\section{Criteria for cuspidality}
\label{fl:sec:arithmetic-cuspidality}

The main result of this section is
Proposition~\ref{fl:prop:full-label-cuspidality}: a function of
finite level whose excursion character is an elliptic parameter is
cuspidal. The idea is that a nonzero constant term along a proper
parabolic subgroup $P=MU$ would produce a parameter $\sigma_M$ of
the Levi subgroup $M$ such that $\sigma$ and the composite
$i\sigma_M$ of $\sigma_M$ with $\widehat M\to\widehat G$ have
conjugate images of Frobenius at almost all places. For a general
reductive group this does not make the two parameters conjugate.
It does make their composites with the adjoint representation of
$\widehat G$ isomorphic, and the spaces of invariants of these two
representations have different dimensions when $\sigma$ is elliptic
and $M$ is proper. The argument uses only the Hecke operators at
unramified places and the classical formula for the constant term
of a convolution (Lemma~\ref{rw:lem:ct-spherical}).

Around this result there are three auxiliary statements, used by
the constructions of \cite{HuII} and
\cite{HuIII}, which deliver their functions with
different amounts of information.
\begin{itemize}
\item Lemma~\ref{fl:lem:faithful-satake-model}: a function known to
satisfy only the Hecke equations of one faithful representation
lies in the spaces of Section~\ref{fl:sec:function-spaces}.
\item Lemma~\ref{fl:lem:hecke-moment-extraction}: from a function
whose pairing with one test function has the right Hecke
eigenvalues, one can extract a function with the right generalized
Hecke character and nonzero pairing.
\item Proposition~\ref{fl:prop:cuspidal-excursion-character}: the
form in which the geometric construction delivers its function, as
a functional on functions with compact support.
\end{itemize}

\subsection{Functions with the Hecke equations of one representation}

Let $u$ be a place and $W$ a finite-dimensional space of
functions stable under the spherical Hecke algebra at $u$. The
joint eigencharacters of the algebra on $W$ correspond, by the
Satake isomorphism, to finitely many points of the dual torus up
to the Weyl group; we call them the \emph{Satake parameters} of
$W$.

\begin{definition}[Unit-valued Satake parameters]
\label{fl:def:unit-satake-support}
The space $W$ has \emph{unit-valued Satake parameters} if every
character of the dual torus takes an $\ell$-adic unit value at
each of its Satake parameters.
\end{definition}

The condition does not depend on the representative modulo the
Weyl group. It does not say that every eigenvalue of a Hecke
operator is a unit: such an eigenvalue is a sum of values of
characters, and can be zero.

\begin{lemma}[From one representation to the whole Hecke algebra]
\label{fl:lem:faithful-satake-model}
Let $r:\widehat G\to\mathrm{GL}(V)$ be a representation with
finite kernel, and $u$ a place outside the level of a smooth
function $h$ of finite level. Let $\mathscr C_r$ be the
subalgebra of the spherical Hecke algebra at $u$ generated by the
operators which correspond to the coefficients of the
characteristic polynomial of $r$ and to the inverse of its
determinant.
\begin{enumerate}
\item If $\mathscr C_rh$ has finite dimension, the orbit of $h$
under the whole spherical Hecke algebra at $u$ has finite
dimension.
\item If moreover, at every joint eigencharacter of $\mathscr C_r$
on $\mathscr C_rh$, the roots of the characteristic polynomial are
$\ell$-adic units, then this orbit has unit-valued Satake
parameters, and it is spanned by a lattice stable under the
integral spherical Hecke algebra.
\item If in addition $h$ is invariant under a central lattice
$\Xi$, it belongs to $\mathcal A_\Xi(G)$.
\end{enumerate}
\end{lemma}

\begin{proof}
(1) Let $\widehat T$ be the dual torus and
$\mathscr L=k[X^*(\widehat T)]$, so that the spherical Hecke
algebra is $\mathscr H=\mathscr L^{W_G}$. We show that
$\mathscr L$ is integral over $\mathscr C_r$. For a weight $\mu$
of $r$ the monomial $t^\mu$ is a root of the characteristic
polynomial of $r$, a monic polynomial with coefficients in
$\mathscr C_r$; and $t^{-\mu}$ is a root of the reciprocal
polynomial, which becomes monic after multiplication by the
inverse of the determinant. Since the kernel of $r$ is finite,
the weights of $r$ generate a subgroup of finite index in
$X^*(\widehat T)$. So for every $\lambda$ some power of
$t^\lambda$ is a product of monomials $t^{\pm\mu}$, and
$t^\lambda$ is integral over $\mathscr C_r$.

