\section{Parameters and Fourier coefficients}
\label{fl:sec:classical-descent}
\label{fl:sec:parameters}

This paper proves Theorem~\ref{fl:thm:classical-realization}: for
the classical groups, every elliptic parameter is the parameter of
a cusp form, in every characteristic. The starting point is L.~Lafforgue's
correspondence for general linear groups, applied to the standard
representation $\varrho=\operatorname{Std}\sigma$ of the parameter.
The form preserved by $\varrho$ determines an Eisenstein series on a larger
classical group; a Fourier coefficient of its residue is a function
on the group we want, the \emph{descent} of Ginzburg, Rallis and
Soudry. For part of the symplectic case the descent is
replaced by a theta lift.

There are two separate questions. A local computation of Jacquet
modules identifies the Hecke eigenvalues of a descent \emph{if the
descent is nonzero}. That it is nonzero is a global statement, which
we prove by comparing Fourier coefficients over $F$ itself. The
comparison uses explicit root subgroups, and this is where
characteristic two needs separate calculations
(Section~\ref{fl:sec:char2-d}). Once a nonzero function with
the right Hecke eigenvalues is found, the results of
\cite{HuI} show that it is cuspidal and has the
parameter $\sigma$, for the excursion operators of all tuples
(Proposition~\ref{fl:prop:prelim-full-parameter}).

The present section collects what is common to all cases: how
$\sigma$ is recovered from $\varrho$
(Lemma~\ref{fl:lem:prelim-form},
Proposition~\ref{fl:prop:prelim-block}), the order of the poles of
the relevant $L$-functions (Lemma~\ref{fl:lem:prelim-pole}), the
recognition statement just mentioned, the case of type $A$
(Theorem~\ref{fl:thm:type-a-realization}), and two general facts
about Fourier coefficients
(Lemmas~\ref{fl:lem:odd-exchange} and~\ref{fl:lem:odd-averaging}).

Throughout the paper $k=\overline{\mathbb Q}_\ell$ with $\ell\ne p$.
Integrals have complex values, compared with $k$ through a fixed
isomorphism $\iota:k\simeq\mathbb C$ which maps the chosen
$q^{1/2}$ to the positive square root of $q$; so the Satake
parameters defined with $q^{1/2}$ are those of normalized
induction with the positive modulus character. All inductions are
normalized.
For a unipotent group $U$ we put
$[U]=U(F)\backslash U(\mathbb A_F)$, with the measure of volume one,
and we fix a nontrivial character
$\psi:F\backslash\mathbb A_F\to\mathbb C^\times$. The irreducible
summands of $\varrho$ are denoted $\varrho_i$. In computations with roots,
$E_{ab}$ is the matrix sending the basis vector $b$ to $a$, and
$x_\alpha(t)$ is the full matrix of the root subgroup, including its
quadratic entries when the root is short. The table shows where each
kind of input is established.
\begin{center}
\begin{tabular}{>{\raggedright\arraybackslash}p{.25\textwidth}
                >{\raggedright\arraybackslash}p{.34\textwidth}
                >{\raggedright\arraybackslash}p{.27\textwidth}}
\toprule
Input & Assertion needed & Location\\ \midrule
Self-dual general-linear blocks & A residual representation with a
nonzero Siegel--Whittaker coefficient &
Proposition~\ref{fl:lem:residual-analytic}\\
Local descent & The parameter and the obstruction to a nonsplit
endpoint & Proposition~\ref{fl:prop:uniform-b-jacquet}\\
Global root operations & A nonzero descent over $F$ &
Propositions~\ref{fl:prop:uniform-b-nonzero},
\ref{fl:prop:even-generic-nonzero},
\ref{fl:prop:char2-direct-generic}\\
Quadratic kernels & Nonzero lifts and spherical
Hecke equations & Corollary~\ref{fl:cor:odd-theta-moment}\\
Common arithmetic & Cuspidality and the excursion character &
Proposition~\ref{fl:prop:prelim-full-parameter}\\ \bottomrule
\end{tabular}
\end{center}

