\section{The smallest Levi containing a parameter}
\label{fl:sec:minimal-levi}

The papers \cite{HuII} and \cite{HuIII} treat
elliptic parameters. A semisimple parameter $\sigma$ that is not
elliptic factors through the dual of a Levi subgroup $M$ of $G$,
and is elliptic as a parameter of $M$ if $M$ is chosen minimal
(Corollary~\ref{fl:cor:elliptic-minimal-levi-parameter}). The
cusp form on $M$ given by the elliptic case is then transferred to
$G$ by Eisenstein series, in
Section~\ref{fl:sec:eisenstein}. The Levi subgroup $M$ is
determined by the centralizer of $\sigma$, and the restriction on
the characteristic in Theorem~\ref{fl:thm:main} is the one needed
for $M$, not for $G$
(Proposition~\ref{fl:prop:levi-root-bound}).

\begin{proposition}[Minimal Levi subgroups containing a reductive subgroup]
\label{fl:prop:reductive-centralizer}
Let $\widehat G$ be a connected reductive group over $k$ and
$H\subset\widehat G$ a closed subgroup with $H^\circ$ reductive.
Let $C=C_{\widehat G}(H)$.
\begin{enumerate}
\item The group $C^\circ$ is reductive.
\item If $S$ is a maximal torus of $C^\circ$, then
\begin{equation}\label{fl:eq:minimal-centralizer}
 \widehat M=C_{\widehat G}(S)
\end{equation}
is a Levi subgroup of $\widehat G$ containing $H$ and minimal with
this property, $Z(\widehat M)^\circ=S$, and
\begin{equation}\label{fl:eq:minimal-ellipticity}
 C_{\widehat M}(H)^\circ=Z(\widehat M)^\circ.
\end{equation}
\item Every minimal Levi subgroup containing $H$ is conjugate to
$\widehat M$ by an element of $C^\circ$.
\end{enumerate}
\end{proposition}

\begin{proof}
(1) In characteristic zero a group whose identity component is
reductive is linearly reductive: an invariant complement under
$H^\circ$ can be averaged over the finite group $H/H^\circ$. So
$H$ is completely reducible in $\widehat G$, and the identity
component of its centralizer is reductive
\cite[Lemma~2.6 and Proposition~3.12]{BMR2005}.

(2) The centralizer of a torus is a Levi subgroup, and
$\widehat M$ contains $H$ because $S\subset C$. We have
$S\subset Z(\widehat M)^\circ\subset C^\circ$, the second
inclusion because $H\subset\widehat M$; as $Z(\widehat M)^\circ$
is a torus containing the maximal torus $S$, they are equal.
Next,
\[
 C_{\widehat M}(H)^\circ=C_C(S)^\circ=C_{C^\circ}(S)=S,
\]
since the centralizer of a maximal torus of the connected
reductive group $C^\circ$ is that torus. Finally, if
$\widehat L\subset\widehat M$ is a Levi subgroup containing $H$,
then $Z(\widehat L)^\circ$ is a torus in $C^\circ$ containing $S$,
hence equal to $S$, and
$\widehat L=C_{\widehat G}(Z(\widehat L)^\circ)=\widehat M$.

(3) Let $\widehat L$ be a minimal Levi subgroup containing $H$.
The torus $Z(\widehat L)^\circ$ lies in $C^\circ$; let $S'$ be a
maximal torus of $C^\circ$ containing it. Then
$C_{\widehat G}(S')\subset C_{\widehat G}(Z(\widehat L)^\circ)
=\widehat L$ is a Levi subgroup containing $H$, so it equals
$\widehat L$ by minimality. The maximal tori $S$ and $S'$ are
conjugate in $C^\circ$.
\end{proof}

\begin{corollary}[Factorization through a minimal Levi subgroup]
\label{fl:cor:elliptic-minimal-levi-parameter}
Let $G$ be split connected reductive and
$\sigma:\Gamma_F\to\widehat G(k)$ a continuous, finitely ramified,
semisimple parameter. After conjugation,
\begin{equation}\label{fl:eq:minimal-factorization}
 \sigma=i\circ\sigma_M,\qquad
 i:\widehat M\hookrightarrow\widehat G,
\end{equation}
where $M$ is a standard Levi subgroup of $G$ and
$\sigma_M:\Gamma_F\to\widehat M(k)$ is semisimple and elliptic. It
is ramified at the same places as $\sigma$ and takes values in a
finite extension of $\mathbb Q_\ell$.
\end{corollary}