The algebra $\mathscr L$ is finitely generated and integral over
$\mathscr C_r$, hence a finitely generated module over it. The
ring $\mathscr C_r$ is Noetherian, so the submodule $\mathscr H$
is finitely generated as well. If $a_1,\dots,a_s$ are generators,
then $\mathscr Hh=\sum_ja_j\,\mathscr C_rh$ has finite dimension.

(2) At a Satake parameter of $\mathscr Hh$ the values of the
weights of $r$ are roots of the characteristic polynomial at a
joint eigencharacter of $\mathscr C_r$, hence units. Since the
weights generate a subgroup of finite index, a power of the value
of any character is a unit, and so is the value itself.

For the lattice, choose finitely many generators $T_1,\dots,T_m$
of the integral spherical Hecke algebra as an algebra over
$\mathcal O$. Their eigenvalues on $\mathscr Hh$ are sums of
values of characters at the Satake parameters, hence integral. So
the characteristic polynomial of each $T_j$ on $\mathscr Hh$ has
integral coefficients, and by the theorem of Cayley and Hamilton
the algebra generated over the ring of integers by the commuting
operators $T_j$ is spanned by finitely many monomials in them.
The image under this algebra of a finite set which spans
$\mathscr Hh$ is the required lattice.

(3) This follows from (2) and
Lemma~\ref{fl:lem:all-place-stable-lattice}.
\end{proof}

For the functions constructed later, the roots in (2) are
eigenvalues of the image of a Frobenius element under a continuous
parameter, and they are units because the image of the parameter
is compact (Lemma~\ref{fl:lem:parameter-satake-integrality}). This
is where the continuity of the parameter is used.

\subsection{Ellipticity implies cuspidality}

The first lemma keeps track of all excursion operators on a
constant term. It depends on Theorem~\ref{fl:thm:normalized-ct}
and is used only in Section~\ref{fl:sec:eisenstein}.

\begin{lemma}[A cuspidal constant term inherits all relations]
\label{fl:lem:minimal-cuspidal-support}
Let $h\ne0$ be a smooth function of finite level on
$G(F)\backslash G(\mathbb A)$ whose orbit under the spherical
Hecke algebra at one place outside the level has finite dimension
and unit-valued Satake parameters. There are a standard Levi
subgroup $M\subset G$ and a nonzero cuspidal function on $M$
which is a joint eigenfunction of the excursion operators of $M$,
for tuples in the Weil group, with a parameter $\sigma_M$ which is
continuous, semisimple and ramified at finitely many places, such
that
\[
 Bh=0\quad\Longrightarrow\quad \nu_{i\sigma_M}(B)=0
\]
for every polynomial $B$ in excursion operators of $G$ with
tuples in the Weil group. Here
$i:\widehat M\hookrightarrow\widehat G$ is the inclusion, and
$\nu_{i\sigma_M}(B)$ is the value of the polynomial when each
operator $S_e$ is replaced by $\nu_{i\sigma_M}(e)$. If $h$ is not
cuspidal, $M$ can be chosen proper.
\end{lemma}

\begin{proof}
\step{1}{a cuspidal constant term} Among the standard parabolic
subgroups, $G$ included, along which the constant term of $h$ is
not zero, choose one whose Levi subgroup $M$ has minimal
semisimple rank, and let $h_M$ be a nonzero slice of the
normalized constant term. The constant term of $h_M$ along a
proper parabolic subgroup of $M$ is a slice of the constant term
of $h$ along a smaller parabolic subgroup of $G$, which vanishes
by the choice of $M$. So $h_M$ is cuspidal. If $h$ is not
cuspidal, some proper parabolic subgroup has a nonzero constant
term, and $M$ is proper.

