\section{Conventions}
\label{fl:sec:parameters-parameters}
\label{fl:app:conventions}

We fix the normalizations of Frobenius and of trace functions used
in the whole paper, define the excursion summand attached to a
parameter, and recall the Chevalley involution. The spaces of
functions on which excursion operators act are constructed in
Sections~\xref{I}{fl:sec:compact-foundations}
and~\xref{I}{rw:sec:function-spaces-section}. The conventions of
\cite[Section~\xnum{I}{fl:sec:parameters-parameters}]{HuI} on the
centre and on class field theory are not needed here, because $G$
is semisimple and simply connected.

\subsection{Frobenius and trace functions}

Let $\deg_F:\Gamma_F\to\widehat{\mathbb Z}$ be the homomorphism
given by the action on the constant field, normalized so that an
arithmetic Frobenius element has degree $1$. The Weil group $W_F$
is the inverse image of $\mathbb Z$; it is dense in $\Gamma_F$. A
geometric Frobenius element at a place $v$ of degree $d_v$ has
degree $-d_v$. We write $\operatorname{Fr}_q$ for the Frobenius
endomorphism of a scheme over $\mathbb F_q$, and $\Frob_q$ for the
action of geometric Frobenius on cohomology.

For a bounded complex $\mathcal K$ with a Weil structure and a
rational point $x$, the trace function is
\[
 t_{\mathcal K}(x)=\sum_i(-1)^i
       \operatorname{Tr}(\Frob_q;H^i(\mathcal K_x)),
\]
and we write $\operatorname{Str}$ for an alternating trace of this
kind. A shift by one changes the sign of the trace function.
Tate twists and shifts are always kept when trace functions are
compared. The Satake equivalence, for $G$ and for all its Levi
subgroups, is normalized by one square root $q^{1/2}\in k$ chosen
once and for all. When $q$ is the square of an integer we take
the positive root; then in all cases there is an isomorphism of
$k$ with $\mathbb C$ which maps $q^{1/2}$ to the positive square
root of $q$.

\subsection{Excursion summands}

\begin{definition}\label{fl:def:full-excursion-summand}
Let $V$ be a finite-dimensional vector space on which the
excursion operators $S_e$ act, for all finite sets $I$, all
functions $f$ and all tuples in $\Gamma_F^I$; they generate a
commutative algebra $\mathcal B$ of endomorphisms. Let
$\mathfrak m_\sigma\subset\mathcal B$ be the ideal generated by
the operators $S_e-\nu_\sigma(e)$. The \emph{excursion summand}
of $\sigma$ in $V$ is the generalized eigenspace
\[
 V_{\nu_\sigma}=\bigcup_{n\ge1}
     \{v\in V:\mathfrak m_\sigma^nv=0\}.
\]
For a space which is a union of subspaces $V$ of this kind, the
excursion summand is the union of the $V_{\nu_\sigma}$.
\end{definition}

The definition involves the excursion operators for all tuples
in the Galois group; the Hecke operators at the unramified places
are among them. A nonzero excursion summand in a space of
finite dimension contains a joint eigenvector of the excursion
operators, with eigenvalues $\nu_\sigma(e)$: the commuting
operators $S_e-\nu_\sigma(e)$ are nilpotent on it. So the
existence of a nonzero summand and the existence of an
eigenfunction are equivalent.

\subsection{The Chevalley involution}

Let $\vartheta_G$ be the Chevalley involution of $\widehat G$ which
preserves the pinning. On the maximal torus of the pinning it is
$t\mapsto w_0(t^{-1})$, where $w_0$ is the longest element of the
Weyl group \cite[Definition~3.1.1]{Li19}. We put
\[
 \sigma^\dagger=\vartheta_G\circ\sigma .
\]
The parameter $\sigma^\dagger$ appears because the pairing
between functions with compact support and arbitrary functions
is bilinear: transposition of an excursion operator replaces
$\sigma$ by $\sigma^\dagger$, see
\xeqref{I}{fl:eq:chevalley-functional}. The letter $\sigma$ always
denotes the parameter of the main theorems.