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

We fix the normalizations of Frobenius, of trace functions, 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~\ref{fl:sec:compact-foundations}
and~\ref{rw:sec:function-spaces-section}.

\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. We fix one square root $q^{1/2}\in k$, 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 Satake equivalence}
\label{rw:sec:satake-convention}

The Satake sheaves, for $G$ and for all its Levi subgroups, are
those of \cite[Th\'eor\`eme~1.17]{VLa18}, normalized to be pure of
weight zero by the square root of $q$ just fixed. The commutativity
constraint is the modified one \cite[Remarque~1.19]{VLa18}. The
sheaf of the irreducible representation of $\widehat G$ of highest
weight $\omega$ is the intersection complex of the space of
modifications $\phi:\mathcal G\to\mathcal G'$ at a point $x$ in
which the position of the source relative to the target is bounded
by $\omega$: for every dominant weight $\lambda$ of $G$, with Weyl
module $V_\lambda$,
\begin{equation}\label{rw:eq:source-bound}
 \phi(\mathcal G_{V_\lambda})\subset
   \mathcal G'_{V_\lambda}(\langle\lambda,\omega\rangle x)
\end{equation}
\cite[D\'efinition~1.2]{VLa18}. For a line bundle and $\omega=r$
the target is the source twisted by $\mathcal O(-rx)$. For
$\mathrm{GL}_r$ and the standard representation the source
contains the target with a quotient of length one, and for
$I=\{1,2\}$ and $\mathrm{Std}\boxtimes\mathrm{Std}^*$ one obtains
Drinfeld's left shtukas, with the zero at the first leg
\cite[Remarque~2.2\,b)]{VLa18}.

Some of the results we use are stated in terms of the loop group,
and we record the translation. Trivialize the target on the formal
disc at $x$, with a uniformizer $t$. A modification is then given
by its source, a point $gG(\mathcal O)$ of the affine Grassmannian;
for $\mathrm{GL}_r$ it is the lattice $g\mathcal O^r$. By
\eqref{rw:eq:source-bound} the modifications bounded by $\omega$
form the closure of the orbit of $t^{-\omega}$. Let
$\mathsf{Sat}^{\rm lp}_G$ be the Satake equivalence in which the
representation of highest weight $\lambda$ corresponds to the
closure of the orbit of $t^\lambda$, as in \cite{MV2007} and
\cite[\S5.3]{Zhu2017}. Fix a maximal torus of $\widehat G$ and an
automorphism $\epsilon_G$ of $\widehat G$ which is $s\mapsto s^{-1}$
on this torus. Then, as tensor functors,
\begin{equation}\label{rw:eq:satake-loop}
 \mathsf{Sat}_G(V)\simeq\mathsf{Sat}^{\rm lp}_G(V\circ\epsilon_G).
\end{equation}
Indeed, two tensor equivalences from the representations of
$\widehat G$ to the same category differ by a tensor autoequivalence
of the category of representations, which, $k$ being algebraically
closed, is composition with an automorphism of $\widehat G$,
determined up to inner automorphisms. Here it sends the
representation of highest weight $\omega$ to that of highest weight
$-w_0\omega$, because $t^{-\omega}$ and $t^{-w_0\omega}$ lie in the
same orbit. So does $\epsilon_G$; and an automorphism which
preserves the isomorphism class of every irreducible representation
is inner, as one sees on the based root datum.

The automorphism $\epsilon_G$ preserves every Levi subgroup
$\widehat M$ which contains the torus, and with
$\epsilon_M=\epsilon_G|_{\widehat M}$ the formula
\eqref{rw:eq:satake-loop} holds for $M$. Restriction of
representations from $\widehat G$ to $\widehat M$ commutes with
composition with $\epsilon_G$ and $\epsilon_M$. Hence the
restriction of Satake sheaves from $G$ to $M$
\cite[Remark~5.3.16]{Zhu2017} corresponds to restriction of
representations in our normalization too, and the summand on which
the centre of $\widehat M$ acts by the weight $\mu$ lies on the
component of $t^{-\mu}$ of the affine Grassmannian of $M$: it
consists of modifications of $M$-bundles which change the degree
from $\nu$ to $\nu-\mu$.