\step{2}{a finite spherical orbit} Let $u$ be the place of the
hypothesis. Take the slice at a point of $G(\prod_v\mathcal O_v)$
whose component at $u$ is one; this is possible because $h$ is
invariant under $G(\mathcal O_u)$. If $B$ is a polynomial in Weil
excursion operators of $G$ with $Bh=0$, then $(i^*B)h_M=0$ by
Theorem~\ref{fl:thm:normalized-ct}. Let $J$ be the annihilator in
$\mathscr H_{G,u}$ of the orbit of $h$. By
Lemma~\ref{rw:lem:ct-spherical} the ideal
$\operatorname{Sat}_P(J)\mathscr H_{M,u}$ annihilates $h_M$. Since
$\mathscr H_{G,u}/J$ has finite dimension and $\mathscr H_{M,u}$
is a finitely generated module over $\mathscr H_{G,u}$, the orbit
$\mathscr H_{M,u}h_M$ has finite dimension. Its elements are cusp
forms, and they are annihilated by every $i^*B$ as above, because
spherical Hecke operators commute with Weil excursion operators.
Let $v$ be a joint eigenvector of $\mathscr H_{M,u}$ in this
orbit. Its Satake parameter, taken modulo the Weyl group of $G$,
is one of the Satake parameters of the orbit of $h$. So all
characters of the dual torus take unit values at it.

\step{3}{a central lattice} Let $\Xi$ be the group generated by the
images of the elements $\lambda(\varpi_u)$, $\lambda\in X_*(T_M)$.
It is a central lattice of $M$. The translations by its
elements are spherical Hecke operators, so $\Xi$ acts on $v$
through a character with unit values. The degree maps $\Xi$
isomorphically onto a subgroup of finite index of $X_*(Q_M)$, and
the group of units of the ring of integers of $k$ is divisible. So
there is a point $c$ of $Z(\widehat M)^\circ(k)$, at which all
characters take unit values, such that the character of
$M(\mathbb A)$ attached to $\psi=c^{\deg_F}$ is the inverse of
this character on $\Xi$. The point $c$ is defined over a finite
extension of $\mathbb Q_\ell$, and $n\mapsto c^n$ extends
continuously to $\widehat{\mathbb Z}$. The function
$v'=T_\psi v$ is cuspidal and invariant under $\Xi$. For a
polynomial $R$ in Weil excursion data of $M$ let
$R^{\psi^{-1}}$ be its image under the substitution
$e\mapsto e^{\psi^{-1}}$ of
\eqref{fl:eq:canonical-finite-twist}. That formula gives
$R^{\psi^{-1}}v'=T_\psi Rv$, which is zero for
$R=i^*B$ with $Bh=0$.

\step{4}{a parameter for $M$} By \cite[Theorem~0.1]{VLa18} the
cusp form $v'$, which is invariant under $\Xi$, is a sum of
components indexed by parameters of $M$. Let $\tau$ be the
parameter of a nonzero component. The projections onto the
components commute with the excursion operators, so the operators
$R^{\psi^{-1}}$ of step~3 annihilate this component. On the
summand of $\tau$ the only eigenvalue of $R^{\psi^{-1}}$ is
$\nu_\tau(R^{\psi^{-1}})$, which is therefore zero. Let $w$
be a joint eigenvector of the excursion operators in the summand,
and put $\sigma_M=\tau\psi^{-1}$. This parameter is
continuous, finitely ramified and semisimple, because $\psi$
is a continuous unramified character with values in the centre.
The cusp form $T_{\psi^{-1}}w$ is not zero, and the Weil
excursion operators act on it through $\sigma_M$. For $B$ with
$Bh=0$,
\[
 \nu_{i\sigma_M}(B)=\nu_{\sigma_M}(i^*B)
 =\nu_\tau\bigl((i^*B)^{\psi^{-1}}\bigr)=0.
\]
\end{proof}

\begin{definition}[Generalized excursion character]
\label{fl:def:locally-generalized-weil-label}
A function $h$ has \emph{generalized excursion character}
$\sigma$ \emph{for Weil tuples} if for every excursion datum $e$
with tuple in $W_F^I$ there is an integer $n_e\ge1$ with
\[
 (S_e^{\mathrm{fun}}-\nu_\sigma(e))^{n_e}h=0.
\]
The exponent may depend on $e$.
\end{definition}