\subsection{Recovering a parameter from its standard representation}

A parameter with values in a classical group is a representation
together with a form. The first lemma says that the representation determines the form
up to isometry. In even orthogonal dimension this leaves the outer
automorphism: two parameters with the same standard representation
are conjugate in the orthogonal group, and not necessarily in the
special orthogonal group.

\begin{lemma}[Forms and simultaneous conjugacy]
\label{fl:lem:prelim-form}
Let $r$ and $r'$ be semisimple representations of a group $\Gamma$
on a finite-dimensional $k$-vector space $V$ which preserve, up to
the same character $\chi$ of $\Gamma$, a nondegenerate symmetric or
alternating form $B$. If $r\simeq r'$, then $r'=grg^{-1}$ for an
isometry $g$ of $B$. If $B$ is symmetric and $\dim V$ is odd, $g$
can be taken of determinant one.
\end{lemma}
\begin{proof}
This is an instance of the following. Let $T$ be a
representation of $\mathrm{GL}(V)$ with an open orbit $\mathcal O$,
let $b\in\mathcal O$, and suppose $r$ and $r'$ both multiply $b$ by
$\chi$. Choose $a\in\mathrm{GL}(V)$ with $r'=ara^{-1}$. Then
$b'=a^{-1}b$ is again multiplied by $\chi$ under $r$. If $b'=cb$
for some $c$ in the centralizer $C$ of $r$, then $g=ac$ stabilizes
$b$ and $grg^{-1}=r'$. So it suffices to show that $C$ acts
transitively on
\[
 \mathcal N=\mathcal O\cap E,\qquad
 E=\{t\in T:r(\gamma)t=\chi(\gamma)t\text{ for all }\gamma\}.
\]
Let $H$ be the Zariski closure of $r(\Gamma)$, a reductive group.
The line $kb$ is stable under $r(\Gamma)$, hence under $H$, and $H$
acts on it through an algebraic character which extends $\chi$; we
denote this character by $\chi$ as well. At a point $t\in\mathcal N$ the map $\xi\mapsto\xi t$ from
$\mathfrak{gl}(V)$ to $T$ is surjective, because $\mathcal O$ is
open, and it is $H$-equivariant when the action of $H$ on $T$ is
twisted by $\chi^{-1}$. Taking invariants under $H$ is exact, so
the Lie algebra $\mathfrak{gl}(V)^H$ of $C$ maps onto $E$. Hence
every orbit of $C$ on $\mathcal N$ is open. But $\mathcal N$ is a
nonempty open subset of the vector space $E$ and therefore
irreducible, so it cannot contain two disjoint nonempty open
subsets. Thus $\mathcal N$ is a single orbit.

Take $T$ to be the space of symmetric or alternating bilinear
forms, in which the nondegenerate forms are an open orbit with
stabilizer the isometry group. In odd dimension $-1$ is an isometry
of determinant $-1$ which acts trivially by conjugation.
\end{proof}

\begin{proposition}[The blocks of an elliptic parameter]
\label{fl:prop:prelim-block}
Let $r$ be a semisimple representation of $\Gamma$ on $V$
preserving a nondegenerate symmetric or alternating form $B$ up to
a character $\chi$; in the symmetric case assume $\dim V\ge3$. The
following are equivalent.
\begin{enumerate}
\item The centralizer of $r$ in the group of similitudes of $B$ is
finite modulo scalars.
\item $V$ is an orthogonal direct sum of pairwise nonisomorphic
irreducible subrepresentations $V_i$, and the restriction of $B$
to each $V_i$ is nondegenerate.
\end{enumerate}
If $B$ is alternating and $\dim V_i=2d_i$, then
$\det V_i=\chi^{d_i}$.
\end{proposition}
\begin{proof}
Write $V=\bigoplus_WW\otimes M_W$, the sum being over the
irreducible representations and $M_W$ the multiplicity spaces. The
form pairs the summand of $W$ with that of
$W^*=W^\vee\otimes\chi$ and with no other.