The normalization is determined by the Hecke operators, in the
following sense.

\begin{lemma}[Change of the Satake normalization]
\label{rw:lem:satake-label}
Let $\delta$ be an automorphism of $\widehat G$. If the Satake
sheaf of every representation $V$ of $\widehat G^I$ is replaced by
that of $V\circ\delta^I$, the excursion operator
$S_{I,f,\boldsymbol\gamma}$ of
Proposition~\ref{fl:prop:canonical-compact} is replaced by
$S_{I,f\circ\delta^I,\boldsymbol\gamma}$. If the operators so
obtained also have property~\textup{(3)} of that proposition, the
compatibility with the Hecke operators at the unramified places,
then $\delta$ is inner, and the excursion operators are unchanged.
\end{lemma}
\begin{proof}
The operator $S_{I,f,\boldsymbol\gamma}$ is constructed from a
representation $V$ of $\widehat G^I$, an invariant vector $x$ and an
invariant linear form $\xi$ with $f(g)=\xi(gx)$. For $V\circ\delta^I$,
with the same $x$ and $\xi$, the function is $f\circ\delta^I$. This
is the first assertion.

Let $v$ be a place and $V$ an irreducible representation of
$\widehat G$, and let $h_V$ be the spherical function at $v$ which
corresponds to $V$ under the Satake isomorphism. By property~(3)
for the two families of operators, the Hecke operators of $h_V$ and
of $h_{V\circ\delta}$ coincide on $C_{N,E}$, for every level $N$
which does not contain $v$. The spherical Hecke algebra at $v$
acts faithfully on the union of these spaces. To see this let $h\ne0$ be
a spherical function and choose $r\in G(F_v)$ with $h(r^{-1})\ne0$.
The set of $\gamma\in G(F)$ which lie in
$r\operatorname{supp}(h)\prod_wG(\mathcal O_w)$ is finite, being the
intersection of a discrete set and a compact set. For each
$\gamma\ne1$ in it there is a level $N$, prime to $v$, with
$\gamma\notin r\operatorname{supp}(h)K_N$; take $N$ which works for
all of them, and let $c$ be the characteristic function of the
class of $1$ in $Y_N$. The value at $r$ of the Hecke operator of
$h$ applied to $c$ is
\[
 \int_{G(F_v)}h(x)c(rx)\,dx=\int_{r^{-1}G(\mathcal O_v)}h(x)\,dx
   =h(r^{-1})\ne0 .
\]
Hence $h_V=h_{V\circ\delta}$, and by the Satake isomorphism
$V\circ\delta$ is isomorphic to $V$. An automorphism with this
property for every $V$ is inner, and then $f\circ\delta^I=f$ for
every function $f$ which is invariant under the diagonal
conjugation.
\end{proof}

By \eqref{rw:eq:satake-loop} and the lemma, the excursion operators
constructed with $\mathsf{Sat}^{\rm lp}_G$ are those of the
parameters $\epsilon_G^{-1}\circ\sigma$: a function with parameter
$\sigma$ in the sense of this paper has parameter
$\epsilon_G^{-1}\circ\sigma$ in that normalization. The automorphism
$\epsilon_G$ differs from the Chevalley involution $\vartheta_G$ of
Section~\ref{rw:sec:chevalley-involution} by an inner automorphism,
so this parameter is conjugate to $\vartheta_G\circ\sigma$.

\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~\ref{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.

\subsection{The Chevalley involution}
\label{rw:sec: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
\eqref{fl:eq:chevalley-functional}. The letter $\sigma$ always
denotes the parameter of the main theorems.