\begin{lemma}[A function which is not cuspidal has a parameter in a Levi subgroup]
\label{rw:lem:noncuspidal-levi-parameter}
Let $h$ be a smooth function of finite level and $S$ a finite set
of places containing the level of $h$. For each place $u\notin S$
let $\mathscr A_u$ be a finitely generated subalgebra of the
spherical Hecke algebra $\mathscr H_{G,u}$ over which
$\mathscr H_{G,u}$ is a finitely generated module, and $\chi_u$ a
character of $\mathscr A_u$ such that for every $T\in\mathscr A_u$
some power of $T-\chi_u(T)$ annihilates $h$. Suppose that $h$ is
not cuspidal, and that at one place $u_0\notin S$ every character
of $\mathscr H_{G,u_0}$ which extends $\chi_{u_0}$ is given by a
Satake parameter at which all characters of the dual torus take
$\ell$-adic unit values. Then there are a proper standard Levi
subgroup $M$ of $G$ and a continuous semisimple parameter
$\sigma_M:\Gamma_F\to\widehat M(k)$, unramified outside $S$, such
that for every $u\notin S$ the character of $\mathscr H_{G,u}$
given by the Satake parameter $i\sigma_M(\operatorname{Frob}_u)$
restricts to $\chi_u$ on $\mathscr A_u$.
\end{lemma}

\begin{proof}
The argument is the classical one for automorphic forms: a nonzero
constant term along a parabolic subgroup which is minimal for this
property is cuspidal (step~1), it satisfies the Hecke equations
obtained from those of $h$ by the Satake transform (step~2), and a
cuspidal function has a parameter by the theorem of V.~Lafforgue
(step~5). Steps~3 and~4 are needed because $h$ is not assumed
invariant under a lattice in the centre: they show that the
constant term becomes invariant under such a lattice after a
twist.

\step{1}{a cuspidal constant term} As in step~1 of the proof
of Lemma~\ref{fl:lem:minimal-cuspidal-support}, choose a proper
standard parabolic subgroup $P=MU$ along which the constant term
of $h$ is not zero, with $M$ of minimal semisimple rank, and a
nonzero slice $h_M$ of the normalized constant term, at a point
$g_0\in G(\prod_v\mathcal O_v)$. Since $h$ is invariant under
$G(\mathcal O_u)$ for $u\notin S$, the slice does not depend on the
components of $g_0$ at these places, and we take them equal to
one. The slice is a cuspidal function on $M(F)\backslash M(\mathbb A)$.
If $K_N$ is the level of $h$, then $h_M$ is invariant under
$M(\mathbb A)\cap g_0K_Ng_0^{-1}$, which contains
$M(\mathcal O_u)$ for every $u\notin S$.

\step{2}{its Hecke equations} Let $u\notin S$, $T\in\mathscr A_u$
and $n\ge1$. By Lemma~\ref{rw:lem:ct-spherical} the slice at $g_0$
of the
normalized constant term of $(T-\chi_u(T))^nh$ is
$(\operatorname{Sat}_P(T)-\chi_u(T))^nh_M$. Hence
\begin{equation}\label{rw:eq:levi-hecke-equations}
 \bigl(\operatorname{Sat}_P(T)-\chi_u(T)\bigr)^nh_M=0
 \qquad(T\in\mathscr A_u,\ n\text{ large}).
\end{equation}