\begin{proof}
Apply Proposition~\ref{fl:prop:reductive-centralizer} to the
Zariski closure $H$ of the image of $\sigma$, and conjugate the
resulting Levi subgroup to the dual of a standard Levi subgroup
of $G$. The closure of the image is still $H$, whose identity
component is reductive, and \eqref{fl:eq:minimal-ellipticity} says that
$\sigma_M$ is elliptic. Continuity and ramification are not
affected, since $\sigma_M$ is the same map with a smaller target.
For the field of values, enlarge that of
Lemma~\ref{fl:lem:arithmetic-finite-coefficients} to contain the
entries of the conjugating element.
\end{proof}

\begin{example}\label{fl:ex:minimal-levi-gl}
For $\widehat G=\mathrm{GL}_n$, write a semisimple parameter as
$\bigoplus_a\rho_a\otimes k^{m_a}$ with pairwise nonisomorphic
irreducible $\rho_a$. Its centralizer is
$\prod_a\mathrm{GL}_{m_a}$, and a maximal torus of it, the
diagonal torus in the multiplicity spaces, gives
\[
 \widehat M
   =\prod_a\prod_{j=1}^{m_a}\mathrm{GL}_{\dim\rho_a}.
\]
For instance $\chi\oplus\chi\oplus\rho$, with $\rho$ irreducible
of dimension two, has minimal Levi subgroup
$\mathrm{GL}_1\times\mathrm{GL}_1\times\mathrm{GL}_2$. The Levi
subgroup $\mathrm{GL}_2\times\mathrm{GL}_2$ also contains the
image, but $\chi\oplus\chi$ is not elliptic in $\mathrm{GL}_2$.
\end{example}

\subsection{The characteristic bound belongs to this Levi}

Recall the bounds of Theorem~\ref{fl:thm:main}:
\begin{equation}\label{fl:eq:root-bound-values}
\begin{gathered}
 b(A_r)=b(B_r)=b(C_r)=b(D_r)=0,\qquad b(G_2)=b(F_4)=3,\\
 b(E_6)=5,\qquad
 b(E_7)=b(E_8)=7,\qquad b(\varnothing)=0,
\end{gathered}
\end{equation}
and, for a reductive group, $b$ is the maximum over the simple
factors of its derived group. Let also
\begin{equation}\label{fl:eq:proper-levi-bound}
 b^{\mathrm{proper}}(G)
    =\max_{M\subsetneq G\ \mathrm{Levi}} b(M),
\end{equation}
which is $0$ if $G$ is a torus.

\begin{proposition}[Bounds for Levi subgroups]
\label{fl:prop:levi-root-bound}
For every Levi subgroup $M\subset G$,
\begin{equation}\label{fl:eq:levi-bound-monotonicity}
 b(M)\le b(G).
\end{equation}
If the root system of $G$ is irreducible, then
\[
\begin{array}{c|ccc}
 \text{type of }G&A,B,C,D,G_2,F_4,E_6&E_7&E_8\\ \hline
 b^{\mathrm{proper}}(G)&0&5&7 .
\end{array}
\]
If the root system of $G$ has at least two irreducible
components, then
\begin{equation}\label{fl:eq:product-proper-bound}
 b^{\mathrm{proper}}(G)=b(G).
\end{equation}
\end{proposition}

\begin{proof}
The Dynkin diagram of a Levi subgroup is obtained from that of
$G$ by removing vertices. A proper connected subdiagram of the
diagram of $F_4$ or $E_6$ is of classical type. A proper connected
subdiagram of $E_7$ has at most six vertices and is of classical
type or of type $E_6$, which occurs. The diagram of $E_8$
contains that of $E_7$, and $b\le7$ for all its proper
subdiagrams. Subdiagrams of classical diagrams and of $G_2$ are
classical. This gives the first two assertions. If there are at
least two components, remove a vertex from a component other than
one on which $b$ is maximal.
\end{proof}

\begin{remark}\label{fl:rem:proper-bound-scope}
A central torus does not count as a component in
\eqref{fl:eq:product-proper-bound}. For a given parameter the
condition that matters is $p>b(M)$ for its minimal Levi subgroup
$M$. For example every parameter of a group of type $F_4$ or $E_6$
which is not elliptic factors through a Levi subgroup all of whose
factors are classical, and is covered in every characteristic.
\end{remark}

