\section{Galois tuples, finite Hecke quotients and the space \texorpdfstring{$\mathcal A^{\mathrm{cont}}(G)$}{A-cont(G)}}
\label{rw:sec:function-spaces-section}

Section~\ref{fl:sec:compact-foundations} gave excursion operators
for tuples in $W_F^I$. A parameter $\sigma$ is defined on
$\Gamma_F$, and its excursion character
$\nu_\sigma(I,f,\boldsymbol\gamma)$ makes sense for all
$\boldsymbol\gamma\in\Gamma_F^I$. To state that a function has
parameter $\sigma$ in this full sense, the operators must be
defined for Galois tuples.

An action of $W_F$ on a vector space over $E$ does not extend to
$\Gamma_F$ in general: Frobenius could act on a line by $\ell$,
and the powers $\ell^n$ do not converge as $n$ varies in
$\widehat{\mathbb Z}$. It does extend, uniquely and continuously,
when it preserves a lattice
(Lemma~\ref{fl:lem:finite-module-weil-extension}). The plan of this
section is therefore the following.
\begin{enumerate}
\item Divide the module of functions with compact support by an
ideal of the integral spherical Hecke algebra at one place. By
Proposition~\ref{fl:prop:classical-one-place-finiteness} the
quotient is a finitely generated $\mathcal O$-module, and
excursion operators for all Galois tuples act on it
(Theorem~\ref{fl:thm:finite-integral-excursion}).
\item Pass to duals. The dual of such a quotient is a space of
functions annihilated by the ideal
(Proposition~\ref{fl:prop:finite-model-duality}). The space
$\mathcal A^{\mathrm{cont}}(G)$ is spanned by the duals of the
quotients with a central lattice and by their twists by characters
of the degree
(Definition~\ref{fl:def:parameter-independent-spaces}); every
finite set of its elements lies in the dual of one quotient
without a central lattice (Theorem~\ref{fl:thm:function-space}). It is
smaller than the union of all such duals
(Remark~\ref{rw:rem:not-all-duals}).
\item Show that the elements of $\mathcal A^{\mathrm{cont}}(G)$ are
automorphic forms
(Proposition~\ref{fl:prop:one-place-automorphicity}), and give the
criterion by which the functions constructed later are recognized
as elements with parameter $\sigma$
(Corollary~\ref{fl:cor:weil-to-galois-nonzero}).
\end{enumerate}

\subsection{From Weil tuples to Galois tuples}

\begin{definition}[Finite Hecke quotients]
\label{fl:def:finite-hecke-model}
An ideal $\mathfrak a\subset\mathscr H_{G,u,\mathcal O}$ is
\emph{cofinite} if the quotient is a finite $\mathcal O$-module.
For $Y_{N,\Xi}=G(F)\backslash G(\mathbb A)/(K_N\Xi)$, put
\[
 \mathcal Q_{N,\Xi,u,\mathfrak a}
 =\bigl(\mathcal O^{(Y_{N,\Xi})}/
              \mathfrak a\mathcal O^{(Y_{N,\Xi})}\bigr)
                 \otimes_{\mathcal O}k.
\]
For an integral character $\chi:\mathscr H_{G,u,\mathcal O}\to\mathcal O$
we also write
\begin{equation}\label{fl:eq:finite-hecke-quotient}
 \mathcal Q_{N,\Xi,u,\chi,n}
   =\mathcal Q_{N,\Xi,u,(\ker\chi)^n}.
\end{equation}
\end{definition}

These spaces are finite-dimensional by
Proposition~\ref{fl:prop:classical-one-place-finiteness}.
The powers in \eqref{fl:eq:finite-hecke-quotient} are cofinite:
the integral Satake algebra is Noetherian, its quotient by
$\ker\chi$ is $\mathcal O$, and the successive quotients of
the powers of this ideal are finite $\mathcal O$-modules.


\begin{lemma}[Integral extension of a Weil action]
\label{fl:lem:finite-module-weil-extension}
Let $\mathcal V$ be a finitely generated $\mathcal O$-module with
an $\mathcal O$-linear action of $W_F$ whose restriction to
$\Gamma_F^{\mathrm{geom}}$ is continuous. The action extends
uniquely to a continuous action of $\Gamma_F$. Submodules stable
under $W_F$ are stable under $\Gamma_F$; homomorphisms which
commute with $W_F$ commute with $\Gamma_F$; and if several actions
of $W_F$ as above commute with one another, so do their
extensions.
\end{lemma}

\begin{proof}
The group $A=\operatorname{Aut}_{\mathcal O}(\mathcal V)$ is
profinite: since $\mathcal V$ is complete and separated for the
$\varpi$-adic topology, it is the inverse limit of its images in
the finite groups
$\operatorname{Aut}(\mathcal V/\varpi^a\mathcal V)$. Let
$r:W_F\to A$ be the action, and choose
$\gamma_{\mathrm{Frob}}\in W_F$ of degree one, so that
\[
 W_F=\Gamma_F^{\mathrm{geom}}\rtimes
           \gamma_{\mathrm{Frob}}^{\mathbb Z},
 \qquad
 \Gamma_F=\Gamma_F^{\mathrm{geom}}\rtimes
           \gamma_{\mathrm{Frob}}^{\widehat{\mathbb Z}} .
\]
In a profinite group the homomorphism
$n\mapsto r(\gamma_{\mathrm{Frob}})^n$ extends to a continuous
homomorphism $z\mapsto r(\gamma_{\mathrm{Frob}})^z$ of
$\widehat{\mathbb Z}$. The relation
\[
 r(\gamma_{\mathrm{Frob}})^z\,r(g)\,r(\gamma_{\mathrm{Frob}})^{-z}
    =r(\gamma_{\mathrm{Frob}}^zg\gamma_{\mathrm{Frob}}^{-z}),
 \qquad g\in\Gamma_F^{\mathrm{geom}},
\]
holds for $z\in\mathbb Z$, and both sides are continuous in $z$;
so it holds for $z\in\widehat{\mathbb Z}$, and
$g\gamma_{\mathrm{Frob}}^z\mapsto
r(g)r(\gamma_{\mathrm{Frob}})^z$ is a continuous homomorphism on
$\Gamma_F$ which extends $r$. It is unique because $W_F$ is dense
in $\Gamma_F$.