\step{3}{a finite-dimensional orbit} Take $u=u_0$. If
$T_1,\dots,T_s$ generate $\mathscr A_{u_0}$ and
$(\operatorname{Sat}_P(T_j)-\chi_{u_0}(T_j))^{n_j}h_M=0$, the
orbit of $h_M$ under $\operatorname{Sat}_P(\mathscr A_{u_0})$ is
spanned by the functions
$\prod_j(\operatorname{Sat}_P(T_j)-\chi_{u_0}(T_j))^{a_j}h_M$ with
$0\le a_j<n_j$. The algebra $\mathscr H_{M,u_0}$ is a finitely
generated module over $\operatorname{Sat}_P(\mathscr H_{G,u_0})$:
in terms of a maximal torus $T$ these are the invariants of $W_M$
and of $W_G$ in the algebra $k[X_*(T)]$, which is a finitely
generated module over its Noetherian subring of invariants of
$W_G$. By hypothesis $\mathscr H_{G,u_0}$ is a finitely generated
module over $\mathscr A_{u_0}$. So the space
\[
 V=\mathscr H_{M,u_0}h_M
\]
has finite dimension. Its elements are cuspidal functions on $M$
of the same level as $h_M$, and they satisfy
\eqref{rw:eq:levi-hecke-equations} at every $u\notin S$, since the
Hecke operators at $u_0$ commute with those at $u$. Let $v\in V$
be a joint eigenvector of $\mathscr H_{M,u_0}$, and
$t\in\widehat T$ its Satake parameter, which is defined up to
$W_M$. By \eqref{rw:eq:levi-hecke-equations} the character of
$\mathscr H_{G,u_0}$ defined by $t$ extends $\chi_{u_0}$, so by
hypothesis all characters of $\widehat T$ take unit values at $t$.

\step{4}{a central lattice and a twist} Let
$T_M=(Z_M^\circ)_{\mathrm{red}}$, and let $\Xi$ be the image in
$C_{T_M}$ of the group of the elements $\lambda(\varpi_{u_0})$,
$\lambda\in X_*(T_M)$, where $\varpi_{u_0}$ is a uniformizer at
$u_0$. The degree maps $\Xi$ isomorphically onto
$d_{u_0}X_*(T_M)$, so $\Xi$ is a central lattice of $M$.
Translation by $\lambda(\varpi_{u_0})$ is the Hecke operator in
$\mathscr H_{M,u_0}$ of the characteristic function of a coset of
$M(\mathcal O_{u_0})$ by a central element, and its Satake
transform is the monomial attached to $\lambda$. Hence $\Xi$ acts
on $v$ by a character $\chi_\Xi$, whose values $\lambda(t)$ are units
of the ring of integers $\mathcal O_k$ of $k$.

Put $Q_M=M/M_{\mathrm{der}}$, a split torus with dual torus
$Z(\widehat M)^\circ$, and let
$\deg_M:M(\mathbb A)\to X_*(Q_M)$ be the degree, defined as for
$T_G$ in Section~\ref{fl:app:conventions}. Since $T_M\to Q_M$ is
an isogeny, $\deg_M$ maps $\Xi$ isomorphically onto a subgroup of
finite index. The group $\mathcal O_k^\times$ is divisible, so
$\chi_\Xi$ extends to a homomorphism
$c:X_*(Q_M)\to\mathcal O_k^\times$, that is, to a point $c$ of
$Z(\widehat M)^\circ$ at which all characters take unit values.
It is defined over a finite extension of $\mathbb Q_\ell$. Let
$\chi_c=c\circ\deg_M$. It is a character of
$M(F)\backslash M(\mathbb A)$, trivial on $M(\mathcal O_u)$ for
all $u$, and $v'=\chi_c^{-1}v$ is a nonzero cuspidal function of
finite level, invariant under $\Xi$. By Harder's theorem
\cite[Chapter~I, \S5.3, p.~104]{Morris1982} its support is compact
modulo $\Xi$.

\step{5}{a parameter of $M$} By \cite[Theorem~0.1]{VLa18},
applied to $M$ and $\Xi$, the space of cuspidal functions of the
level of $v'$ which are invariant under $\Xi$ is the direct sum of
subspaces $\mathfrak H_\tau$ indexed by continuous semisimple
parameters $\tau$ of $M$, unramified outside $S$. The projections
onto the summands are polynomials in excursion operators, which
commute with the Hecke operators. For $u\notin S$ every
$T\in\mathscr H_{M,u}$ has on $\mathfrak H_\tau$ the single
eigenvalue $\widehat T(\tau(\operatorname{Frob}_u))$, where
$\widehat T$ denotes the Satake transform of $T$, a function on
$\widehat T/W_M$. Let $w\ne0$ be the component of $v'$ in one of
the spaces $\mathfrak H_\tau$.