Assume (1). If some $W\not\simeq W^*$ occurs, then for every
$t\in k^\times$ the transformation equal to $t$ on $W\otimes M_W$,
to $t^{-1}$ on $W^*\otimes M_{W^*}$ and to $1$ on the other
summands is an isometry commuting with $r$. These form a torus
which meets the scalars in a finite group, contrary to (1); in the
symmetric case this uses $\dim V\ge3$ when there is no other
summand. So every $W$ that occurs satisfies $W\simeq W^*$. By
Schur's lemma the form on $W\otimes M_W$ is then $b_W\otimes b_M$,
where $b_W$ is the invariant form of $W$, unique up to a scalar,
and $b_M$ is a nondegenerate symmetric or alternating form on
$M_W$. The isometry group of $b_M$ commutes with $r$, and it is
infinite as soon as $\dim M_W\ge2$. Hence $\dim M_W=1$, the form
$b_M$ is symmetric, and $b_W$ has the symmetry of $B$. This is
(2).

Assume (2). A similitude commuting with $r$ acts on each $V_i$ by a
scalar $c_i$, and $c_i^2$ is its multiplier, the same for all $i$.
So it is a scalar times an element of a finite group of signs.

For the last assertion, a similitude of an alternating form in
dimension $2d$ with multiplier $c$ has determinant $c^d$.
\end{proof}

\begin{remark}
Ellipticity excludes multiplicities; it does not make the standard
representation irreducible. In particular an orthogonal elliptic
parameter has at most one line of invariants under $\Gamma_F$,
which will determine the rank of the quadratic theta lift in
Section~\ref{fl:sec:odd-theta}.
\end{remark}

\subsection{Poles of \texorpdfstring{$L$}{L}-functions}

\begin{lemma}[The order of the pole at $s=1$]
\label{fl:lem:prelim-pole}
Let $V$ be a semisimple lisse sheaf on a nonempty open
$U\subset X$, pure of weight zero. Then the order of the pole of
$L^U(s,V)$ at $s=1$ is $\dim V^{\Gamma_F}$. The function has
neither zeros nor poles at real $s>1$, nor at $s=1$ if
$V^{\Gamma_F}=0$.
\end{lemma}
\begin{proof}
By the Grothendieck--Lefschetz formula,
\[
 L^U(s,V)=\prod_{j=0}^2
 \det\bigl(1-q^{-s}\operatorname{Frob}\bigm|
 H^j_c(U_{\overline{\mathbb F}_q},V)\bigr)^{(-1)^{j+1}}.
\]
The eigenvalues of Frobenius on $H^0_c$ and $H^1_c$ have absolute
values at most $1$ and $q^{1/2}$
\cite[Theorem~3.3.1 and \S3.3.10]{Del80}, so these two factors
have no zero for real $s\ge1$. By Poincar\'e duality $H^2_c$ is the
space of coinvariants of the geometric fundamental group on $V$,
twisted by $-1$. Its eigenvalues have absolute value $q$, so the
factor for $j=2$ has no zero for $s>1$, and its order of vanishing
at $s=1$ is the multiplicity of the eigenvalue $1$ of Frobenius on
the geometric coinvariants. Since $V$ is semisimple and the
geometric monodromy group is normal in the arithmetic one, the
geometric invariants are a direct summand of $V$, isomorphic to
the coinvariants, on which the quotient group acts. That quotient
is reductive with a dense cyclic subgroup, hence diagonalizable, so
Frobenius is semisimple there and the multiplicity of the
eigenvalue $1$ is $\dim V^{\Gamma_F}$.
\end{proof}

We shall apply the lemma to the blocks $V_i$ of an elliptic
parameter. By Schur's lemma the square $L$-function of the same sign
as the form, $L(s,\Lambda^2V_i\otimes\chi^{-1})$ or
$L(s,\operatorname{Sym}^2V_i\otimes\chi^{-1})$, has a simple pole
at $s=1$, and for $i\ne j$ the function
$L(s,V_i\otimes V_j\otimes\chi^{-1})$ has neither a pole nor a zero
there. This is what prevents cancellation in the Eisenstein
coefficient of Section~\ref{fl:sec:common-descent}.