\section{Eisenstein transfer of the excursion character}
\label{fl:sec:eisenstein}

Let $\varphi_M$ be an eigenfunction of the excursion operators on a
Levi subgroup $M$, with parameter $\sigma_M$. We want an
eigenfunction on $G$ with parameter $i\sigma_M$
(Theorem~\ref{fl:thm:eisenstein-full-label}). The Eisenstein
series induced from $\varphi_M$ would be one, but it may have a
pole at the point in question. We avoid choosing a value or a
residue. The Eisenstein series are used, in the region of
convergence and for a family of unramified twists of $\varphi_M$,
only to prove that every relation among the excursion operators on
functions with compact support holds for the values given by
$i\sigma_M$ (Lemma~\ref{fl:lem:eisenstein-relation-specialization}):
for the twists this follows from the constant term, and a relation
is a Laurent polynomial in the twist, so it holds at the
untwisted point as well. When $M\ne G$ the same is true of the
relations which hold only modulo the cuspidal functions with
compact support, because a convergent Eisenstein series is
orthogonal to these functions
(Lemma~\ref{rw:lem:eisenstein-relation-mod-cusp}). Hence $i\sigma_M$
defines a character of the algebra generated by the excursion
operators on the quotient of the space of functions with compact
support by the cuspidal ones. This quotient is a finitely generated
faithful module over that algebra, so the character occurs in a
quotient of it (Lemma~\ref{fl:lem:faithful-finite-character-fibre}).
The function obtained is orthogonal to the cuspidal functions with
compact support, and a cuspidal function with this property is zero
(Lemma~\ref{rw:lem:compact-cusp-tests}).

\begin{theorem}[Transfer from a Levi subgroup]
\label{fl:thm:eisenstein-full-label}
Let $P=MU$ be a parabolic subgroup of a split connected reductive
group $G$, with dual inclusion
$i:\widehat M\hookrightarrow\widehat G$. Let
$\sigma_M:\Gamma_F\to\widehat M(k)$ be continuous, finitely
ramified and semisimple, and suppose there is a nonzero smooth
function $\varphi_M$ of finite level on $M$ on which the Weil
excursion operators act through $\sigma_M|_{W_F}$. Then there is
a nonzero automorphic form
$\varphi\in\mathcal A^{\mathrm{cont}}(G)$ with connected central
character $\omega_{i\sigma_M}$ such that
\begin{equation}\label{fl:eq:eisenstein-full-label}
 S^{G,\mathrm{fun}}_{I,f,\boldsymbol\gamma}\varphi
    =f\bigl((i\sigma_M(\gamma_j))_{j\in I}\bigr)\varphi
 \qquad(\boldsymbol\gamma\in\Gamma_F^I).
\end{equation}
For every central lattice $\Xi$ on which
$\omega_{i\sigma_M}$ is trivial, $\varphi$ lies in the dual of a
finite quotient defining $\mathcal A_\Xi(G)$, with the first power
of the spherical ideal. If $M\ne G$, the form $\varphi$ can be
chosen not cuspidal.
\end{theorem}

In the assertion on the lattices the ideal is the spherical ideal
$(\ker\chi_{i\sigma_M,u})^\star$ at a place $u$ outside the level, over a
sufficiently large finite coefficient field. Here $\varphi$ is
an eigenfunction, and not only a generalized one. The function
$\varphi_M$ may be a twist of a cuspidal eigenfunction by a
continuous central character.

\subsection{Relations among excursion operators}

For a parameter $\tau$ of $G$ and a polynomial $r$ in excursion
data, $\nu_\tau(r)$ denotes the value of $r$ when each datum
$(I,f,\boldsymbol\gamma)$ is replaced by
$f((\tau(\gamma_j))_j)$.