For $f\in\mathscr H_{M,u}$ and a function $\phi$ on
$M(\mathbb A)$ the definition of $T_f$ gives
$T_f(\chi_c^{-1}\phi)=\chi_c^{-1}\,T_{\chi_c^{-1}f}\phi$. So
\eqref{rw:eq:levi-hecke-equations}, which holds for $v$, gives
\[
 \bigl(T_{\chi_c\operatorname{Sat}_P(T)}-\chi_u(T)\bigr)^nv'=0
 \qquad(u\notin S,\ T\in\mathscr A_u,\ n\text{ large}),
\]
and the same holds for $w$. Comparing with the eigenvalue on
$\mathfrak H_\tau$ we find
\begin{equation}\label{rw:eq:twisted-satake-values}
 \chi_u(T)=\widehat{\chi_c\operatorname{Sat}_P(T)}
            \bigl(\tau(\operatorname{Frob}_u)\bigr)
 \qquad(u\notin S,\ T\in\mathscr A_u).
\end{equation}
Multiplication by $\chi_c$ on $\mathscr H_{M,u}$ corresponds under
the Satake isomorphism to a translation of the dual torus. Indeed,
the Satake transform of a function supported on the double coset
of $\lambda(\varpi_u)$ is a combination of monomials, the exponents
of which all have the image $\bar\lambda$ of $\lambda$ in
$X_*(Q_M)$; and on this double coset $\chi_c$ is equal to
$c(d_u\bar\lambda)=\bar\lambda(c^{d_u})$. So the Satake transform of
$\chi_cf$ is the translate by $c^{d_u}$ of that of $f$. Moreover
the Satake
transform of $\operatorname{Sat}_P(T)$ is that of $T$, regarded as
a function on $\widehat T/W_M$. So the right-hand side of
\eqref{rw:eq:twisted-satake-values} is the value of the Satake
transform of $T$ at
$c^{d_u}\tau(\operatorname{Frob}_u)$.

Recall that a geometric Frobenius element at $u$ has degree
$\deg_F(\operatorname{Frob}_u)=-d_u$. Since $c$ lies in a compact
subgroup of $Z(\widehat M)^\circ(k)$, the homomorphism
$\gamma\mapsto c^{-\deg_F(\gamma)}$ is defined and continuous,
and
\[
 \sigma_M(\gamma)=\tau(\gamma)\,c^{-\deg_F(\gamma)}
\]
is a continuous parameter of $M$, unramified outside $S$. It is
semisimple: the Zariski closures of the images of $\tau$ and of
$\sigma_M$ have the same image in
$\widehat M/Z(\widehat M)^\circ$, and a closed subgroup of
$\widehat M$ whose image there has reductive identity component
has reductive identity component itself. Finally
$\sigma_M(\operatorname{Frob}_u)
=c^{d_u}\tau(\operatorname{Frob}_u)$, so
\eqref{rw:eq:twisted-satake-values} is the assertion.
\end{proof}

\begin{proposition}[Ellipticity implies cuspidality]
\label{fl:prop:full-label-cuspidality}
Let $\sigma:\Gamma_F\to\widehat G(k)$ be a continuous semisimple
elliptic parameter, and let $h$ be a smooth function of finite
level with generalized excursion character $\sigma$ for Weil
tuples. Then $h$ is cuspidal. If $h$ is invariant under a central
lattice, its support is compact modulo this lattice.

More generally, let $S$ be a finite set of places containing the
level of $h$ and the places where $\sigma$ is ramified, and for
$u\notin S$ let $\chi_u$ be the character of $\mathscr H_{G,u}$
given by the Satake parameter $\sigma(\operatorname{Frob}_u)$. If
for every $u\notin S$ and every $T\in\mathscr H_{G,u}$ some power
of $T-\chi_u(T)$ annihilates $h$, then $h$ is cuspidal.
\end{proposition}

\begin{proof}
We prove the second assertion first. The Hecke operators give only
the conjugacy classes of the Frobenius elements, and the point is
that this suffices: it determines the composite of a parameter
with the adjoint representation, and the invariants of this
representation detect ellipticity.