The remaining assertions also follow from density. A submodule of
$\mathcal V$ is closed, so its stabilizer in $\Gamma_F$ is closed;
the set of elements of $\Gamma_F$ which commute with a given
homomorphism, or with a given automorphism of $\mathcal V$, is
closed as well.
\end{proof}

\begin{theorem}[Continuous excursion actions on finite quotients]
\label{fl:thm:finite-integral-excursion}
Let $\mathcal Q$ be one of the finite Hecke quotients of
Definition~\ref{fl:def:finite-hecke-model}, with or without a
central lattice.
\begin{enumerate}
\item For every excursion datum $e=(I,f,\boldsymbol\gamma)$ with
$\boldsymbol\gamma\in\Gamma_F^I$ there is an operator $S_e$ on
$\mathcal Q$, continuous in $\boldsymbol\gamma$, which for
$\boldsymbol\gamma\in W_F^I$ is induced by the operator of
Proposition~\ref{fl:prop:canonical-compact}. These operators
satisfy the same relations as on cusp forms.
\item The operators commute with the maps between such quotients
induced by changes of level, of central lattice, of coefficient
field and of the ideal.
\item There is an $\mathcal O$-lattice spanning $\mathcal Q$ that
is stable under the integral spherical Hecke algebras at all
places outside the level.
\end{enumerate}
\end{theorem}

\begin{proof}
The proof has five steps. Steps~1, 3 and~4 are formal: the maps
which define the excursion operators pass to the quotient, the
action of the Weil group extends by
Lemma~\ref{fl:lem:finite-module-weil-extension}, and identities
extend by density. The one point which needs an argument is
step~2, the continuity of the action of the geometric fundamental
group on a module which is defined as a quotient of an infinite
limit.
We treat the quotient without a central lattice; the argument with a central lattice
is the same, and is in fact the case considered by Xue.

\step{1}{finite modules at the generic point} Let $I$ be a finite
set and $V_{\mathcal O}$ a representation of $\widehat G^I$ on a
free $\mathcal O$-module of finite rank. Besides the module
$H_\eta=H^0_{N,I,V_{\mathcal O}}$ we consider the corresponding
module $H_\Delta$ at the geometric generic point of the diagonal.
The creation map $\mathcal O^{(Y_N)}\to H_\Delta$, the
specialization isomorphism $H_\Delta\to H_\eta$, the action of
$W_F^I$ on $H_\eta$ and the annihilation map
$H_\Delta\to\mathcal O^{(Y_N)}$ all commute with
$\mathscr H_{G,u,\mathcal O}$, because the Hecke correspondences
at $u$ act on the stacks of shtukas with legs outside $u$
compatibly with all the structures. Here the specialization
isomorphism and the action of $W_F^I$ are those of
\cite[Propositions~8.1.2 and~8.2.6]{XueIntegral2023}, extended to
the modules without a central lattice by
Lemma~\ref{fl:lem:unperiodized-specialization}. So all these maps
pass to the quotients by $\mathfrak a$. By
Proposition~\ref{fl:prop:classical-one-place-finiteness} the
module $H_\eta$ is finitely generated over
$\mathscr H_{G,u,\mathcal O}$, and therefore
\[
 \mathcal V=H_\eta/\mathfrak aH_\eta
\]
is a finitely generated $\mathcal O$-module.

\step{2}{continuity} This is the point at which integral
coefficients are needed. We show that the subgroup
$(\Gamma_F^{\mathrm{geom}})^I$ of $W_F^I$ acts continuously on
$\mathcal V$. Fix $a\ge1$, choose finitely many generators of
$\mathcal V/\varpi^a\mathcal V$ and lift them to $H_\eta$. The
module $H_\eta$ is the inductive limit, over the bounds $\mu$ on the
Harder--Narasimhan polygon and over finite sets of degrees, of the
stalks at
$\bar\eta_I$ of the cohomology sheaves of stacks of finite type
over an open subset of $X^I$. So the chosen lifts come from one
such stalk. It is the stalk of a constructible sheaf, on which the
geometric fundamental group of the generic point of $X^I$ acts
continuously; an open subgroup of that group therefore fixes the
lifts modulo $\varpi^a$. The geometric fundamental group maps onto
$(\Gamma_F^{\mathrm{geom}})^I$: over the algebraically closed
field of constants, a product of connected finite coverings of the
geometric generic points of the factors is integral, which gives
the surjectivity on every finite quotient. A continuous surjective
homomorphism of profinite groups is open, and the action on
$\mathcal V$ factors through $W_F^I$. Hence an open subgroup of
$(\Gamma_F^{\mathrm{geom}})^I$ acts trivially on
$\mathcal V/\varpi^a\mathcal V$. Since $a$ is arbitrary, the
action is continuous.