\begin{lemma}[Relations among excursion operators]
\label{fl:lem:eisenstein-relation-specialization}
In Theorem~\ref{fl:thm:eisenstein-full-label} assume that
$\varphi_M$ is cuspidal. There is a level $K$ of $G$ such that,
with $\mathcal C=C_c(G(F)\backslash G(\mathbb A)/K,k)$, every
polynomial $r$ in Weil excursion data whose operator on
$\mathcal C$ is zero satisfies $\nu_{i\sigma_M}(r)=0$.
\end{lemma}

\begin{proof}
\step{1}{the level and the section} Let
$K_0=G(\prod_v\mathcal O_v)$, and let $K\subset K_0$ be a
principal level such that $M(\mathbb A)\cap K$ fixes $\varphi_M$.
By the Iwasawa decomposition there are representatives $g_a$ of
$P(\mathbb A)\backslash G(\mathbb A)/K$ in $K_0$, and we take $1$
as one of them. Define $\Phi$ by
$\Phi(umk)=\delta_P(m)^{1/2}\varphi_M(m)$ on $P(\mathbb A)K$ and
by $\Phi=0$ on the other double cosets. This is well defined: the
image of $P(\mathbb A)\cap K$ in $M(\mathbb A)$ is
$M(\mathbb A)\cap K$, which fixes $\varphi_M$ and on which
$\delta_P$ is trivial. Its restrictions
\[
 \varphi_{M,a}(m)=\delta_P(m)^{-1/2}\Phi(mg_a)
\]
are equal to $\varphi_M$ for $g_a=1$ and to zero for the other
representatives, and the excursion operators
of $M$ act on them through $\sigma_M$. If $M=G$ there is nothing
more to do: an operator which is zero on $\mathcal C$ is zero on
all functions of level $K$ by
Lemma~\ref{fl:lem:finite-row-testing}, in particular on
$\varphi_M$. Assume $M\ne G$.

\step{2}{the family of twists} For $c$ in the torus
$Z(\widehat M)^\circ$ let $\chi_c$ be the unramified character of
$M(\mathbb A)$ attached to the homomorphism $c^{\deg_F}$ of $W_F$,
and let $\Phi_c$ be the section obtained by multiplying the
$\varphi_{M,a}$ by $\chi_c$. By
Proposition~\ref{fl:prop:compact-central-twists} the excursion
operators act on the twisted functions through the Weil parameter
$\sigma_Mc^{\deg_F}$. This parameter need not extend continuously
to $\Gamma_F$, which is why Weil tuples are used. For a fixed $r$,
\[
 c\longmapsto \nu_{i(\sigma_M c^{\deg_F})}(r)
\]
is a Laurent polynomial in $c$: the degrees of the Weil elements
in $r$ are integers and the functions $f$ are regular.

\step{3}{Eisenstein series} By
Corollary~\ref{fl:cor:weil-to-galois-nonzero} the function
$\varphi_M$ takes values in a finite extension of $\mathbb Q_\ell$
and has central character $\omega_{\sigma_M}$. We transport it to
complex coefficients by $\iota$; the identities between values of
functions that express the action of the excursion operators are
finite sums and are preserved. For $c$ in a nonempty open subset
of the region where the twist is sufficiently positive, the
Eisenstein series $E_P(\Phi_c)$ converges absolutely.

\step{4}{the constant term} Let $r$ be zero on $\mathcal C$, hence
on all functions of level $K$. The excursion operators are
compatible with the normalized constant term along $P$, so
\[
 (i^*r)\,\mathsf{CT}^{\mathrm{nor}}_P E_P(\Phi_c)=0,
\]
where $i^*r$ is the corresponding polynomial in excursion data of
$M$. Since $\varphi_M$ is cuspidal, the constant term is a sum
over $W(M,M)=N_G(M)/M$,
\[
 \mathsf{CT}^{\mathrm{nor}}_{P,g_a}E_P(\Phi_c)
    =v_U\varphi_{M,c,a}
       +\sum_{\substack{w\in W(M,M)\\w\ne1}}
           \varphi_{M,w,c,a},
 \qquad v_U=\operatorname{vol}(U(F)\backslash U(\mathbb A))\ne0.
\]
The first term comes from the identity double coset; the others
are images of intertwining operators, on which the connected
centre of $M$ acts through $w(\omega_{\sigma_M}\chi_c)$
\cite[(6.2), (6.4), (6.17), and \S2.2]{G2BernsteinLapid24}. The
double cosets whose intersection with the Levi subgroup is a
proper parabolic subgroup contribute constant terms of
$\varphi_M$, which vanish.