Suppose that $h$ is not cuspidal. At every $u\notin S$ the
Satake parameter $\sigma(\operatorname{Frob}_u)$ is unit-valued,
because $\sigma$ is continuous
(Lemma~\ref{fl:lem:parameter-satake-integrality}). So
Lemma~\ref{rw:lem:noncuspidal-levi-parameter} applies, with
$\mathscr A_u=\mathscr H_{G,u}$,
and gives $M$ and $\sigma_M$. For $u\notin S$ the semisimple parts
of $\sigma(\operatorname{Frob}_u)$ and of
$i\sigma_M(\operatorname{Frob}_u)$ define the same character of
$\mathscr H_{G,u}$; so they are conjugate in $\widehat G$, and
have the same trace in the adjoint representation of $\widehat G$
on its Lie algebra $\widehat{\mathfrak g}$. The two
representations
\[
 \operatorname{Ad}\circ\sigma,\qquad
 \operatorname{Ad}\circ i\sigma_M
\]
of $\Gamma_F$ on $\widehat{\mathfrak g}$ are continuous and
unramified outside $S$, so by the theorem of Chebotarev their
characters are equal. Both are semisimple, because the identity
components of the Zariski closures of the images of $\sigma$ and
of $i\sigma_M$ are reductive and $k$ has characteristic zero. So
they are isomorphic, and their spaces of invariants have the same
dimension. The invariants of the first form the Lie algebra of the
centralizer of the image of $\sigma$, whose dimension is
$\dim Z(\widehat G)$ because $\sigma$ is elliptic. The invariants
of the second contain the Lie algebra of $Z(\widehat M)^\circ$,
and $\dim Z(\widehat M)>\dim Z(\widehat G)$ because $M$ is proper.
This is a contradiction. So $h$ is cuspidal.

Let now $h$ have generalized excursion character $\sigma$ for Weil
tuples, and let $S$ be the set of places where $h$ or $\sigma$ is
ramified. For $u\notin S$ every element $T$ of $\mathscr H_{G,u}$
is a linear combination $\sum_ja_jS_{e_j}$ of excursion operators
with tuples in the Weil group, and
$\sum_ja_j\nu_\sigma(e_j)=\chi_u(T)$
(Proposition~\ref{fl:prop:canonical-compact}(3)). These operators
commute, and $(S_{e_j}-\nu_\sigma(e_j))^{n_j}h=0$; expanding the
power of $T-\chi_u(T)=\sum_ja_j(S_{e_j}-\nu_\sigma(e_j))$ with
exponent $\sum_jn_j$ shows that it annihilates $h$. So the
hypothesis of the second assertion holds, and $h$ is cuspidal. The
assertion on the support is the theorem of Harder quoted above.
\end{proof}

\begin{remark}
Equality of Satake parameters at the unramified places gives only
local conjugacy of $\sigma$ and $i\sigma_M$, which for a general
reductive group is weaker than conjugacy. The proof does not need
more: local conjugacy already determines the composite with every
representation of $\widehat G$ up to isomorphism, and the adjoint
representation detects ellipticity. Identifying the parameter of a
cuspidal function is a different matter, and uses the excursion
operators of all tuples: in \cite{HuII} through
Proposition~\ref{fl:prop:cuspidal-excursion-character}, and in
\cite{HuIII} through
Proposition~\xref{III}{fl:prop:prelim-full-parameter}.
Lemma~\ref{rw:lem:noncuspidal-levi-parameter} is
stated for a subalgebra $\mathscr A_u$ because the construction of
\cite{HuIII} controls only the Hecke operators of
the standard representation.
\end{remark}

\subsection{Extracting a function from one pairing}
\label{fl:sec:hecke-moment-extraction}

\begin{lemma}[Extraction from a pairing]
\label{fl:lem:hecke-moment-extraction}
Fix a level $K$ and a central lattice $\Xi$, and let
$\mathscr H^{\mathrm{good}}$ be the algebra generated by the
spherical Hecke algebras at the places outside the level. Let $v$
be a function on $G(F)\backslash G(\mathbb A)/K\Xi$, let $c$ be a
function with compact support on the same set, and let
$\eta:\mathscr H^{\mathrm{good}}\to k$ be a character, such that
\[
 [c,Tv]=\eta(T)[c,v]\quad(T\in\mathscr H^{\mathrm{good}}),
 \qquad [c,v]\ne0.
\]
Then there is a function $h$ on the same set with $[c,h]\ne0$
which is annihilated by a power of a maximal ideal of the
spherical Hecke algebra at one place and by a power of
$T-\eta(T)$ for every $T\in\mathscr H^{\mathrm{good}}$. It is an
automorphic form. If the Satake parameter of $\eta$ at
that place is unit-valued, $h$ belongs to $\mathcal A_\Xi(G)$.
\end{lemma}