\step{3}{extension to Galois tuples} For each $i\in I$,
Lemma~\ref{fl:lem:finite-module-weil-extension} extends the action
of the $i$-th factor of $W_F^I$ on $\mathcal V$ to an action of
$\Gamma_F$. By the same lemma these extensions commute, and
together they define a continuous action of $\Gamma_F^I$ on
$\mathcal V$. For $x\in V_{\mathcal O}$ and
$\xi\in V_{\mathcal O}^*$ invariant under the diagonal, and
$\boldsymbol\gamma\in\Gamma_F^I$, we let $S_e$ be the composite of
the creation map defined by $x$, the specialization isomorphism,
the action of $\boldsymbol\gamma$ on $\mathcal V$, the inverse of
specialization and the annihilation map defined by $\xi$. It is an
endomorphism of
$\mathcal O^{(Y_N)}/\mathfrak a\mathcal O^{(Y_N)}$, continuous in
$\boldsymbol\gamma$, and for a tuple in $W_F^I$ it is induced by
the operator of Proposition~\ref{fl:prop:canonical-compact}.

A function $f$ on $\widehat G^I$ defined over $E$ is of the form
$f((g_i))=\xi((g_i)x)$ for a representation $V$ over $E$. Such a
representation is a direct sum of irreducible representations, and
each of them contains the Weyl module over $\mathcal O$ of the same
highest weight; so $V$ contains a lattice $V_{\mathcal O}$ which is
a representation of the split group scheme $\widehat G^I$ over
$\mathcal O$. Suitable nonzero multiples $ax$ and $a'\xi$ lie in
$V_{\mathcal O}$ and in its dual. We define $S_e$ on $\mathcal Q$
as $(aa')^{-1}$ times the operator defined by $ax$ and $a'\xi$.

\step{4}{relations and compatibilities} For tuples in $W_F^I$ the
operators just defined are induced by those of
Proposition~\ref{fl:prop:canonical-compact}. So for such tuples
they do not depend on the choices made, they satisfy the relations
of that proposition, and they commute with the maps induced by
changes of level, central lattice, coefficient field and ideal. Each of
these statements is an equality between two continuous functions
of $\boldsymbol\gamma\in\Gamma_F^I$ with values in a
finite-dimensional space, or follows from such equalities, and
$W_F^I$ is dense in $\Gamma_F^I$. This proves (1) and (2).

\step{5}{the lattice} For a place $v\notin N$ the algebra
$\mathscr H_{G,v,\mathcal O}$ commutes with
$\mathscr H_{G,u,\mathcal O}$, so it acts on
$\mathcal O^{(Y_N)}/\mathfrak a\mathcal O^{(Y_N)}$. The image of
this module in $\mathcal Q$ is a finitely generated
$\mathcal O$-module which spans $\mathcal Q$ and is stable under
all these algebras.
\end{proof}

We also record that the action of $\Gamma_F^I$ on the space
$\mathcal V\otimes_{\mathcal O}E$ of step~1 is unramified outside
$N$:
\begin{equation}\label{fl:eq:finite-model-unramified-action}
 \Gamma_F^I\longrightarrow\pi_1(X-N)^I
      \longrightarrow\operatorname{Aut}_E(\mathcal V\otimes E).
\end{equation}
This is a statement with coefficients in $E$, and that is all that
the construction of parameters uses. With a central lattice it is
\cite[Proposition~5.0.4]{XueSmoothCoh2020}, which is formulated
with coefficients in $E$. Without a central lattice, let $v\notin N$ and
consider an inertia group at $v$ in one of the factors. It is
contained in $\Gamma_F^{\mathrm{geom}}$, so it preserves the degree
on $H_\eta\otimes E$, and it acts trivially on the $m\Xi_u$-coinvariants of
$H_\eta\otimes E$ for every $m$; by
Lemma~\ref{fl:lem:graded-central-period}(3), applied after
tensoring with $E$, it acts trivially on $H_\eta\otimes E$, hence
on $\mathcal V\otimes E$. The kernel of the continuous
action of $\Gamma_F^I$ is a closed normal subgroup, and the
quotient of $\Gamma_F$ by the closed normal subgroup generated by
the inertia groups at the places outside $N$ is $\pi_1(X-N)$.

\begin{remark}
The construction works with the modules $\mathcal Q$ themselves and
not with their semisimplifications under the Hecke algebra, so
nilpotent parts of the operators and generalized eigenspaces are
kept; this is used in
Definition~\ref{fl:def:full-excursion-summand}. Note also that the
existence of a stable lattice implies that the Hecke eigenvalues
are integral. At a Satake parameter, every character of the dual
torus and its inverse are integral over the ring of invariants of
the Weyl group, so their values are units.
\end{remark}

Recall the involution $h\mapsto h^\star$,
$h^\star(g)=h(g^{-1})$, of the spherical Hecke algebra. It is the
transposition for the pairing $[\ ,\ ]$: for a Hecke operator $T$
one has $[Tc,h]=[c,T^\star h]$. It preserves the integral algebra
and maps cofinite ideals to cofinite ideals.

\begin{proposition}[Dual quotients]
\label{fl:prop:finite-model-duality}
The pairing $[\ ,\ ]$ identifies
\begin{equation}\label{fl:eq:periodic-annihilator-dual}
 \mathcal Q_{N,\Xi,u,\mathfrak a}^*
 =\{h\in k^{Y_{N,\Xi}}:\mathfrak a^\star h=0\}.
\end{equation}
The same assertion holds without a central lattice. For tuples in $W_F^I$
the transpose of the operator $S_{e^\star}$ on the quotient is the
restriction of the operator $S_e^{\mathrm{fun}}$ of
Definition~\ref{fl:def:function-excursion-action}.
\end{proposition}

\begin{proof}
The pairing identifies the dual of $k^{(Y_{N,\Xi})}$ with
$k^{Y_{N,\Xi}}$. The functional $c\mapsto[c,h]$ factors through
$\mathcal Q_{N,\Xi,u,\mathfrak a}$ if and only if $[Tc,h]=0$ for
all $T\in\mathfrak a$ and all $c$. Since
$[Tc,h]=[c,T^\star h]$, this means $T^\star h=0$ for all
$T\in\mathfrak a$. The last assertion is the definition
\eqref{fl:eq:function-action}.
\end{proof}

By means of this proposition we transpose the operators of
Theorem~\ref{fl:thm:finite-integral-excursion} for all Galois
tuples: on the space \eqref{fl:eq:periodic-annihilator-dual} we
define $S_e^{\mathrm{fun}}$ as the transpose of $S_{e^\star}$.