\step{5}{specialization} The group $W(M,M)$ acts faithfully on
$T_M$, whose centralizer is $M$. The isogeny $T_M\to Q_M$ commutes
with the action of $W(M,M)$. For each $w\ne1$ there is therefore a
cocharacter of $T_M$ whose image in $X_*(Q_M)$ is moved by $w$;
let $a_w\in T_M(\mathbb A)$ be its value at a uniformizer at some
place. Then the degrees of $a_w$ and of $w^{-1}a_w$ in $X_*(Q_M)$
are different, so for $\chi_w=w(\omega_{\sigma_M}\chi_c)$ and
$\chi_1=\omega_{\sigma_M}\chi_c$ the equality
$\chi_1(a_w)=\chi_w(a_w)$ is a monomial equation in $c$ which is
not an identity. We remove from the torus the finitely many proper
closed subsets defined by these equations. Let $T_{a_w}$ be the
translation by $a_w$. The operator
\[
 e_c=\prod_{w\ne1}
       \frac{T_{a_w}-\chi_w(a_w)}
            {\chi_1(a_w)-\chi_w(a_w)}
\]
is the identity on the first term of the constant term and kills
the others, and it commutes with excursion operators. Applying it
to the equation of step~4 gives
\[
 \nu_{i(\sigma_M c^{\deg_F})}(r)\,v_U\varphi_{M,c,a}=0 .
\]
For the index $a$ of step~1 the function is not zero, so the
scalar vanishes for all $c$ in a Zariski dense subset of the
torus. Being a Laurent polynomial it vanishes identically, and at
$c=1$ we get $\nu_{i\sigma_M}(r)=0$.
\end{proof}

\begin{remark}
The specialization at $c=1$ is that of a Laurent polynomial, so
it is valid even if the Eisenstein series has a pole there.
\end{remark}

For a level $K$ and
$\mathcal C=C_c(G(F)\backslash G(\mathbb A)/K,k)$, let
$\mathcal C_{\rm cusp}\subset\mathcal C$ be the subspace of
cuspidal functions.

\begin{lemma}[Cuspidal functions with compact support]
\label{rw:lem:compact-cusp-tests}
\begin{enumerate}
\item The subspace $\mathcal C_{\rm cusp}$ is stable under the
operators $S_e$, for excursion data with tuples in the Weil group.
\item Let $h$ be a cuspidal function of level $K$. If $[c,h]=0$
for every $c\in\mathcal C_{\rm cusp}$, then $h=0$.
\end{enumerate}
\end{lemma}

\begin{proof}
(1) On $\mathcal C$ the operator $S_e$ is $S_e^{\mathrm{fun}}$, which
commutes with the normalized constant terms along all parabolic
subgroups, by Theorem~\ref{fl:thm:normalized-ct}. So it maps
cuspidal functions to cuspidal functions.

(2) Let $Q_G=G/G_{\mathrm{der}}$, and let
$\deg_G:G(\mathbb A)\to X_*(Q_G)$ be the degree. For
$\nu\in X_*(Q_G)$ let $h_\nu$ be the product of $h$ and of the
characteristic function of the elements of degree $\nu$. The
unipotent radicals of the parabolic subgroups lie in
$G_{\mathrm{der}}$, on whose adelic points the degree is zero; so
$h_\nu$ is cuspidal. Its support is compact. Indeed, let $\Xi$ be
a central lattice. The translates of $h_\nu$ by the elements of
$\Xi$ are supported in different degrees, because the degree is
injective on $\Xi$ and $T_G\to Q_G$ is an isogeny. So their sum is a
cuspidal function of level $K$ which is invariant under $\Xi$. By
Harder's theorem \cite[Chapter~I, \S5.3, p.~104]{Morris1982} the
support of this sum is compact modulo $\Xi$, and its part in
degree $\nu$, which is the support of $h_\nu$, is compact.