\begin{proof}
Let $C$ be the space of functions with compact support on
$G(F)\backslash G(\mathbb A)/K\Xi$ with values in $k$, let $u$ be
a place outside the level and
$\mathscr H=\mathscr H_{G,u,k}$.

\step{1}{a finite algebra} By
Proposition~\ref{fl:prop:classical-one-place-finiteness}, $C$ is
a finitely generated module over $\mathscr H$. Let $A$ be the
image of $\mathscr H^{\mathrm{good}}$ in
$\operatorname{End}_{\mathscr H}(C)$. Evaluation on a finite set
of generators embeds $\operatorname{End}_{\mathscr H}(C)$ into a
finite sum of copies of $C$; so $A$ is a finitely generated
module over the Noetherian ring $\mathscr H$.

\step{2}{a maximal ideal} Write $L_v(d)=[d,v]$ for $d\in C$ and
$\eta^\star(T)=\eta(T^\star)$. Transposing the hypothesis we get
\[
 L_v(Tc)=\eta^\star(T)L_v(c),\qquad T\in\mathscr H^{\mathrm{good}}.
\]
Since $L_v(c)\ne0$, the character $\eta^\star$ vanishes on the
operators which act by zero on $C$. So it is a character of $A$;
let $\mathfrak m$ be its kernel. The image of $c$ in the
localization $C_{\mathfrak m}$ is not zero: otherwise $sc=0$ for
some $s\notin\mathfrak m$, while
$L_v(sc)=\eta^\star(s)L_v(c)\ne0$.

\step{3}{a finite quotient} The ideal
$\mathfrak j=\mathfrak m\cap\mathscr H$ is maximal, because $A$
is integral over the image of $\mathscr H$. By the intersection
theorem of Krull, applied to the finitely generated module
$C_{\mathfrak m}$ over the Noetherian local ring
$A_{\mathfrak m}$, there is an $n$ such that the image of $c$ is
not in $\mathfrak j^nC_{\mathfrak m}$. The space
$C/\mathfrak j^nC$ has finite dimension and is a module over the
Artinian ring $A/\mathfrak j^nA$, which is a product of local
rings; the factor attached to $\mathfrak m$ cuts out the direct
summand $C_{\mathfrak m}/\mathfrak j^nC_{\mathfrak m}$. Choose a
linear form on this summand which does not vanish on the image of
$c$, and extend it by zero on the other summands. On the summand
every $T-\eta^\star(T)$ is nilpotent, so the linear form is
annihilated by a power of the transpose of $T-\eta^\star(T)$.

\step{4}{the function} By the pairing, the linear form is
$d\mapsto[d,h]$ for a function $h$. Then $[c,h]\ne0$; the
function $h$ is annihilated by the ideal $(\mathfrak j^n)^\star$
of $\mathscr H$; and it is annihilated by a power of $T-\eta(T)$
for every $T\in\mathscr H^{\mathrm{good}}$. Its orbit under
$\mathscr H$ has finite dimension, so it is an
automorphic form by
Proposition~\ref{fl:prop:one-place-automorphicity}. The only
Satake parameter of this orbit is that of $\eta$ at $u$; if it is
unit-valued, the orbit is spanned by a stable lattice as in the
proof of Lemma~\ref{fl:lem:faithful-satake-model}(2), and
$h\in\mathcal A_\Xi(G)$ by
Lemma~\ref{fl:lem:all-place-stable-lattice}.
\end{proof}

\begin{remark}
The power $\mathfrak j^n$ cannot be replaced by $\mathfrak j$. For
$C=k[t]$ and $c=t$, the functional with value $1$ on $t$ and $0$ on
the other monomials satisfies the hypothesis for the character
$t\mapsto0$, but $c$ vanishes in $C/tC$; it survives in $C/t^2C$.
\end{remark}