. To pass to the whole
module we apply Lemma~\ref{fl:lem:graded-central-period}. The
algebra $\mathscr H_{G,u,\mathcal O}$ is graded by $\Lambda_G$,
according to the change of the degree of a bundle, and
$H^j_{N,I,V_{\mathcal O}}$ is a graded module over it. By
Lemma~\ref{fl:lem:central-hecke-units} the translations by the
elements of $\Xi_u$ are central homogeneous units of the algebra,
and by Lemma~\ref{fl:lem:central-degree-stacks} the cohomology of
the quotient by $\Xi_u$ is the module of coinvariants. So part (2)
of Lemma~\ref{fl:lem:graded-central-period} gives the finite
generation of $H^j_{N,I,V_{\mathcal O}}$.

For another central lattice $\Xi$, the same identification shows
that the cohomology of the quotient by $\Xi$ is the module of
$\Xi$-coinvariants of $H^j_{N,I,V_{\mathcal O}}$, a quotient of a
finitely generated module. All of this is done with coefficients in $\mathcal O$.
\end{proof}

\begin{remark}\label{fl:rem:deep-periods}
An identity between operators cannot always be checked on the
coinvariants under one lattice. For example, $z_\lambda-1$ becomes zero after quotienting by
$\Lambda$ although it is nonzero before the quotient. Exact-degree
maps can be compared on the coinvariants under one lattice; an identity involving several
degree shifts is compared at all $m\Xi_u$ and then uses
Lemma~\ref{fl:lem:graded-central-period}(3). This distinction will
be used for the excursion identities.
\end{remark}

\subsection{The Weil action and its transpose}

Recall that $W_F=\deg_F^{-1}(\mathbb Z)$ is dense in $\Gamma_F$ and
contains every inertia group. In this section an excursion datum
has its tuple in $W_F^I$. Its function is an invariant regular
function
$f\in E[\widehat G\backslash\widehat G^I/\widehat G]$.
It can be expressed as $f((g_i))=\xi((g_i)x)$, with $x$ and
$\xi$ invariant under the diagonal group in a representation and
its dual.  The corresponding operator creates $x$, transports it
along $(\gamma_i)\in W_F^I$, and contracts with $\xi$.

\begin{definition}[Excursion data]
\label{fl:def:weil-excursion-datum}
Write $e=(I,f,\boldsymbol\gamma)$ for such a datum, and let
$\vartheta_G$ be the pinned Chevalley involution of $\widehat G$.
Define
\begin{equation}\label{fl:eq:star-label}
 e^\star=(I,f^\dagger,\boldsymbol\gamma^{-1}),
 \qquad
 f^\dagger((g_i))=f((\vartheta_G(g_i^{-1}))).
\end{equation}
\end{definition}

\begin{lemma}[Specialization and the Weil action for all degrees]
\label{fl:lem:unperiodized-specialization}
The specialization map from the cohomology at the geometric generic
point of the diagonal to $H^j_{N,I,V_{\mathcal O}}$ is an
isomorphism, and it commutes with central translations. The action
of partial Frobenius maps and of the fundamental group on
$H^j_{N,I,V_{\mathcal O}}$ factors through $W_F^I$.
\end{lemma}

\begin{proof}
Let $H_\Delta$ and $H_\eta$ be the cohomology at the geometric
generic point of the diagonal and at $\bar\eta_I$. The
specialization map $H_\Delta\to H_\eta$ preserves the degree
$\delta$ and commutes with central translations. For each $\delta$
it fits into the square
\[
\begin{tikzcd}[column sep=large]
 H_{\Delta,\delta}\arrow[r,"\mathrm{sp}^*"]
   \arrow[d,"\sim"'] & H_{\eta,\delta}\arrow[d,"\sim"]\\
 (H_\Delta/\Xi_u)_{\bar\delta}
   \arrow[r,"\mathrm{sp}^*"']&(H_\eta/\Xi_u)_{\bar\delta},
\end{tikzcd}
\]
in which $H/\Xi_u$ is the cohomology of the quotient by $\Xi_u$,
or equivalently the module of coinvariants, and the vertical maps
are the isomorphisms of Lemma~\ref{fl:lem:central-degree-stacks}.
The lower map is an isomorphism by
\cite[Proposition~8.1.2]{XueIntegral2023}, so the upper one is.
Taking the sum over $\delta$ gives the first assertion.