Suppose that $h_\nu\ne0$, and let $A$ be its support, a finite
subset of $G(F)\backslash G(\mathbb A)/K$. A function supported in
$A$ is cuspidal if and only if its values satisfy a system of
linear equations, whose coefficients are volumes of subsets of the
spaces $U(F)\backslash U(\mathbb A)$ which are invariant under
compact open subgroups; with a suitable normalization of the
measures they are rational numbers. So the space of cuspidal
functions supported in $A$ is $D\otimes_{\mathbb Q}k$, where $D$ is
the space of those with rational values. The weights of the points
in the pairing are positive rational numbers. Hence the pairing is
positive definite on $D$, and its extension to
$D\otimes_{\mathbb Q}k$ is nondegenerate, the determinant of its
matrix in a basis of $D$ being a nonzero rational number. As
$h_\nu$ belongs to $D\otimes_{\mathbb Q}k$, there is $c$ in this
space with $[c,h_\nu]\ne0$. Then $c\in\mathcal C_{\rm cusp}$, and
$[c,h]=[c,h_\nu]$ because $c$ is supported in degree $\nu$. This
contradicts the hypothesis. So $h_\nu=0$ for every $\nu$, and
$h=0$.
\end{proof}

\begin{lemma}[Relations modulo cuspidal functions]
\label{rw:lem:eisenstein-relation-mod-cusp}
In Lemma~\ref{fl:lem:eisenstein-relation-specialization} suppose
that $M\ne G$, and let $K$ be the level of its proof. Every
polynomial $r$ in Weil excursion data whose operator maps
$\mathcal C$ into $\mathcal C_{\rm cusp}$ satisfies
$\nu_{i\sigma_M}(r^\star)=0$, where $r^\star$ is the polynomial
obtained from $r$ by replacing each excursion datum $e$ by
$e^\star$.
\end{lemma}

\begin{proof}
We use the sections $\Phi_c$ and the Eisenstein series of steps~2
and~3 of that proof, with complex coefficients as there. Let $c$ be
in the region of absolute convergence. For $d\in\mathcal C_{\rm cusp}$,
\[
 [d,E_P(\Phi_c)]=0 .
\]
Indeed, the pairing is a finite sum of values of the Eisenstein
series, each of which is an absolutely convergent sum over
$P(F)\backslash G(F)$. Unfolding, it is the integral over
$P(F)\backslash G(\mathbb A)$ of $d\,\Phi_c$. The section $\Phi_c$
is invariant on the left under $U(\mathbb A)$, so integrating
first over $U(F)\backslash U(\mathbb A)$ replaces $d$ by its
constant term along $P$, which is zero.

Now let $r$ map $\mathcal C$ into $\mathcal C_{\rm cusp}$. Since
$(e^\star)^\star=e$, the definition \eqref{fl:eq:function-action}
gives, for every $b\in\mathcal C$,
\[
 [b,r^\star E_P(\Phi_c)]=[rb,E_P(\Phi_c)]=0,
\]
where $r^\star$ acts through the operators $S_e^{\mathrm{fun}}$.
Taking for $b$ the characteristic functions of points, we get
$r^\star E_P(\Phi_c)=0$. This is the only property of the
polynomial which is used in steps~4 and~5 of the proof of
Lemma~\ref{fl:lem:eisenstein-relation-specialization}. Applied to
$r^\star$ they give $\nu_{i(\sigma_Mc^{\deg_F})}(r^\star)=0$ for all
$c$ in a Zariski dense subset of the torus, hence identically, and
at $c=1$ we get $\nu_{i\sigma_M}(r^\star)=0$.
\end{proof}

\subsection{A nonzero character quotient}

\begin{lemma}[A character of a faithful finite module occurs in a quotient]
\label{fl:lem:faithful-finite-character-fibre}
Let $\mathcal B$ be a commutative $k$-algebra and $\mathcal C$ a
finitely generated faithful $\mathcal B$-module. For every
character $\eta:\mathcal B\to k$, the vector space
$\mathcal C/(\ker\eta)\mathcal C$ is nonzero and finite
dimensional.
\end{lemma}

