\section{Results used from the literature}
\label{rw:sec:inputs}

The proofs rest on a small number of deep theorems. We collect
here, in the form in which they are used, those which enter the
present paper, so that the reader can see what is imported and
where. Precise references to the individual statements are repeated
at each point of use. The results used only in the two
constructions are listed in the sequels.

\begin{extinput}[General linear groups]\label{rw:in:llafforgue}
Let \(\varrho\) be a continuous irreducible \(n\)-dimensional
\(\overline{\mathbb Q}_\ell\)-representation of \(\Gamma_F\),
unramified at almost all places, whose determinant has finite order.
\begin{enumerate}
\item There is a cuspidal automorphic representation \(\pi\) of
\(\mathrm{GL}_n(\mathbb A_F)\) whose Satake parameters at the
unramified places are the eigenvalues of geometric Frobenius
elements in \(\varrho\)
\cite[Introduction, Theorem \textup{(i)}$_n$--\textup{(ii)}$_n$]{LLa02}.
\item The Frobenius eigenvalues of \(\varrho\) at the unramified
places are algebraic numbers all of whose complex absolute values
are one \cite[Theorem~VII.6]{LLa02}.
\item The representation \(\pi\) is tempered at every place
\cite[Theorem~VI.10(i)]{LLa02}.
\end{enumerate}
\end{extinput}

In part (1) the Hecke operators act by right translation, as in
Section~\ref{fl:sec:parameters-parameters}. The statement does not
depend on the conventions of \cite{LLa02} for Hecke operators and
for Frobenius elements: a change of either convention replaces
$\varrho$ by its dual and $\pi$ by its contragredient, and the
theorem applies to all irreducible representations.

Part (2) is the source of the purity theorem of
Section~\ref{fl:sec:arithmetic-purity}. Parts (1) and (3) are not
used in this paper; they are the starting point of \cite{HuIII}.

\begin{extinput}[Excursion operators on cusp forms]\label{rw:in:vlafforgue}
For a split reductive group \(G\) and a level \(N\), the excursion
operators \(S_{I,f,\boldsymbol\gamma}\) form a commutative algebra
of endomorphisms of the space of cusp forms of level \(N\) on the
quotient by a central lattice. They commute with the Hecke algebra,
and at an unramified place \(v\) the operator attached to a
representation \(V\) of \(\widehat G\) and a Frobenius element at
\(v\) is the Hecke operator corresponding to \(V\) under the Satake
isomorphism. Every character of this algebra has the form
\(\nu_\sigma\) for a semisimple parameter \(\sigma\), unique up to
conjugacy
\cite[Theorem~0.1, Definition--Proposition~11.3 and Proposition~11.7]{VLa18}.
\end{extinput}

\begin{extinput}[Shtukas beyond the cuspidal part]\label{rw:in:xue}
We use the following results of Xue on the cohomology of stacks of
shtukas with compact support, without passage to the cuspidal part.
\begin{enumerate}
\item With integral coefficients, and after dividing by a central
lattice, the cohomology at the generic point of the space of legs
is a finitely generated module over the spherical Hecke algebra at
any one unramified place \cite[Theorem~5.0.4]{XueIntegral2023}.
\item The specialization map of the sheaf, from its stalk at the
geometric generic point of the diagonal to its stalk at the
geometric generic point of the space of legs, is an isomorphism,
and the action of partial
Frobenius maps and the fundamental group factors through
\(W_F^I\)
\cite[Propositions~8.1.2 and~8.2.6]{XueIntegral2023}; after
\(\ell\) is inverted, the action is unramified outside the level
\cite[Proposition~5.0.4]{XueSmoothCoh2020}.
\item Excursion operators act on functions with compact support and
satisfy the same relations as on cusp forms
\cite[\S3.4]{XueHecke2020}.
\item The constant-term map along a parabolic subgroup \(P=MU\)
is realized by a morphism from the cohomology of shtukas for \(G\)
to that for \(M\), compatible with partial Frobenius maps and with
creation and annihilation
\cite[Theorem~3.2.6 and Proposition~3.5.3]{XueCusp2020},
\cite[Proposition~4.1.3]{XueHecke2020}.
\end{enumerate}
\end{extinput}

\begin{extinput}[Automorphic forms over function fields]
\label{rw:in:function-field-forms}
\begin{enumerate}
\item Cusp forms of fixed level on a group over a function field
have compact support modulo the centre
\cite[Chapter~I, \S5.3 and Proposition~5.3]{Morris1982}.
\item Eisenstein series over function fields continue
meromorphically, with the constant-term formulas of
\cite[(6.2), (6.4), (6.17)]{G2BernsteinLapid24}. In the same
work automorphic forms are characterized by finite systems of
difference equations satisfied by the cuspidal projections of
their constant terms
\cite[\S9.1]{G2BernsteinLapid24}.
\end{enumerate}
\end{extinput}