\subsection{The spaces of functions}
\label{fl:sec:function-spaces}

The quotients $\mathcal Q$ depend on a level, a place, an ideal and
possibly a central lattice. We now assemble their duals into spaces of
functions that do not depend on such choices, and in particular do
not depend on the parameter one wishes to realize.

\begin{definition}
\label{fl:def:finite-integral-orbit}
A finite collection of functions of level $N$ has a \emph{finite
integral spherical orbit} at a place $u\notin N$ if, after a finite
extension of $E$, it is contained in a finitely generated
$\mathcal O_E$-module of functions that is stable under
$\mathscr H_{G,u,\mathcal O_E}$.
\end{definition}

\begin{lemma}[Criterion for lying in the dual of a finite quotient]
\label{fl:lem:all-place-stable-lattice}
At fixed level and central lattice, a finite collection of functions lies
in the dual of one finite Hecke quotient if and only if it has a
finite integral spherical orbit at one place outside the level. It
then has such an orbit at every place outside the level. The same
holds for the quotients without a central lattice.
\end{lemma}

\begin{proof}
Suppose that the functions lie in a finitely generated
$\mathcal O_E$-module $L$ of functions stable under
$\mathscr H_{G,u,\mathcal O_E}$. The image of this algebra in the
endomorphisms of $L$ is a finitely generated $\mathcal O_E$-module,
so the kernel $\mathfrak b$ of the action is a cofinite ideal. As
$\mathfrak bL=0$, formula \eqref{fl:eq:periodic-annihilator-dual}
shows that $L$ is contained in the dual of the quotient defined by
the ideal $\mathfrak b^\star$.

Conversely, let the functions lie in the dual of a finite quotient
$\mathcal Q$. By
Theorem~\ref{fl:thm:finite-integral-excursion}(3) there is a
lattice in $\mathcal Q$ stable under the integral spherical
algebras at all places outside the level. The functions in
$\mathcal Q^*$ which have integral values on this lattice form a
lattice in $\mathcal Q^*$, stable under the same algebras because
they are stable under $\star$. After a finite extension of $E$,
the given functions lie in a multiple of it. So they have a finite
integral spherical orbit at every place outside the level.

It follows that finitely many functions which lie in the duals of
finite quotients for different places and ideals lie in the dual
of a single one: by what has been proved we may use the same place
for all of them, and then the intersection of the ideals, which is
cofinite because the quotient by it embeds into the product of the
quotients.
\end{proof}

\begin{lemma}[Integrality of parameter values]
\label{fl:lem:parameter-satake-integrality}
Let $\sigma$ be a continuous semisimple parameter, unramified at a
place $v$. The character of the spherical Hecke algebra at $v$
defined by the Satake parameter $\sigma(\operatorname{Frob}_v)$
has integral values on the integral Hecke algebra, after a finite
extension of $E$. The same holds for $\vartheta_G\sigma$.
\end{lemma}

\begin{proof}
Under the normalized Satake isomorphism the integral Hecke algebra
is spanned over $\mathcal O$ by the characters of the
representations of $\widehat G$; recall that $q^{1/2}$ is a unit
of $\mathcal O$. So it is enough to see that the trace of
$\sigma(\operatorname{Frob}_v)$ in a representation $V$ of
$\widehat G$ is integral. The image of $\sigma$ is compact and
contained in $\widehat G(E')$ for a finite extension $E'$ of $E$.
A compact subgroup of $\operatorname{GL}(V)(E')$ stabilizes a
lattice, so the traces of its elements are integral. The
involution $\vartheta_G$ exchanges a representation and its dual,
and the argument applies to $\vartheta_G\sigma$.
\end{proof}

\begin{definition}[The spaces $\mathcal A_\Xi(G)$, $\mathcal A^{\mathrm{per}}(G)$ and $\mathcal A^{\mathrm{cont}}(G)$]
\label{fl:def:parameter-independent-spaces}
Define spaces of functions on $G(F)\backslash G(\mathbb A_F)$,
as sums inside the space of all functions of finite level, by
\begin{equation}\label{fl:eq:periodic-function-space}
 \mathcal A_\Xi(G)=\bigcup_N\sum_{E,u,\mathfrak a}
           \mathcal Q_{N,\Xi,u,\mathfrak a}^*,
 \qquad
 \mathcal A^{\mathrm{per}}(G)=\sum_\Xi\mathcal A_\Xi(G),
\end{equation}
and allow twists by characters of the degree with values in the
centre by setting
\begin{equation}\label{fl:eq:continuous-centre-space}
 \mathcal A^{\mathrm{cont}}(G)
     =\sum_{\psi=c^{\deg_F}}T_\psi\mathcal A^{\mathrm{per}}(G),
 \qquad c\text{ a unit point of }Z(\widehat G)^\circ.
\end{equation}
Here the dual of each quotient is regarded as a space of functions
by \eqref{fl:eq:periodic-annihilator-dual}, $\Xi$ runs through the
central lattices, and $T_\psi$ is
multiplication by the character of $G(\mathbb A_F)$ attached to
$\psi$ (Proposition~\ref{fl:prop:compact-central-twists}). In
words: $\mathcal A_\Xi(G)$ consists of the $\Xi$-invariant
functions of finite level with a finite integral spherical orbit
at one place, and $\mathcal A^{\mathrm{cont}}(G)$ is spanned by
the twists of such functions, for varying $\Xi$, by characters of
the degree.
\end{definition}