\begin{proof}
If $(\ker\eta)\mathcal C=\mathcal C$, then by Nakayama's lemma
\cite[Tag~00DV(1)]{StacksProject} some element of $1+\ker\eta$
annihilates $\mathcal C$; it is nonzero, contradicting
faithfulness. The quotient is spanned over
$\mathcal B/\ker\eta=k$ by the images of finitely many generators.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{fl:thm:eisenstein-full-label}]
\step{1}{reduction to cuspidal $\varphi_M$} The Hecke operators at
an unramified place act on $\varphi_M$ through the Satake parameter
of $\sigma_M$ at that place. Since $\sigma_M$ is continuous, the
eigenvalues of this parameter in every representation of
$\widehat M$ are $\ell$-adic units; so the eigenvalues of the Hecke
operators, which are traces, are integral over $\mathcal O$. (They
need not be units: a sum of units can vanish.) By
Lemma~\ref{fl:lem:minimal-cuspidal-support}, applied in $M$, there
are a Levi subgroup $M_0\subset M$ and a cuspidal function on
$M_0$ which is an eigenfunction of the excursion operators with a
parameter $\sigma_{M_0}$, and which satisfies every relation
satisfied by $\varphi_M$. With
$j:\widehat M_0\hookrightarrow\widehat M$, the parameters
$j\sigma_{M_0}$ and $\sigma_M$ then have the same values on all
excursion data, and are conjugate because they are continuous and
semisimple. So we may replace $M$ by $M_0$ and assume $\varphi_M$
cuspidal. If $M=G$ the theorem follows from
Corollary~\ref{fl:cor:weil-to-galois-nonzero}.

\step{2}{a character of the excursion algebra} Now $M\ne G$. Take
$K$ and $\mathcal C$ as in
Lemma~\ref{fl:lem:eisenstein-relation-specialization}, and put
$\overline{\mathcal C}=\mathcal C/\mathcal C_{\rm cusp}$. By
Lemma~\ref{rw:lem:compact-cusp-tests}(1) the Weil excursion
operators act on $\overline{\mathcal C}$; let $\mathcal B$ be the
image of the algebra of polynomials in Weil excursion data in the
endomorphisms of $\overline{\mathcal C}$. We claim that
$\eta=\nu_{\vartheta_Gi\sigma_M}$ is a character of $\mathcal B$,
that is, vanishes on every $r$ whose operator on
$\overline{\mathcal C}$ is zero. Such an $r$ maps $\mathcal C$ into
$\mathcal C_{\rm cusp}$, and by
Lemma~\ref{rw:lem:eisenstein-relation-mod-cusp}
\[
 \eta(r)=\nu_{\vartheta_G i\sigma_M}(r)
           =\nu_{i\sigma_M}(r^\star)=0.
\]
In particular $\overline{\mathcal C}\ne0$, since $\eta(1)=1$.

\step{3}{a function which is not cuspidal} At a place $u$ outside
the level the spherical Hecke algebra acts on
$\overline{\mathcal C}$ through $\mathcal B$, and
$\overline{\mathcal C}$ is finitely generated over it, because
$\mathcal C$ is
(Proposition~\ref{fl:prop:classical-one-place-finiteness}); hence
it is finitely generated over $\mathcal B$, and it is a faithful
$\mathcal B$-module by definition. By
Lemma~\ref{fl:lem:faithful-finite-character-fibre} there is a
nonzero linear form on
$\overline{\mathcal C}/(\ker\eta)\overline{\mathcal C}$. We regard
it as a linear form on $\mathcal C$ which vanishes on
$\mathcal C_{\rm cusp}$. Through the pairing
$[\,\cdot\,,\cdot\,]$ it is given by a function $\varphi\ne0$ of
level $K$, and for every Weil excursion datum $e$ and every
$c\in\mathcal C$,
\[
 [c,S_e^{\mathrm{fun}}\varphi]
    =[S_{e^\star}c,\varphi]
    =\eta(e^\star)[c,\varphi]
    =\nu_{i\sigma_M}(e)[c,\varphi].
\]
Taking for $c$ the characteristic functions of points gives
$S_e^{\mathrm{fun}}\varphi=\nu_{i\sigma_M}(e)\varphi$. Moreover
$[c,\varphi]=0$ for every $c\in\mathcal C_{\rm cusp}$. If $\varphi$
were cuspidal it would be zero, by
Lemma~\ref{rw:lem:compact-cusp-tests}(2). So $\varphi$ is not
cuspidal.

\step{4}{from Weil to Galois tuples}
Corollary~\ref{fl:cor:weil-to-galois-nonzero} applies to $\varphi$
and the continuous parameter $i\sigma_M$. It shows that $\varphi$
is an automorphic form in
$\mathcal A^{\mathrm{cont}}(G)$, that
\eqref{fl:eq:eisenstein-full-label} holds for all Galois tuples,
that the connected centre acts through $\omega_{i\sigma_M}$, and
that for each lattice $\Xi$ as in the statement $\varphi$ lies in
the dual of the corresponding finite quotient, with the first
power of the ideal.
\end{proof}