\subsection{Recognizing the parameter of a function}

The constructions of this paper produce complex-valued functions.
The next lemma moves them to the coefficient field $k$.

\begin{lemma}[Transport of coefficients]\label{fl:lem:prelim-coeff}
Let $\varphi$ be a complex cusp form of finite level, invariant under a
central lattice. Then $\iota^{-1}\circ\varphi$ is a cusp form with
values in a finite extension of $\mathbb Q_\ell$. It satisfies the
same Hecke equations as $\varphi$, and its Whittaker coefficients are the
images of those of $\varphi$.
\end{lemma}
\begin{proof}
By Harder's theorem
\cite[Theorem~1.2.1 and Corollary~1.2.3]{Harder74} the support of
$\varphi$ at a given level is compact modulo the centre, and the lattice
makes it finite modulo $G(F)$ and the level subgroup. So $\varphi$ takes
finitely many values, and their images under $\iota^{-1}$ lie in a
finite extension of $\mathbb Q_\ell$. Hecke operators, constant
terms and Whittaker coefficients at a given level are finite sums
whose coefficients are rational numbers, powers of the chosen
square root of $q$ and roots of unity; $\iota^{-1}$ respects such
sums.
\end{proof}

A smooth complex function that is not yet known to be cuspidal can
also be composed with $\iota^{-1}$: its constant terms, its Fourier
coefficients and the Hecke operators are still finite sums. Only the
argument given above for the finiteness of the field of values
uses cuspidality.

\begin{proposition}[Recognition by standard Hecke data]
\label{fl:prop:prelim-full-parameter}
Suppose that the dual group of $G$ is $\mathrm{Sp}_{2n}$,
$\mathrm{SO}_d$ with $d\ge3$, $\mathrm{GSp}_{2n}$, or
$\mathrm{GSO}_{2n}$ with $n\ge2$. Let
$\sigma:\Gamma_F\to\widehat G(k)$ be continuous, finitely ramified
and semisimple, with multiplier of finite order in the similitude
cases. Let $h\ne0$ be a smooth function of finite level on
$G(F)\backslash G(\mathbb A_F)$, and let $S$ be a finite set of
places containing the places where $h$ or $\sigma$ is ramified.
Suppose that at every place outside $S$ the Hecke operators
attached to the standard representation act on $h$ through the
coefficients of the characteristic polynomial of
$\operatorname{Std}\sigma$ of Frobenius, possibly as generalized
eigenvalues; in the similitude cases assume the same for the
multiplier, and that $h$ has the corresponding central character of
finite order. (The set $S$ may be larger than the set of ramified
places: in the applications it also contains the places where an
auxiliary additive character, a section or a level is ramified.)
\begin{enumerate}
\item If $\sigma$ is elliptic, then $h$ is cuspidal.
\item If $h$ is cuspidal, the cuspidal excursion summand of
$\sigma$ is nonzero. If moreover a Whittaker coefficient of $h$ is
nonzero, this summand contains a function with a nonzero Whittaker
coefficient.
\end{enumerate}
\end{proposition}
\begin{proof}
\step{1}{parameters with the same standard data} Let $\sigma'$ be
a continuous semisimple parameter with the same characteristic
polynomials of the standard representation at almost all places,
and the same multiplier. By Chebotarev and Brauer--Nesbitt,
$\operatorname{Std}\sigma'\simeq\operatorname{Std}\sigma$.
Lemma~\ref{fl:lem:prelim-form} then shows that $\sigma'$ is
conjugate to $\sigma$ in the stabilizer of the tensor, that is, in
$\widehat G$ or, in even orthogonal type, in the full orthogonal or
similitude group.

