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

We fix the normalizations of Frobenius, of the Satake equivalence
and of class field theory used in the whole paper, and define the
excursion summand attached to a parameter. 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}.

\subsection{Frobenius and the Satake equivalence}

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 $1$. A
geometric Frobenius element at a place $v$ of degree $d_v$ has
degree $-d_v$. 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{The centre and class field theory}

Let $T_G=(Z_G^\circ)_{\mathrm{red}}$ and
\[
 C_{T_G}=T_G(F)\backslash T_G(\mathbb A_F),\qquad
 \deg(t)(\chi)=\sum_v d_v\,v(\chi(t_v))
\]
for a character $\chi$ of $T_G$. This defines a homomorphism
$\deg:C_{T_G}\to X_*(T_G)$. The degree of the bundle attached to
$t$ is $-\deg(t)$.

\begin{definition}\label{fl:def:central-period-lattice}
A \emph{central lattice} is a free subgroup
$\Xi\subset C_{T_G}$ on which $\deg$ is injective and whose image
under $\deg$ has finite index in $X_*(T_G)$. A function on
$G(F)\backslash G(\mathbb A_F)$ is \emph{invariant under $\Xi$}
if it is invariant under translation by the elements of $\Xi$.
\end{definition}

The quotient of $C_{T_G}$ by a central lattice and an open
compact subgroup is finite.

Class field theory is normalized by the arithmetic Artin map,
$\operatorname{Art}_v(\varpi_v)=\Frob_v^{\mathrm{arith}}$, and a
character $\chi$ of $\Gamma_F$ corresponds to the character
\[
 \operatorname{CFT}_{\Gm}(\chi)(t)
       =\chi(\operatorname{Art}(t)^{-1})
\]
of the idele class group; for a split torus one takes the product
over a basis of characters. Right translation by $a$ is denoted
$T_a$, $(T_ah)(g)=h(ga)$. With these conventions the excursion
operator of a character $\lambda$ of $\widehat T_G$ and a
geometric Frobenius element at $v$ is $T_{\lambda(\varpi_v)}$
\cite[Proposition~6.2]{VLa18}, and a parameter of the form
$c^{\deg_F}$ corresponds to the character $t\mapsto c^{-\deg(t)}$.
Let $\alpha_G:\widehat G\to\widehat T_G$ be the homomorphism dual
to the inclusion $T_G\to G$. The \emph{central character} of a
parameter $\sigma$ is
\[
 \omega_\sigma=\operatorname{CFT}_{T_G}(\alpha_G\sigma),
\]
a character of $C_{T_G}$. This is the character which occurs in
Theorem~\xref{I}{fl:thm:central-period-realization}.

\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.