The partial Frobenius morphisms of the stacks of shtukas act on
the cohomology groups once the limit over $\mu$ has been taken. A
weight of the centre of $\widehat G$ is an element of
$X^*(Z(\widehat G))=\pi_1(G)$, and it has an image in $\Lambda_G$
under the natural map $\pi_1(G)\to\Lambda_G$; this map is
surjective with finite kernel, and it is not an isomorphism in
general (for $\mathrm{PGL}_2$ the source is $\mathbb Z/2$ and the
target is zero). On the part of the Satake sheaf on which the
centre of $\widehat G$ at a leg acts by a given weight, the partial
Frobenius at that leg shifts the degree $\delta$ by the image of
that weight. These maps commute with
one another and their product is the total Frobenius, which acts
by an isomorphism; so each of them is an isomorphism. Together
with the action of the fundamental group of an open subset of
$X^I$ they define, as in \cite{VLa18,XueIntegral2023}, an action
of a group which maps onto $W_F^I$. An element of the kernel has
degree zero at every leg, so it preserves the degree $\delta$, and
it acts trivially on the coinvariants by
\cite[Proposition~8.2.6]{XueIntegral2023}. By the injectivity of
the vertical maps of the square it acts trivially on
$H^j_{N,I,V_{\mathcal O}}$.
\end{proof}

\begin{remark}[On the proof of the integral form of Drinfeld's lemma]
\label{rw:rem:xue-drinfeld}
The proposition of Xue just quoted rests on
\cite[Lemma~8.2.4]{XueIntegral2023}: if $A$ is a finitely generated
$\mathcal O$-algebra and $M$ a finitely generated $A$-module with a
continuous $A$-linear action of the Frobenius--Weil group, then the
kernel of the map to $W_F^I$ acts trivially on $M$. The proof given
there reduces to the quotients $M/\mathfrak m^nM$, for the maximal
ideals $\mathfrak m$ of $A$, and uses that $A/\mathfrak m^n$ is a
torsion $\mathcal O$-module of finite type. That holds when the
uniformizer $\varpi$ lies in $\mathfrak m$, but not otherwise: for
$A=\mathcal O[t,t^{-1}]$ and $\mathfrak m=(\varpi t-1)$ the quotient
$A/\mathfrak m$ is $E$. The lemma is nevertheless true, and the
missing case is the following. If $\varpi\notin\mathfrak m$, then
$A/\mathfrak m^n$ is an Artinian local algebra of finite type over
$E$, hence of finite dimension over $E$, and so is
$Q=M/\mathfrak m^nM$. By the definition of continuity
\cite[Definition~8.2.3]{XueIntegral2023}, $M$ is a union of
$\mathcal O$-submodules of finite type which are stable under the
geometric part of the group and on which it acts continuously.
Finitely many of them contain lifts of a basis of $Q$; the image of
their sum is a lattice in $Q$ on which the action is continuous.
So the action on $Q$ is a continuous action on a vector space of
finite dimension over $E$, to which Drinfeld's lemma with
coefficients in $E$ applies \cite[\S3.2]{XueHecke2020}. In both
cases the kernel acts trivially on $M/\mathfrak m^nM$, and the end
of Xue's proof, which uses Krull's intersection theorem at each
maximal ideal, applies without change.
\end{remark}

