\section{Proofs of the realization theorems}
\label{fl:sec:semisimple-proof}
\label{fl:sec:reductive-proof}

We now prove the three theorems of the introduction. The geometric
construction (Theorem~\ref{rw:thm:intro-geometric}) and the
classical one (Theorem~\ref{rw:thm:intro-classical}) both
give, for an
elliptic parameter of a simply connected simple group, a nonzero
cuspidal excursion summand. Between them they cover every type
under the hypothesis on the characteristic. Theorem~A follows by
taking products and passing to an isogenous group
(Section~\ref{fl:sec:arithmetic-central}); Theorem~B by passing to
a minimal Levi subgroup and applying
Theorem~\ref{fl:thm:eisenstein-full-label}; Theorem~C from the
central character.

\subsection{Semisimple groups}
\label{rw:sec:proof-semisimple}

The classical types are covered in every positive characteristic
by Theorem~\ref{rw:thm:intro-classical}. For the exceptional
types we use the geometric construction, which requires that $p$
does not divide the order of the Weyl group:
\[
\begin{array}{c|c|c}
\text{type}&|W|&\text{prime divisors}\\ \hline
G_2&2^2\,3&2,3\\
F_4&2^7\,3^2&2,3\\
E_6&2^7\,3^4\,5&2,3,5\\
E_7&2^{10}\,3^4\,5\,7&2,3,5,7\\
E_8&2^{14}\,3^5\,5^2\,7&2,3,5,7.
\end{array}
\]
For these types the condition \(p\nmid|W_G|\) is the condition
\(p>b(\Phi)\) of \eqref{fl:eq:root-bound-values}.

\begin{proof}[Proof of Theorem~\ref{fl:thm:semisimple-elliptic}]
If \(G\) is trivial, the constant function one is the required
form. Otherwise let \(G^{\mathrm{sc}}=\prod_{a\in A}G_a\) be the
simply connected cover of $G$, with the \(G_a\) simple, and let
\[
 \mathsf a:\widehat G\longrightarrow
       \widehat{G^{\mathrm{sc}}}
       =\prod_{a\in A}\widehat G_a,\qquad
 \sigma_a=\pr_a\mathsf a\sigma .
\]

\step{1}{the parameters of the factors} The map \(\mathsf a\) is a central
isogeny of groups over a field of characteristic zero, so it
induces an isomorphism of Lie algebras compatible with the adjoint
actions, and
\[
 \Lie C_{\widehat G}(\sigma)
   \simeq\bigoplus_{a\in A}
                 \Lie C_{\widehat G_a}(\sigma_a).
\]
The left-hand side is zero because $\sigma$ is elliptic. So each
\(\sigma_a\) is elliptic. It is semisimple because the image of a
reductive group under a homomorphism is reductive, and it is
continuous and finitely ramified.

\step{2}{the factors} For each $a$ the cuspidal excursion summand of $\sigma_a$ on $G_a$ is nonzero at some finite level: for
a factor of type $G_2$, $F_4$, $E_6$, $E_7$ or $E_8$ by
Theorem~\ref{rw:thm:intro-geometric}, whose hypothesis
holds since $p>b(G)\ge b(G_a)$, and for the other factors by
Theorem~\ref{rw:thm:intro-classical}. The summand
has finite dimension, so it contains a joint eigenvector \(v_a\)
of the excursion operators (see the remark after
Definition~\ref{fl:def:full-excursion-summand}).

\step{3}{the product} Let \(v=\bigotimes_av_a\), a function on
$G^{\mathrm{sc}}(F)\backslash G^{\mathrm{sc}}(\mathbb A_F)$. It is
not zero: evaluate it at a point whose components are points where
the \(v_a\) do not vanish. It is cuspidal: a proper parabolic
subgroup of a product is proper in at least one factor, and the
constant term of $v$ along it has as a factor a constant term of
the corresponding cusp form $v_a$. It is an eigenvector of the
excursion operators with character \(\nu_{\mathsf a\sigma}\): for every
finite set $I$ of legs, the ring of functions on
$\widehat{G^{\mathrm{sc}}}{}^I$ invariant under left and right
diagonal translation is the tensor product of the corresponding
rings of the factors, since each factor acts on its own
coordinates only; so every excursion operator of
$G^{\mathrm{sc}}$ is a sum of products of excursion operators of
the factors, and it acts on $v$ by its value at $\mathsf a\sigma$.