\step{2}{cuspidality} Suppose that $\sigma$ is elliptic and that
$h$ is not cuspidal. Let $r$ be the standard representation of
$\widehat G$, and for $u\notin S$ let $\mathscr A_u$ be the
subalgebra of the spherical Hecke algebra at $u$ generated by the
operators which correspond to the coefficients of the
characteristic polynomial of $r$, to the inverse of its
determinant and, in the similitude cases, to the multiplier and
its inverse. The spherical Hecke algebra is a finitely generated
module over $\mathscr A_u$, by the proof of
Lemma~\xref{I}{fl:lem:faithful-satake-model}(1). (The determinant of
$r$ is one in the isometry cases and a power of the multiplier in
the similitude cases.) By hypothesis $\mathscr A_u$ acts on $h$
through the generalized character $\chi_u$ given by the standard
data of $\sigma(\operatorname{Frob}_u)$. A Satake parameter over
$\chi_u$ has in the standard representation the eigenvalues of
$\operatorname{Std}\sigma(\operatorname{Frob}_u)$, which are
$\ell$-adic units because $\sigma$ is continuous; since the
weights of the standard representation, together with the
multiplier in the similitude cases, generate a subgroup of finite
index in the character group of the dual torus, every character of
the dual torus takes a unit value at such a parameter.

So Lemma~\xref{I}{rw:lem:noncuspidal-levi-parameter} applies. It
gives a proper standard Levi subgroup $M$ and a continuous
semisimple parameter $\sigma_M$ of $M$, unramified outside $S$,
such that $i\sigma_M$ and $\sigma$ have the same characteristic
polynomials of the standard representation, and the same
multiplier, at the Frobenius elements of all places outside $S$.
By step~1, $\sigma$ is conjugate to $i\sigma_M$ in $\widehat G$
or, in even orthogonal type, in the full orthogonal or similitude
group. Conjugation by an element of that group preserves
$\widehat G$ and maps $i(\widehat M)$ onto a proper Levi subgroup
of $\widehat G$. So the image of $\sigma$ lies in a proper Levi
subgroup of $\widehat G$, whose connected centre centralizes it
and has positive dimension modulo $Z(\widehat G)$. This
contradicts ellipticity.

\step{3}{the excursion summand} Assume $h$ cuspidal. The space of
cusp forms of its level and central character is finite dimensional
and is the direct sum of its generalized excursion summands.
Excursion operators commute with Hecke operators, so every
component of $h$ satisfies the same Hecke equations, and by step~1
the parameter of every nonzero component is $\sigma$ or its outer
conjugate. Some component is nonzero, and if a Whittaker
coefficient of $h$ is nonzero then it is nonzero on some component.

\step{4}{the outer automorphism} In even orthogonal type let $s$
be the isometry, defined over $F$, which exchanges the two vectors
of the last hyperbolic pair and fixes the other vectors of the
basis. Conjugation by $s$ is an automorphism of the special
orthogonal group over $F$ which induces the outer automorphism of
the dual group. It lifts to $\mathrm{GSpin}$. An element of the
Clifford group which represents $s$ is odd, so it does not lie in
$\mathrm{GSpin}$; but conjugation by it preserves the even
Clifford group $\mathrm{GSpin}$, and fixes the scalars. This holds
in characteristic two as well, where $s$ has Dickson invariant
one. By Lemma~\xref{I}{fl:lem:arithmetic-central-pullback} this
automorphism maps the summand of a parameter onto the summand of
the outer conjugate of the parameter, and it maps a Whittaker
coefficient to the Whittaker coefficient for the character which
it transports. Hence the two summands are both nonzero.
\end{proof}

\subsection{Type \texorpdfstring{$A$}{A} and lifting along a central torus}
\label{fl:sec:type-a}