\begin{proposition}[Excursion operators on functions with compact support]
\label{fl:prop:canonical-compact}
For every excursion datum $e=(I,f,\boldsymbol\gamma)$ with
$\boldsymbol\gamma\in W_F^I$ there is an operator $S_e$ on
$C_{N,E}$ with the following properties.
\begin{enumerate}
\item The operators $S_e$ commute with each other and satisfy the
relations of \cite[\S3.4]{XueHecke2020}, which are those of
\cite[Proposition~10.8]{VLa18}:
\begin{itemize}
\item for fixed $I$ and $\boldsymbol\gamma$, the map
$f\mapsto S_{I,f,\boldsymbol\gamma}$ is a homomorphism of
algebras;
\item for a map $\zeta:I\to J$ and
$f^\zeta((g_j)_{j\in J})=f((g_{\zeta(i)})_{i\in I})$, one has
$S_{J,f^\zeta,(\gamma_j)_{j\in J}}
=S_{I,f,(\gamma_{\zeta(i)})_{i\in I}}$;
\item for
$\widetilde f((g_i),(g'_i),(g''_i))=f((g_ig_i'^{-1}g''_i))$,
\[
 S_{I\sqcup I\sqcup I,\widetilde f,
 (\gamma_i)\times(\gamma'_i)\times(\gamma''_i)}
 =S_{I,f,(\gamma_i\gamma_i'^{-1}\gamma''_i)}.
\]
\end{itemize}
On the coinvariants under any central lattice they induce Xue's
excursion operators.
\item They commute with changes of level and with right
translations.
\item For an unramified place $v$, a representation $V$ of
$\widehat G$ and a geometric Frobenius element at $v$, the operator
attached to the character of $V$ is the Hecke operator
corresponding to $V$ under the normalized Satake isomorphism.
\item Each $S_e$ changes the degree in $\Lambda_G$ by one of
finitely many values.
\end{enumerate}
\end{proposition}

\begin{proof}
Write $f((g_i))=\xi((g_i)x)$ as above. The operator $S_e$ is the
composite of four maps: the creation map defined by $x$, from
$C_{N,E}$ to the cohomology at the generic point of the diagonal
with coefficients $V$; the specialization isomorphism of
Lemma~\ref{fl:lem:unperiodized-specialization}; the action of
$\boldsymbol\gamma\in W_F^I$ given by the same lemma; and the
inverse specialization followed by the annihilation map defined by
$\xi$. Creation and annihilation come from the isomorphisms of
coalescence of legs.

Each of these maps commutes with central translations, and on the
$m\Xi_u$-coinvariants the composite is Xue's
excursion operator. That operator does not depend on the choice of
$(V,x,\xi)$, and the operators commute and satisfy the relations
of coalescence
\cite[Lemma~3.4.2, Construction~3.4.6 and
Lemmas~3.4.9, 3.4.10, 3.4.12 and~3.4.14]{XueHecke2020}. To deduce the same statements
on $C_{N,E}$, apply the difference of the two sides of an identity
to a function with compact support. The result has components in
finitely many degrees and vanishes in the $m\Xi_u$-coinvariants
for every $m$, so it is zero by
Lemma~\ref{fl:lem:graded-central-period}(3). This proves (1), and
(2) follows in the same way from the corresponding maps between
stacks of shtukas. Property (3) holds on all coinvariants by the
compatibility of excursion operators with Hecke operators on
cohomology with compact support
\cite[Proposition~7.2.10]{XueIntegral2023}, which extends
\cite[Proposition~6.2]{VLa18}, and is transferred in the same way.

For (4), decompose $x$ according to the weights of the centre of
$\widehat G^I$. A weight is a family of characters of
$Z(\widehat G)$, that is, of elements of $\pi_1(G)$; let
$(\lambda_i)_{i\in I}$ be their images in $\Lambda_G$. Since $x$ is invariant under the diagonal, $\sum_i\lambda_i=0$.
The degree is preserved by specialization and by the action of
the fundamental group. The partial Frobenius morphism at the leg $i$ maps
shtukas of degree $\delta$ to shtukas of degree
$\delta-\lambda_i$, so the pullback by it shifts the degree of a
cohomology class by $\lambda_i$. Hence $S_e$ shifts
the degree by an
element of the finite set
\[
 \Bigl\{-\sum_i\deg_F(\gamma_i)\lambda_i:
          (\lambda_i)\text{ a weight of }x\Bigr\}.
\]
For a geometric Frobenius element at a place $v$ the supports are
shifted by $+\deg(v)\lambda_i$. For a translation by a central
element this is also the shift of the supports in
Lemma~\ref{fl:lem:central-hecke-units}.
\end{proof}