\begin{theorem}[Excursion operators on $\mathcal A^{\mathrm{cont}}(G)$]
\label{fl:thm:function-space}
\begin{enumerate}
\item For every excursion datum $e$ with tuple in $\Gamma_F^I$ the
operators of Theorem~\ref{fl:thm:finite-integral-excursion}, on the
duals of the finite quotients, define an operator
$S_e^{\mathrm{fun}}$ on $\mathcal A^{\mathrm{cont}}(G)$ that
preserves $\mathcal A_\Xi(G)$ and
$\mathcal A^{\mathrm{per}}(G)$. It does not depend on the way a
function is written as a sum of elements of such duals, and for
tuples in $W_F^I$ it is the operator of
Definition~\ref{fl:def:function-excursion-action}.
\item The three spaces are stable under right translations and
Hecke operators, which commute with the operators
$S_e^{\mathrm{fun}}$.
\item They contain the cusp forms that are invariant under a
central lattice, respectively the twists of such
forms, and on cusp forms $S_e^{\mathrm{fun}}$ is Lafforgue's
operator.
\item For a twist $\psi=c^{\deg_F}$ as in
\eqref{fl:eq:continuous-centre-space}, and for a central parameter
$\psi$ of finite order after shrinking the lattice to the kernel of
its class-field character,
\begin{equation}\label{fl:eq:transported-full-action}
 S_e^{\mathrm{fun}}(T_\psi h)=T_\psi S_{e^\psi}^{\mathrm{fun}}h.
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
\step{1}{one quotient for finitely many functions} We first show
that every finite collection of elements of
$\mathcal A^{\mathrm{cont}}(G)$ lies in the dual of one finite
quotient without a central lattice. It is enough to consider functions of
the form $T_\psi h'$ with $h'$ in the dual of a quotient
$\mathcal Q_{N,\Xi,u,\mathfrak a}$; we may take the same level for
all of them, and by Lemma~\ref{fl:lem:all-place-stable-lattice}
the same place $u$ outside that level. The function $h'$ has a finite integral
spherical orbit at $u$. Conjugation by $T_\psi$ multiplies
the Hecke operator of a double coset whose elements have valuation
$\lambda$ in $Q_G(F_u)$ by $c^{-\deg(u)\lambda}$, which is a unit.
So it is an automorphism of the integral Hecke algebra, and
$T_\psi h'$ has a finite integral spherical orbit at $u$ as
well. The assertion follows from
Lemma~\ref{fl:lem:all-place-stable-lattice}, in its version
without a central lattice.

\step{2}{definition and independence} On the dual of a finite
quotient without a central lattice, $S_e^{\mathrm{fun}}$ is the transpose
of $S_{e^\star}$. Suppose that a function $h$ lies in the duals of two
finite quotients, with or without a central lattice. For tuples in $W_F^I$
the two operators map $h$ to the same function, namely the
function $S_e^{\mathrm{fun}}h$ of
Definition~\ref{fl:def:function-excursion-action}, which is
defined without reference to a quotient
(Proposition~\ref{fl:prop:finite-model-duality}). For tuples in
$\Gamma_F^I$ the two images of $h$ are continuous functions of
the tuple with values in the sum of the two duals, a space of
functions of finite dimension, and they agree on the dense subset
$W_F^I$; so they agree. By step~1 this defines
$S_e^{\mathrm{fun}}$ on $\mathcal A^{\mathrm{cont}}(G)$. Applied
to a quotient by a central lattice $\Xi$, the same remark shows that
$\mathcal A_\Xi(G)$ is preserved.

\step{3}{twists} The twists by characters of the degree lie in
duals of finite quotients, by step~1. For a central twist of
finite order, enlarge the level so that it contains the places
where the twist is ramified, and replace $\Xi$ by the kernel
of the class-field character of the twist on it. At a place $u$
outside the level this character is constant on each double coset,
so conjugation by the twist multiplies each Hecke operator of a
double coset by a unit. Hence these twists lie in duals of finite
quotients as well. For tuples in $W_F^I$,
\eqref{fl:eq:transported-full-action} is the transpose of
\eqref{fl:eq:canonical-finite-twist}. Both sides are continuous in
$\boldsymbol\gamma\in\Gamma_F^I$: the datum $e^\psi$ depends
on $\boldsymbol\gamma$ through the elements
$\psi(\gamma_i)$, which are continuous in $\gamma_i$ because
$\psi$ is of finite order, or because $c$ is a unit point.
Indeed, the translate of $f$ by an element of the central torus
at each leg is the sum of the components of $f$ of given central
weights, each multiplied by the value of the corresponding
character at $\psi(\gamma_i)$. So
the formula holds for all Galois tuples. This proves (4), and (1)
follows since each $T_\psi\mathcal A^{\mathrm{per}}(G)$ is
preserved.

\step{4}{translations} Let $g\in G(\mathbb A_F)$ and let $h$ be an
element of one of the three spaces, of level $N$. Let $N'$ be a
level of the translate of $h$ by $g$, and let $u\notin N\cup N'$ be a
place with $g_u\in G(\mathcal O_u)$. By
Lemma~\ref{fl:lem:all-place-stable-lattice} the function $h$ has a
finite integral spherical orbit at $u$. Right translation by $g$
commutes with the Hecke algebra at $u$ and with the centre, so the
translate has such an orbit as well and lies in the same space.
For tuples in $W_F^I$ the excursion operators commute with
translations by Proposition~\ref{fl:prop:canonical-compact}(2),
and for Galois tuples by continuity. A Hecke operator is a finite
linear combination of right translations.