\begin{lemma}[Lifting along a central torus]\label{fl:lem:central-torus-lift}
Let $1\to Z\to H'\to H\to1$ be an exact sequence of algebraic
groups over $k$ with $H$ semisimple, $H'$ connected reductive and
$Z$ a central torus. Every continuous, finitely ramified parameter
$\tau:\Gamma_F\to H(k)$ lifts to a continuous, finitely ramified
parameter $\widetilde\tau:\Gamma_F\to H'(k)$ whose image in the
abelianization of $H'$ is finite. If $\tau$ is semisimple, or
elliptic, or strongly pure, then so is $\widetilde\tau$, ellipticity
being understood modulo the centre.
\end{lemma}
\begin{proof}
A continuous lift, ramified at finitely many additional places,
exists by \cite[Lemma~5.2 and Proposition~5.3]{Conrad2011}, and it
takes values in a finite extension of $\mathbb Q_\ell$ by
Lemma~\xref{I}{fl:lem:arithmetic-finite-coefficients}. Since $H$ is
semisimple,
$Z$ is the connected centre of $H'$ and maps onto the
abelianization of $H'$ with finite kernel. Multiplying the lift by
a character with values in $Z$ that factors through the degree, as
in Lemma~\xref{I}{fl:lem:finite-coefficients}, we make its image in
the abelianization finite without changing its projection to $H$
or its restriction to inertia.

If $\tau$ is semisimple, the unipotent radical of the Zariski
closure of the image of $\widetilde\tau$ maps to a unipotent normal
subgroup of the reductive closure of the image of $\tau$, hence
trivially; it lies in the torus $Z$ and is trivial. The centralizer
of $\widetilde\tau$ maps into that of $\tau$, and the inverse image
of the centre of $H$ is the centre of $H'$; this gives ellipticity.
For purity, let $r$ be an irreducible representation of $H'$. The
representation $\operatorname{End}(r)$ factors through $H$, so at
an unramified place the ratios of the Frobenius eigenvalues of
$r\widetilde\tau$ are algebraic of absolute value one under every
embedding; and $\det r$ factors through the abelianization, so the
product of the eigenvalues is a root of unity. Hence each
eigenvalue is algebraic of absolute value one.
\end{proof}

\begin{theorem}[Type $A$]\label{fl:thm:type-a-realization}
Every continuous, finitely ramified, semisimple elliptic parameter
with values in $\mathrm{PGL}_n(k)$ has a nonzero cuspidal excursion summand on $\mathrm{SL}_n(\mathbb A_F)$.
\end{theorem}
\begin{proof}
By Lemma~\ref{fl:lem:central-torus-lift} the parameter lifts to a
representation $\widetilde\tau$ of dimension $n$ with determinant
of finite order. It is irreducible: if it were a sum of two
subrepresentations, scaling them independently would give a torus
in the centralizer of the original parameter. By L.~Lafforgue's
theorem \cite[Introduction, assertion $(\mathrm{ii})_n$]{LLa02}
there is a cuspidal representation of
$\mathrm{GL}_n(\mathbb A_F)$ with parameter $\widetilde\tau$ and
central character of finite order. Every excursion summand that
meets it has, by the compatibility of excursion operators with
Hecke operators and by Chebotarev, the parameter $\widetilde\tau$.
Choose a function in such a summand, translate it so that it is
nonzero at the identity, and restrict it to
$\mathrm{SL}_n(\mathbb A_F)$. The restriction is cuspidal because
the unipotent radicals of the parabolic subgroups of
$\mathrm{SL}_n$ and $\mathrm{GL}_n$ coincide, and by
Lemma~\xref{I}{fl:lem:arithmetic-central-pullback} it lies in the
summand of the image of $\widetilde\tau$ in $\mathrm{PGL}_n$.
\end{proof}

\subsection{Two facts about Fourier coefficients}
\label{fl:sec:odd-tools}

For a character $\zeta$ of $[U]$ write
\[
 \mathcal F_{U,\zeta}f(g)=\int_{[U]}f(ug)\zeta(u)^{-1}\,du.
\]
Let $\mathcal V$ be a space of automorphic functions stable under
right translation. We write $(U,\zeta)\sim(U',\zeta')$ if
$\mathcal F_{U,\zeta}$ vanishes identically on $\mathcal V$ exactly
when $\mathcal F_{U',\zeta'}$ does. This relates the two
coefficients only as conditions on the whole space; it does not
compare them on an individual function. The global arguments of
this paper consist of chains of such equivalences, obtained from the
following lemma and from conjugation by elements of $G(F)$.