\begin{proposition}[The transpose of an excursion operator]
\label{fl:prop:compact-excursion-transpose}
For functions $c,h$ with compact support,
\begin{equation}\label{fl:eq:full-transpose}
 [S_ec,h]=[c,S_{e^\star}h].
\end{equation}
In the basis of point masses, every $S_e$ has finite rows and columns.
\end{proposition}

\begin{proof}
The proof rests on the pairing constructed by Li on the cohomology
with compact support of the stacks of shtukas with bounded
modification, before any passage to the cuspidal part
\cite[Remark~5.1.3 and (5-1)--(5-2)]{Li19}. It pairs the
cohomology with coefficients $V$ with the cohomology whose
coefficients are the dual of $V$ twisted by the Chevalley
involution, and it is invariant under the partial Frobenius
morphisms \cite[Proposition~5.2.3 and Corollary~5.2.4]{Li19}. The
transposition formula at the end of that paper is stated for cusp
forms only; the argument for all functions with compact support is
the following. The cup product which enters the pairing requires
the diagonal of the stacks of shtukas with bounded modification,
divided by a central lattice, to be proper. It is, because over the
complement of a sufficiently large level these stacks are quotients
of quasi-projective schemes by finite groups
\cite[Remarque~4.3]{VLa18}.

Write $a_x$ for the creation map defined by $x$, $b_\xi$ for the
annihilation map defined by $\xi$, and $\mathrm{sp}$ for the
specialization from the diagonal, so that
\[
 S_e=b_\xi\,\mathrm{sp}^{-1}\,
       \boldsymbol\gamma\;\mathrm{sp}\,a_x .
\]
Li's pairing is defined on the cohomology with coefficients in
$E$. This is enough, because the identity to be proved is between
operators on $C_{N,E}$. Three properties of
the pairing $\langle-,-\rangle_I$ in degree zero, between
$H^0_{N,I,V_{\mathcal O}}\otimes_{\mathcal O}E$ and the same space for
the dual coefficients twisted by the Chevalley
involution, are used. First, the annihilation map
$b_\xi$ is the transpose of the creation map $a_\xi$ on the dual
coefficients, and likewise for $x$; these are statements about
maps of coefficients, so they are compatible with specialization.
Secondly, the action of the fundamental group preserves the
pairing, because the pairing is a morphism of sheaves. Thirdly,
the partial Frobenius morphisms preserve it, by the result of Li
quoted above. The last two properties say that $\boldsymbol\gamma$
preserves the pairing. Hence
\begin{align*}
 [S_ec,h]_N
 &=\langle\boldsymbol\gamma\,\mathrm{sp}\,a_xc,
                     \mathrm{sp}\,a_\xi h\rangle_I\\
 &=\langle\mathrm{sp}\,a_xc,
             \boldsymbol\gamma^{-1}\mathrm{sp}\,a_\xi h\rangle_I\\
 &=[c,b_x\,\mathrm{sp}^{-1}
             \boldsymbol\gamma^{-1}\mathrm{sp}\,a_\xi h]_N ,
\end{align*}
where the inverse of the specialization map exists by
Lemma~\ref{fl:lem:unperiodized-specialization}. The operator in
the last line is defined by the dual representation, the vectors
$\xi$ and $x$ with their roles exchanged, and the tuple
$\boldsymbol\gamma^{-1}$; its function is $f^\dagger$, so it is
$S_{e^\star}$.

