\subsection{The form used by the geometric construction}
\label{fl:sec:arithmetic-excursion}

The geometric construction of \cite{HuII} produces
a function $\varphi$ and proves an eigenvalue equation for the
functional $c\mapsto[c,\varphi]$ on functions with compact support.
Transposing this equation into one for $\varphi$ itself introduces
the Chevalley involution, by \eqref{fl:eq:chevalley-functional}.
The following proposition records the outcome.

\begin{proposition}[From a functional to a cusp form]
\label{fl:prop:cuspidal-excursion-character}
Let $G/F$ be split semisimple, let $N$ be a level, and let
$\varphi\ne0$ be a function on
$\operatorname{Bun}_{G,N}(\mathbb F_q)$. Let
$\sigma_{\mathrm{coeff}}:\pi_1(X-\operatorname{supp}N)\to\widehat G(k)$
be continuous, semisimple and elliptic. Suppose that the linear
form $L_\varphi=[\ ,\varphi]$ on functions with compact support
satisfies
\begin{equation}\label{fl:eq:functional-excursion}
 L_\varphi S_e=\nu_{\sigma_{\mathrm{coeff}}}(e)L_\varphi
\end{equation}
for every excursion datum $e$ with tuple in the Weil group. Then
$\varphi$ is cuspidal with compact support, and
\[
 S^{\mathrm{fun}}_{I,f,\boldsymbol\gamma}\varphi
  =f\bigl(((\vartheta_G\sigma_{\mathrm{coeff}})(\gamma_i))_{i\in I}\bigr)
      \varphi
 \qquad\text{for all }\boldsymbol\gamma\in\Gamma_F^I .
\]
\end{proposition}

\begin{proof}
For a function $c$ with compact support and a tuple in the Weil
group, \eqref{fl:eq:function-action} and
\eqref{fl:eq:chevalley-functional} give
\[
 [c,S_e^{\mathrm{fun}}\varphi]
   =[S_{e^\star}c,\varphi]
   =\nu_{\sigma_{\mathrm{coeff}}}(e^\star)[c,\varphi]
   =\nu_{\vartheta_G\sigma_{\mathrm{coeff}}}(e)[c,\varphi].
\]
Taking for $c$ the characteristic functions of points, whose
weights in the pairing are not zero, we get
$S_e^{\mathrm{fun}}\varphi
=\nu_{\vartheta_G\sigma_{\mathrm{coeff}}}(e)\varphi$. The
parameter $\vartheta_G\sigma_{\mathrm{coeff}}$ is continuous,
finitely ramified, semisimple and elliptic, like
$\sigma_{\mathrm{coeff}}$. So $\varphi$ is cuspidal by
Proposition~\ref{fl:prop:full-label-cuspidality}, with compact
support because $G$ is semisimple, and the equations hold for all
Galois tuples by Corollary~\ref{fl:cor:weil-to-galois-nonzero}.
The level is $N$ throughout.
\end{proof}

\subsection{The Chevalley involution and excursion summands}

The parameters $\sigma$ and $\vartheta_G\sigma$ play symmetric
roles for the existence of cusp forms. On the spaces of cusp forms
invariant under a central lattice this is the transposition formula of W.-W.~Li
\cite[Lemma~5.3.3 and Proposition~5.3.4]{Li19}.

\begin{lemma}[Duality of excursion summands]
\label{rw:lem:chevalley-dual-summand}
Let $N$ be a level, $\Xi$ a central lattice, and
$C^{\mathrm{cusp}}$ the space of cusp forms of level $N$
invariant under $\Xi$, with values in $k$.
\begin{enumerate}
\item The pairing $[\ ,\ ]$ of
$Y_{N,\Xi}=G(F)\backslash G(\mathbb A)/K_N\Xi$ is nondegenerate
on $C^{\mathrm{cusp}}$.
\item For all excursion data $e$ with tuples in $\Gamma_F^I$ and
all $c,h\in C^{\mathrm{cusp}}$,
$[S_ec,h]=[c,S_{e^\star}h]$.
\item The pairing identifies the excursion summand of
$\vartheta_G\sigma$ in $C^{\mathrm{cusp}}$ with the dual of the
excursion summand of $\sigma$. In particular one is zero if and
only if the other is.
\end{enumerate}
\end{lemma}

\begin{proof}
(1) By Harder's theorem the cusp forms of level $N$ invariant
under $\Xi$ are supported on a finite subset $Y_0$ of
$Y_{N,\Xi}$. For a function of level $N$ the constant term at a
point is a finite sum of values of the function with positive
rational coefficients, if the Haar measures are chosen so that
open compact subgroups have rational volumes. So
$C^{\mathrm{cusp}}$ is the space of solutions in $k^{Y_0}$ of a
system of linear equations with rational coefficients, and it is
spanned by the space $C^{\mathrm{cusp}}_{\mathbb Q}$ of solutions
with rational values. The weights $w_N(y)$ of the pairing are
positive rational numbers, so the pairing is positive definite on
$C^{\mathrm{cusp}}_{\mathbb Q}$. Hence it is nondegenerate on
$C^{\mathrm{cusp}}_{\mathbb Q}$ and on
$C^{\mathrm{cusp}}=C^{\mathrm{cusp}}_{\mathbb Q}\otimes k$.

(2) The pairing with the weights $1/|\operatorname{Aut}(y)|$ is
the pairing by integration for the measure which gives $K_N$ the
volume one, as in \cite[Remark~2.2.3]{Li19}. Dividing by the
lattice does not change the stabilizers: the lattice acts freely on
the adelic double quotient, because the degree, with values in
$X_*(Q_G)$, is injective on it. The identity for all Galois tuples
is then \cite[Lemma~5.3.3]{Li19}, with
$e^\star=(I,f^\dagger,\boldsymbol\gamma^{-1})$.

(3) Let $V_\nu$ and $V_{\nu'}$ be the generalized eigenspaces in
$C^{\mathrm{cusp}}$ of two characters of the algebra generated by
the excursion operators. Suppose that $\nu(e)\ne\nu'(e^\star)$
for some $e$. Then $S_{e^\star}-\nu(e)$ is invertible on
$V_{\nu'}$, while a power of $S_e-\nu(e)$ annihilates $V_\nu$; by
(2), $[V_\nu,V_{\nu'}]=0$. Since $C^{\mathrm{cusp}}$ is the
direct sum of the generalized eigenspaces, (1) shows that the
pairing between $V_\nu$ and $V_{\nu'}$ is perfect when
$\nu'(e^\star)=\nu(e)$ for all $e$. For $\nu=\nu_\sigma$ this
character $\nu'$ is $\nu_{\vartheta_G\sigma}$, because
$\nu_\sigma(e^\star)=\nu_{\vartheta_G\sigma}(e)$
by \eqref{fl:eq:chevalley-functional}.
\end{proof}