\step{5}{cusp forms} By Harder's theorem the cusp forms of level
$N$ invariant under $\Xi$ are supported on a finite subset of
$Y_{N,\Xi}$. Those with values in $\mathcal O$ form a finitely
generated module stable under the integral Hecke algebras, so cusp
forms have a finite integral spherical orbit and belong to
$\mathcal A_\Xi(G)$. On the cuspidal part of the cohomology the
operators of Proposition~\ref{fl:prop:canonical-compact} are those
of \cite{VLa18} by construction, and the latter are continuous in
the Galois tuple; so the two actions agree on cusp forms.
\end{proof}

\begin{proposition}[A finite spherical orbit at one place implies automorphy]
\label{fl:prop:one-place-automorphicity}
A smooth function of finite level on
$G(F)\backslash G(\mathbb A_F)$ whose orbit under the spherical
Hecke algebra at one place outside the level is finite-dimensional
is an automorphic form in the sense of
\cite[\S9.1]{G2BernsteinLapid24}. No growth condition is assumed.
\end{proposition}

\begin{proof}
Let $\varphi$ be the function, $u$ the place, and $\mathfrak b$ an
ideal of finite codimension in the spherical algebra
$\mathscr H_{G,u}$ over $k$ which annihilates $\varphi$.

\step{1}{constant terms} Let $P=MU$ be a parabolic subgroup and
$\varphi_P$ the constant term of $\varphi$ along $P$. Since
$\varphi$ has finite level, $\varphi_P$ is determined by the
finitely many functions
\[
 m\longmapsto\delta_P(m)^{-1/2}\varphi_P(mx)
\]
on $M(F)\backslash M(\mathbb A_F)$, where $x$ runs through a set
of representatives of
$G(\prod_v\mathcal O_v)/K_N$ with component one at $u$. The
computation of the constant term of a convolution
(Lemma~\ref{rw:lem:ct-spherical}) shows that each
of these functions is annihilated by the image of $\mathfrak b$
under the normalized Satake homomorphism
$\mathscr H_{G,u}\to\mathscr H_{M,u}$. In terms of a maximal split
torus $T$ of $M$ this homomorphism is the inclusion
\[
 k[X_*(T)]^{W_G}\subset k[X_*(T)]^{W_M} .
\]
The algebra $k[X_*(T)]$ is a finitely generated module over the
first ring, which is Noetherian, and the second ring is a
submodule of it. Hence $\mathscr H_{M,u}$ is a finitely generated
module over $\mathscr H_{G,u}$, the ideal generated by the image
of $\mathfrak b$ has finite codimension in $\mathscr H_{M,u}$, and
each of the functions above has a finite-dimensional orbit under
$\mathscr H_{M,u}$.

\step{2}{the centre of the Levi subgroup} The criterion of
\cite[\S9.1, Lemma, p.~227]{G2BernsteinLapid24} is formulated in
terms of a lattice $A_P$ in the centre of $M(\mathbb A_F)$, which
in the function field case consists of the elements
$\lambda(\varpi_{v_0})$ for a place $v_0$ chosen once and for all
and $\lambda$ a cocharacter of the connected centre of $M$
\cite[\S6.2]{G2BernsteinLapid24}. We take $v_0=u$. The
translations by these elements belong to $\mathscr H_{M,u}$, so by
step~1 the translates of $\varphi_P$ by $A_P$ span a space $W$ of
finite dimension. The elements of $A_P$ act on $W$ by commuting
operators. Hence there are characters
$\lambda_1,\dots,\lambda_n$ of $A_P$, with $n=\dim W$, such that
\[
 \prod_{i=1}^n\bigl(a_i-\lambda_i(a_i)\bigr)\varphi_P=0
 \qquad\text{for all }a_1,\dots,a_n\in A_P ,
\]
where $a_i$ stands for the translation by $a_i$: decompose $W$
into generalized eigenspaces and use that a product of $d$
commuting nilpotent endomorphisms of a space of dimension $d$ is
zero. This holds for every $P$, and it is the hypothesis of the
criterion, which shows that $\varphi$ is automorphic.

\step{3}{coefficients} The criterion is stated for functions with
complex values. Let $K\subset k$ be a finitely generated subfield
which contains the chosen square root of $q$ and the coefficients
of a set of generators of $\mathfrak b$, and let $\mathfrak b_K$
be the ideal generated by these generators over $K$. By
Proposition~\ref{fl:prop:classical-one-place-finiteness} the
quotient of the space of functions with compact support on $Y_N$
with values in $K$ by $\mathfrak b_K^\star$ has finite dimension:
this can be seen after extension of scalars to $k$. By
transposition $\varphi$ is a linear form on the extension of
scalars of this quotient. So $\varphi$ is a combination, with
coefficients in $k$, of functions with values in $K$ which are
annihilated by $\mathfrak b_K$. The field $K$ embeds into
$\mathbb C$, and the first two steps apply to each of these
functions. The space of automorphic forms with coefficients in $k$
contains their span.
\end{proof}

\begin{corollary}
\label{fl:cor:excursion-space-automorphicity}
Every element of $\mathcal A^{\mathrm{cont}}(G)$ is an
automorphic form.
\end{corollary}