Two points remain to be justified, since Li's pairing is defined
for a fixed bound $\mu$ and after division by a lattice in the
centre. The vectors which occur in the calculation are finite in
number, so they lie in the cohomology for some bound $\mu$, and the
pairings for different bounds are compatible. For the lattice,
choose $m$ so large that the finitely many degrees which occur are
distinct modulo $m\Lambda_u$. By
Lemma~\ref{fl:lem:central-degree-stacks} the parts of the stacks
of these degrees, with their Satake sheaves, are identified with
the corresponding parts of the quotient by $m\Xi_u$, with the same
automorphism groups. So the calculation may be made in the
quotient, where the pairing is defined, and
\eqref{fl:eq:full-transpose} follows.

For the last assertion write $S_e\delta_y=\sum_xa_{xy}\delta_x$.
Each column of the matrix $(a_{xy})$ is finite because
$S_e\delta_y$ has compact support. Let $(a^\star_{xy})$ be the
matrix of $S_{e^\star}$ and $w_N(x)\ne0$ the weight of the point
$x$ in the pairing. Then \eqref{fl:eq:full-transpose} says
$w_N(x)a_{xy}=w_N(y)a^\star_{yx}$, so the row of $(a_{xy})$ at $x$
is finite because the column of $(a^\star_{xy})$ at $x$ is.
\end{proof}

By this proposition the operators extend from functions with
compact support to all functions.

\begin{definition}[Action on functions]
\label{fl:def:function-excursion-action}
For any function $h$ on $Y_N$, define $S_e^{\mathrm{fun}}h$ by
\begin{equation}\label{fl:eq:function-action}
 [c,S_e^{\mathrm{fun}}h]=[S_{e^\star}c,h].
\end{equation}
\end{definition}
The operator $S_{e^\star}$ maps the characteristic function of a
point to a function with compact support, so this formula
determines the value of $S_e^{\mathrm{fun}}h$ at each point.
On functions with
compact support it agrees with $S_e$, by
\eqref{fl:eq:full-transpose}.  If
$L_h(c)=[c,h]$ and
$\nu_\sigma(e)=f((\sigma(\gamma_i)))$, then
\begin{equation}\label{fl:eq:chevalley-functional}
 S_e^{\mathrm{fun}}h=\nu_\sigma(e)h\quad\text{for all }e
 \quad\Longleftrightarrow\quad
 L_hS_e=\nu_{\vartheta_G\sigma}(e)L_h\quad\text{for all }e.
\end{equation}
Indeed $\nu_\sigma(e^\star)=\nu_{\vartheta_G\sigma}(e)$.
The Chevalley involution appears because the pairing is bilinear
and not sesquilinear. It is the reason why the constructions of
\cite{HuII} and \cite{HuIII} start
from $\sigma^\dagger=\vartheta_G\sigma$ when the function they
produce is tested through the pairing.

\begin{proposition}[Central twisting]
\label{fl:prop:compact-central-twists}
Let $\psi:\Gamma_F\to Z(\widehat G)^\circ$ be a continuous
homomorphism with finite image. Take a level on which the
character attached to $\psi$ by class field theory is trivial,
and let $T_\psi$ be the multiplication by this character. It
satisfies
\begin{equation}\label{fl:eq:canonical-finite-twist}
 S_eT_\psi=T_\psi S_{e^\psi},
 \qquad
 f^\psi((g_i))=f((g_i\psi(\gamma_i))).
\end{equation}
The formula also holds for $\psi=c^{\deg_F}$ on $W_F$, with any
$c\in Z(\widehat G)^\circ(k)$.
\end{proposition}

\begin{proof}
Choose the modulus $N$ so that the class-field character of
$\psi$ is trivial on the level subgroup of $Q_G$.
By geometric class field theory with modulus there is a
multiplicative Weil local system $\mathcal L_\psi$ of rank
one on $\Bun_{Q_G,N}$: on the generalized Picard group of degree
zero it is given by the Lang isogeny, and on the other components
by the character of the degree. Its trace function is the
character by which $T_\psi$ multiplies. We describe what it
induces on the cohomology of shtukas, before coinvariants are taken.