\begin{lemma}[Exchange of root subgroups]\label{fl:lem:odd-exchange}
Let $U$ be a unipotent group with a character $\zeta$ of $[U]$, and
let $X$ and $Y$ be vector groups which normalize $U$, preserve
$\zeta$, meet $U$ trivially and satisfy $[X,Y]\subset U$. Suppose
that $\zeta$ extends to characters of $UX$ and of $UY$, trivial on
$X$ and on $Y$, and that $(x,y)\mapsto\zeta([x,y])$ is a perfect
pairing between $X$ and $Y$ defined over $F$. Then
\[
 (UX,\zeta)\sim(U,\zeta)\sim(UY,\zeta).
\]
\end{lemma}
\begin{proof}
Since $\mathcal F_{UX,\zeta}$ is the integral of
$\mathcal F_{U,\zeta}$ over $[X]$, the vanishing of the latter
implies that of the former. Conversely, for fixed $g$ the function
$x\mapsto\mathcal F_{U,\zeta}f(xg)$ on $[X]$ has a Fourier
expansion, which is a finite sum because $f$ is smooth. By the
perfect pairing its characters are $x\mapsto\zeta([x,y])$ with
$y\in Y(F)$. Write $[x,y]=xyx^{-1}y^{-1}$. The function
$\mathcal F_{U,\zeta}f$ is equivariant on the left under
$U(\mathbb A_F)$ and invariant on the left under $Y(F)$, so
$\mathcal F_{U,\zeta}f(xy^{-1}g)=
\zeta([x,y])^{-1}\mathcal F_{U,\zeta}f(xg)$.
Hence the coefficient of this character is
$\mathcal F_{UX,\zeta}f(y^{-1}g)$. Consequently, if
$\mathcal F_{UX,\zeta}$ vanishes on $\mathcal V$, then all the
coefficients vanish, and so does
$\mathcal F_{U,\zeta}$. The statement for $Y$ follows by symmetry.
\end{proof}

The second fact is local. It computes twisted Jacquet modules of
induced representations and is the tool for
Proposition~\ref{fl:prop:uniform-b-jacquet}.

\begin{lemma}[Twisted Jacquet modules of induced representations]
\label{fl:lem:odd-averaging}
\label{fl:cor:jacquet-coset-decomposition}
Let $G$ be a reductive group over a nonarchimedean local field,
$P=MU$ and $P'$ parabolic subgroups, $\zeta$ a character of $U$,
$H\subset M$ a subgroup preserving $\zeta$, and $\chi$ a character
of $P'$. For a representative $w$ of a double coset in
$P'\backslash G/P$, taken in the Weyl group of a maximal split torus
contained in both Levi subgroups, write $P\cap w^{-1}P'w=M_w\ltimes U_w$ and
\[
 Z_w=\{M_w\gamma\in M_w\backslash M:\zeta\text{ is trivial on }
 \gamma^{-1}U_w\gamma\}.
\]
Assume that the conditions defining $Z_w$ are closed conditions on
the matrix entries of $\gamma$.
\begin{enumerate}
\item The double coset of $w$ contributes to the twisted Jacquet
module $J_{U,\zeta}(i_{P'}^G\chi)$ only through the fibres over $Z_w$.
\item Suppose that $Z_w$ is empty for all double cosets but one,
and that for this one it is a finite disjoint union of closed
orbits $M_w\gamma_jH\simeq B_j\backslash H$ admitting local sections.
Put $U_j=U\cap (w\gamma_j)^{-1}P'w\gamma_j$. Then, as a representation of
$H$,
\[
 J_{U,\zeta}(i_{P'}^G\chi)\simeq
  \bigoplus_j \operatorname{c-Ind}_{B_j}^H\left(
 (\chi\delta_{P'}^{1/2})\circ\operatorname{Ad}(w\gamma_j)\,
 \left|\det\operatorname{Ad}(\cdot)|_{\Lie U/\Lie U_j}\right|\right).
\]
\end{enumerate}
\end{lemma}
\begin{proof}
The proof has three parts: exactness of twisted coinvariants, the
computation on a single fibre, and the passage from one fibre to a
family of fibres, which is the only delicate point.