\begin{proof}
By step~1 of the proof of Theorem~\ref{fl:thm:function-space} an
element of this space lies in the dual of a finite quotient
without a central lattice, so its orbit under the spherical Hecke algebra
at one place has finite dimension.
Proposition~\ref{fl:prop:one-place-automorphicity} applies.
\end{proof}

\begin{remark}\label{rw:rem:not-all-duals}
The dual of a finite quotient without a central lattice need not be
contained in $\mathcal A^{\mathrm{cont}}(G)$. For example,
take $G=\mathbb G_m$ over $F=\mathbb F_q(t)$, the trivial level and
a Hecke place of degree one. The quotient of the functions with
compact support on $\mathbb Z$ by the square of the
augmentation ideal of the group algebra of the translations has a
dual which contains the function $n\mapsto n$. On the span of this
function and the constants, translation acts by a nontrivial
Jordan block, whereas it acts semisimply on every element of
$\mathcal A^{\mathrm{cont}}(G)$: on a function with a central lattice it has
finite order, and a twist by a character of the degree multiplies
it by a scalar. For this reason the subspaces $V$ used below are
intersections with $\mathcal A^{\mathrm{cont}}(G)$.
\end{remark}

For a continuous semisimple parameter $\sigma$, let
$\mathfrak m_\sigma$ be the ideal generated by the operators
$S_e^{\mathrm{fun}}-\nu_\sigma(e)$ in the algebra of excursion
operators of the space considered. Every finite collection of
elements of $\mathcal A^{\mathrm{cont}}(G)$ lies in a subspace $V$
of finite dimension stable under the excursion operators, namely
the intersection of $\mathcal A^{\mathrm{cont}}(G)$ with the dual
of a finite quotient: the intersection is stable because both
spaces are
(Theorem~\ref{fl:thm:function-space}(1)). Applying
Definition~\ref{fl:def:full-excursion-summand} to these subspaces
gives the summand
\begin{equation}\label{fl:eq:generalized-block}
 \mathcal A^{\mathrm{cont}}(G)_\sigma
   =\bigcup_V\{h\in V:\mathfrak m_\sigma^a h=0
                             \text{ for some }a\ge1\}
\end{equation}
of $\mathcal A^{\mathrm{cont}}(G)$; a joint eigenfunction is an
element with $a=1$.

\begin{corollary}[Recognizing an eigenfunction for all Galois tuples]
\label{fl:cor:weil-to-galois-nonzero}
Let $\sigma$ be a continuous, finitely ramified, semisimple
parameter, and let $h$ be a nonzero smooth function of finite level
such that $S_e^{\mathrm{fun}}h=\nu_\sigma(e)h$ for every excursion
datum $e$ with tuple in $W_F^I$. Then:
\begin{enumerate}
\item $h$ belongs to $\mathcal A^{\mathrm{cont}}(G)$ and is an
automorphic form;
\item $S_e^{\mathrm{fun}}h=\nu_\sigma(e)h$ for every excursion
datum with tuple in $\Gamma_F^I$;
\item $T_G(\mathbb A)$ acts on $h$ by the character
$\omega_\sigma$;
\item for every central lattice $\Xi$ on which
$\omega_\sigma$ is trivial, $h$ lies in the dual of
$\mathcal Q_{N,\Xi,u,\chi,1}$ for a suitable level $N$, place $u$
and integral character $\chi$.
\end{enumerate}
\end{corollary}

\begin{proof}
We may enlarge the level $N$ so that $\sigma$ is unramified
outside $N$ and $\omega_\sigma$ is trivial on
$T_G(\mathbb A)\cap K_N$.

\step{1}{the central character} By the proof of
Lemma~\ref{fl:lem:central-hecke-units}, the translation by an
element of $T_G(\mathbb A)$ is, on functions of level $N$, a
product of translations by elements $\lambda(\varpi_v)$ with
$v\notin N$, and such a translation is the Hecke operator of the
representation $\alpha_G^*\lambda$ of $\widehat G$. By
Proposition~\ref{fl:prop:canonical-compact}(3) a Hecke operator at
$v$ is an excursion operator with a Frobenius element at $v$, so
$h$ is an eigenvector of these translations. The eigenvalue is
$\omega_\sigma(\lambda(\varpi_v))$. Indeed, by
Section~\ref{fl:sec:parameters-parameters} the excursion operator
$S_e$ attached to $\alpha_G^*\lambda$ and to a geometric Frobenius
element $\operatorname{Frob}_v$ is the translation
$T_{\lambda(\varpi_v)}$ on functions with compact support; its
transpose is $T_{\lambda(\varpi_v)^{-1}}$, so
$S_e^{\mathrm{fun}}=T_{\lambda(\varpi_v)}$ on all functions. Its
eigenvalue on $h$ is
\[
 \nu_\sigma(e)=\lambda\bigl(\alpha_G\sigma(\operatorname{Frob}_v)\bigr)
  =\lambda\bigl(\alpha_G\sigma(\operatorname{Art}(\varpi_v)^{-1})\bigr)
  =\omega_\sigma(\lambda(\varpi_v)),
\]
because $\operatorname{Art}(\varpi_v)$ is an arithmetic Frobenius
element. Since $\omega_\sigma$ is trivial on $T_G(F)$ and on
$T_G(\mathbb A)\cap K_N$, this proves (3).