Consider the part of the stack of shtukas on which the weights at
the legs are fixed, and let $\lambda_i\in X_*(Q_G)$ be the image of
the weight at the leg $i$. On the underlying $Q_G$-bundles the
modification at the leg $i$ is the tensor product with the bundle
attached to $-\lambda_ix_i$, with its canonical level structure
away from $x_i$. Let $\mathcal W_i$ be the local system on $X-N$
obtained by pulling back $\mathcal L_\psi$ along the
Abel--Jacobi map of $+\lambda_ix_i$. Its character is
$(\lambda_i\circ\psi)^{-1}$. Indeed, at a place $v\notin N$
the image of the Abel--Jacobi map is represented by
$\lambda_i(\varpi_v)^{-1}$, and with the normalization of class
field theory of Section~\ref{fl:sec:parameters-parameters} this
gives the value at the geometric Frobenius elements, which are
dense.

Let $E_0,\ldots,E_s$ be the successive $Q_G$-bundles of a shtuka.
Since $\mathcal L_\psi$ is multiplicative, its fibre at the
source of a modification is the tensor product of its fibre at the
target with the fibres $\mathcal W_i(x_i)$ at the legs of the
modification. Composing these isomorphisms along the chain, then
the isomorphism $E_s\simeq{}^\tau E_0$ of the shtuka, then the
Weil structure of $\mathcal L_\psi$ at $E_0$, we get
\[
 \mathcal L_\psi(E_0)
    \ \xrightarrow{\sim}\
 \mathcal L_\psi(E_0)\otimes\bigotimes_{i\in I}\mathcal W_i(x_i),
\]
that is, a section which trivializes the pullback of
$\mathcal W=\boxtimes_i\mathcal W_i$ to the stack of shtukas.

This section is compatible with the partial Frobenius morphisms.
Let $b$ be the isomorphism of lines given by the first
modification and $d$ the composite of those given by the others.
For the original shtuka the section is $b$, followed by $d$,
followed by the Weil structure at $E_0$. For its image under the
partial Frobenius it is $d$, followed by the pullback of $b$ by
Frobenius, followed by the Weil structure at $E_1$. The Weil
structure is natural, so the last two maps can be replaced by the
Weil structure at $E_0$ followed by $b$. The two composites differ
by a cyclic permutation of three isomorphisms between lines, so
they define the same section. Hence multiplication by the section
is a map from the cohomology of shtukas to its tensor product
with $\mathcal W$ which commutes with the fundamental group and
with the partial Frobenius maps.

On the diagonal, creation and annihilation involve only families
of weights with $\sum_i\lambda_i=0$. For these the product of the
Abel--Jacobi twists is canonically trivial, and the section is the
function by which $\mathcal L_\psi$ multiplies on the stack
without legs, that is, $T_\psi$. Transport the trivialization
of $\mathcal W$ on the diagonal to its generic stalk by the chosen
specialization map. In this basis, the equivariance just proved
says that moving the multiplication by the section across the
action of a tuple $(\gamma_i)$ multiplies the result by
$\prod_i\lambda_i(\psi(\gamma_i))$. Summing over the families
of weights which occur in the matrix coefficient $f$ gives
\eqref{fl:eq:canonical-finite-twist}.

For $\psi=c^{\deg_F}$, the operator $T_\psi$ multiplies
the functions supported in degree $\delta$ by a monomial in $c$.
By Proposition~\ref{fl:prop:canonical-compact}(4), applied to a
function supported in one degree, the two sides of
\eqref{fl:eq:canonical-finite-twist} are Laurent polynomials in
$c$. They agree when $c$ has finite order, by the first part, and
the points of finite order are Zariski dense in the torus; so they
agree for all $c$. Finally, multiplication by a function is its
own transpose for $[\ ,\ ]$, and
$(e^\psi)^\star=(e^\star)^{\psi^{-1}}$, because the
Chevalley involution is the inversion on the connected centre.
So the transpose of the formula is the same formula for
$S_e^{\mathrm{fun}}$, with the same $\psi$.
\end{proof}