\step{4}{the isogenous group $G$} By
Theorem~\ref{fl:thm:arithmetic-central-bridge}, applied to
$G=G^{\mathrm{sc}}/C$ and to the summand of $\mathsf a\sigma$ which
contains $v$, the cuspidal excursion summand of \(\sigma\) on
\(G\) is nonzero. The theorem gives the parameter
$\sigma$ itself, and not only its image $\mathsf a\sigma$. A joint
eigenvector in this summand is the form \(\varphi_\sigma\).
\end{proof}

\begin{proof}[Proof of Corollary~\ref{rw:cor:intro-hecke}]
Let $S$ contain the places at which $\sigma$ is ramified and those
at which the component of $K$ is not $G(\mathcal O_v)$. For
$v\notin S$ and a representation $V$ of $\widehat G$, the excursion
operator attached to $V$ and $\Frob_v$ is the Hecke operator
$h_{V,v}$ which corresponds to $V$ under the Satake isomorphism
(Input~\ref{rw:in:vlafforgue}). So
\[
 h_{V,v}\varphi_\sigma=\operatorname{Tr}(\sigma(\Frob_v);V)\,
 \varphi_\sigma .
\]
The representation of $G(\mathbb A_F)$ generated by
$\varphi_\sigma$ is contained in the space of cusp forms, so it is
semisimple. Let $\pi$ be an irreducible constituent on which
$\varphi_\sigma$ has a nonzero projection. The projection commutes
with $G(\mathbb A_F)$, hence with the Hecke operators. Its value on
$\varphi_\sigma$ is therefore, for every $v\notin S$, a nonzero
vector of $\pi$ fixed by $G(\mathcal O_v)$ on which the spherical
Hecke algebra at $v$ acts through the same character.
\end{proof}

\subsection{Reductive groups and the central character}

\begin{proof}[Proof of Theorem~\ref{fl:thm:main}]
By Corollary~\ref{fl:cor:elliptic-minimal-levi-parameter} we may
write \(\sigma=i\sigma_M\), where
\(i:\widehat M\hookrightarrow\widehat G\) is the inclusion of a
minimal Levi subgroup containing the image and \(\sigma_M\) is
elliptic as a parameter of $M$.

\step{1}{the derived group of $M$} Let
\(\beta_M:\widehat M\to\widehat{M_{\mathrm{der}}}\) be the
projection dual to the inclusion of the derived group; its kernel
is the connected centre of $\widehat M$. The parameter
$\beta_M\sigma_M$ is elliptic. Indeed, if it were not, it would
factor through a proper Levi subgroup of
$\widehat{M_{\mathrm{der}}}$ by
Proposition~\ref{fl:prop:reductive-centralizer}; the inverse image
of that subgroup in $\widehat M$ is a proper Levi subgroup of
$\widehat M$, defined by the same set of simple roots, and it
contains the image of $\sigma_M$, which contradicts the minimality
of $\widehat M$.

\step{2}{a cusp form on $M$} Since \(p>b(M)\),
Theorem~\ref{fl:thm:semisimple-elliptic} gives a nonzero cuspidal excursion summand of \(\beta_M\sigma_M\) on
\(M_{\mathrm{der}}\); if \(M_{\mathrm{der}}\) is trivial, the
constant function one. By Lemma~\ref{fl:lem:derived-extension}
there is a nonzero cuspidal joint eigenfunction $\varphi_M$ on
\(M\) with parameter \(\sigma_M\) and connected central character
\(\omega_{\sigma_M}\). That lemma allows a central character of
infinite order, which it treats by a twist by a character of the
degree; so no condition on the central character is needed here.

\step{3}{transfer to $G$} If \(M=G\), then $\sigma$ is elliptic
and $\varphi_M$ is the required cusp form. Otherwise
Theorem~\ref{fl:thm:eisenstein-full-label} gives a function
\(\varphi_\sigma\in\mathcal A^{\mathrm{cont}}(G)\) which is not
cuspidal, in particular not zero, and which satisfies \eqref{fl:eq:main-full-label} for all tuples of elements
of $\Gamma_F$ and has connected central character
\(\omega_\sigma\). It is an automorphic form by
Corollary~\ref{fl:cor:excursion-space-automorphicity}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{fl:thm:central-period-realization}]
(1) The form \(\varphi_\sigma\) constructed above has connected
central character \(\omega_\sigma\). If this character is trivial
on \(\Xi\), then \(\varphi_\sigma\) is invariant under $\Xi$, and
by Corollary~\ref{fl:cor:weil-to-galois-nonzero} and the last
assertion of Theorem~\ref{fl:thm:eisenstein-full-label} it lies in
\(\mathcal A_\Xi(G)\).

(2) If $\omega_\sigma|_\Xi=1$, the cusp form of (1) is a nonzero
element of the summand. Conversely, suppose that the summand is
not zero. By Proposition~\ref{fl:prop:connected-central-character}
the connected centre acts on it through \(\omega_\sigma\). On the
other hand the translation by an element \(\xi\) of \(\Xi\) is the
identity on functions on \(G(F)\backslash G(\mathbb A_F)/\Xi\).
Since the summand is not zero, \(\omega_\sigma(\xi)=1\).
\end{proof}

The characteristic of $F$ enters the proof of
Theorem~\ref{fl:thm:main} only through the realization on
\(M_{\mathrm{der}}\), which requires \(p>b(M)\).
Proposition~\ref{fl:prop:levi-root-bound} gives \(b(M)\le b(G)\)
and the sharper bounds for proper Levi subgroups.