\step{2}{the case of a central character trivial on a lattice}
Suppose that $\omega_\sigma$ is trivial on $\Xi$. Then $h$ is
invariant under $\Xi$. Let $u\notin N$. By
\eqref{fl:eq:chevalley-functional} and
Proposition~\ref{fl:prop:canonical-compact}(3), the linear form
$c\mapsto[c,h]$ on functions with compact support is an
eigenvector of the Hecke algebra at $u$, for the character $\chi$
defined by $\vartheta_G\sigma(\operatorname{Frob}_u)$. This
character is integral by
Lemma~\ref{fl:lem:parameter-satake-integrality}. The linear form
vanishes on the image of $\ker\chi$, so $h$ lies in the dual of
$\mathcal Q_{N,\Xi,u,\chi,1}$, which is (4), and in particular
$h\in\mathcal A_\Xi(G)$. On this space of finite dimension both
$S_e^{\mathrm{fun}}h$ and $\nu_\sigma(e)h$ are continuous in the
tuple, and they agree on the dense subset $W_F^I$ of $\Gamma_F^I$;
this is (2). The function is an automorphic form by
Corollary~\ref{fl:cor:excursion-space-automorphicity}.

\step{3}{the general case} Write
$\sigma=\psi\sigma^{\mathrm{fin}}$ as in
Lemma~\ref{fl:lem:finite-coefficients}. By
Proposition~\ref{fl:prop:compact-central-twists} the function
$T_\psi^{-1}h$ satisfies the hypothesis of the corollary for
$\sigma^{\mathrm{fin}}$, whose central character has finite order
and is therefore trivial on some central lattice. By step~2, $T_\psi^{-1}h$ lies in
$\mathcal A^{\mathrm{per}}(G)$ and satisfies (2) for
$\sigma^{\mathrm{fin}}$. Hence
$h\in\mathcal A^{\mathrm{cont}}(G)$, and (2) for $h$ and $\sigma$
follows from \eqref{fl:eq:transported-full-action}.
\end{proof}

\subsection{An example: the multiplicative group}
\label{rw:sec:example-gm}

For $G=\mathbb G_m$ the statements of this section are class field
theory, and it may help to see them in this form. Fix a level $N$.
The functions of level $N$ are the functions on the abelian group
\[
 A_N=F^\times\backslash\mathbb A_F^\times/K_N ,
\]
which the degree maps onto $\mathbb Z$ with finite kernel. The
dual group is $\mathbb G_m$, and a function on $\widehat G^I$
invariant under the diagonal translations is a combination of
monomials $\prod_iz_i^{n_i}$ with $\sum_in_i=0$. For the monomial
$z_1^nz_2^{-n}$ and the tuple $(\operatorname{Frob}_v,1)$, with
$v\notin N$, the excursion operator is the translation
$T_{\varpi_v^n}$, where $\operatorname{Frob}_v$ is a geometric
Frobenius element (Section~\ref{fl:sec:parameters-parameters}).

\emph{Eigenfunctions.} A character $\chi:A_N\to k^\times$ is an
eigenfunction of all the translations, and
$T_{\varpi_v^n}\chi=\chi(\varpi_v)^n\chi$. No character has compact
support. This is the reason for which the excursion operators,
which are defined first on functions with compact support, have to
be transposed to spaces of functions without this property.

\emph{The space $\mathcal A^{\mathrm{cont}}(G)$.} A function of
level $N$ which is invariant under a central lattice $\Xi$ is
a function on the finite group $A_N/\Xi$, and every such function
has a finite integral orbit under the translations. So the part of
$\mathcal A^{\mathrm{per}}(G)$ of level $N$ is spanned by the
characters of $A_N$ of finite order. A twist by $c^{\deg_F}$
multiplies a function by $t\mapsto c^{-\deg(t)}$, and a character
of $A_N$ whose values are $\ell$-adic units is the product of a
character of finite order and of a character of this form, with
$c$ a unit. Hence the part of $\mathcal A^{\mathrm{cont}}(G)$ of
level $N$ is spanned by the characters of $A_N$ with values in the
$\ell$-adic units. These are the characters
$\operatorname{CFT}_{\mathbb G_m}(\sigma)$ attached to the
continuous characters $\sigma$ of $\Gamma_F$: a character of the
idele class group extends continuously to the profinite group
$\Gamma_F^{\mathrm{ab}}$ if and only if its values are units.

\emph{Excursion summands.} Let $\sigma$ be a continuous character
of $\Gamma_F$ and $\chi=\operatorname{CFT}_{\mathbb G_m}(\sigma)$.
By Corollary~\ref{fl:cor:weil-to-galois-nonzero}(3) a joint
eigenfunction of the excursion operators with parameter $\sigma$
is a multiple of $\chi$. Conversely $\chi$ is such an
eigenfunction. Indeed, write it as a twist of a character $\chi_0$
of finite order. The excursion operators commute with the
translations, for which the line of $\chi_0$ is a joint eigenspace
in the functions on $A_N/\Xi$; so $\chi_0$ is a joint eigenfunction
of the excursion operators, and its parameter is determined by the
values $\chi_0(\varpi_v)$ at the Frobenius elements, by the theorem
of Chebotarev. The twist is
treated by \eqref{fl:eq:transported-full-action}. So the excursion
summand of $\sigma$ is the line of $\chi$, and the decomposition
of $\mathcal A^{\mathrm{cont}}(G)$ into excursion summands is its
decomposition into characters.

\emph{What is left out.} The character $t\mapsto c^{-\deg(t)}$,
for $c$ which is not a unit, is an eigenfunction of all the
translations, but the corresponding character of the Weil group
does not extend continuously to $\Gamma_F$, and the function is
not in $\mathcal A^{\mathrm{cont}}(G)$. The function $\deg$ is
killed by $(T_a-1)^2$ for every $a$, without being a sum of
eigenfunctions (Remark~\ref{rw:rem:not-all-duals}). For a group
which is not a torus the eigenfunctions with a given parameter
are no longer determined in this way; what
Sections~\ref{fl:sec:constant-term}
and~\ref{fl:sec:arithmetic-cuspidality} add is a control of their
constant terms.