\step{1}{exactness and a single fibre}
A contracting cocharacter exhausts $U$ by compact open subgroups
$U_n$. Coinvariants for $(U_n,\zeta)$ are the image of the
projector that averages against $\zeta^{-1}$, which is exact in
characteristic zero, and $J_{U,\zeta}$ is their direct limit; so
$J_{U,\zeta}$ is exact. For a closed subgroup $L\subset U$, the
twisted coinvariants of $\operatorname{c-Ind}_L^U1$ are zero if
$\zeta|_L$ is nontrivial, and if $\zeta|_L$ is trivial they are
one-dimensional, given by integration over $L\backslash U$ against
$\zeta^{-1}$. Indeed a function with compact support modulo $L$ is
supported in $LK_0$ for a compact open subgroup $K_0$, and
averaging over a sufficiently large $U_n$ proves both statements.

\step{2}{families of fibres}
By step~1 and the filtration of the induced representation by
the double cosets, these can be treated one at a time. The piece
attached to $P'wP$ is the space of sections with compact support
of the inducing line bundle on $(M_w\ltimes U_w)\backslash P$.
Over a chart of $M_w\backslash M$ choose a continuous section
$b\mapsto\gamma(b)$ of $M\to M_w\backslash M$. Writing the points of
the fibre over $b$ as $wv\gamma(b)$ identifies this fibre with
$U_w\backslash U$ and trivializes the line bundle on it. In these
coordinates $u\in U$ acts on the fibre by right translation by
$\gamma(b)u\gamma(b)^{-1}$, and the character becomes
$\zeta_b(v)=\zeta(\gamma(b)^{-1}v\gamma(b))$.

Let a section have vanishing integrals over the fibres above
$Z_w$, and let $b$ be a point of the projection of its support. On
a small compact open chart around $b$ the supports in the fibres
lie in the image of one compact open subgroup $K'$ of $U$. If
$b\notin Z_w$, enlarge $K'$ so that it contains an element of
$U_w$ on which $\zeta_b$ is not trivial. In both cases the average
of the section over $K'$ against $\zeta_b$ vanishes at $b$, by
step~1. It vanishes near $b$ as well. Indeed, after shrinking the
chart, the character $\zeta_{b'}$ on the compact set $K'$ and the
section, in the trivialization, do not depend on $b'$; and since
the normalizer of a compact open subgroup of $U$ is open in $M$,
we may also assume that $\gamma(b')^{-1}K'\gamma(b')=\gamma(b)^{-1}K'\gamma(b)$ on
the chart. So one compact open subgroup of $U$ kills the section
over the chart. The projection of the support is compact; cover it
by finitely many such charts. A subgroup $U_n$ which is large
enough then kills the section.

Conversely, a function with compact support on $Z_w$ extends to
the base, by means of a finite partition into open and closed
charts. On each chart choose $K'$ so small that $\zeta_b$ is
trivial on it for all $b$ in the chart, and multiply the extension
by the characteristic function of the image of $K'$ in
$U_w\backslash U$, divided by its volume. The integrals over the
fibres of the section obtained are the given function on $Z_w$.
This proves (1), and identifies the contribution of the double
coset.

\step{3}{the surviving layer}
Under the hypothesis of (2) the map
\[
 f\longmapsto\left[h\longmapsto
       \int_{U_j\backslash U}f(w\gamma_juh)\zeta(u)^{-1}\,du\right]
\]
identifies the contribution of the $j$th orbit with a space of
functions on $B_j\backslash H$. Substituting $u=bu'b^{-1}$ for
$b\in B_j$ gives its transformation law: the factor
$\chi\delta_{P'}^{1/2}$ of the induced representation evaluated at
$w\gamma_jb(w\gamma_j)^{-1}$, times the modulus of the change of variables on
$U_j\backslash U$. This is the compact induction of the statement.
If $B_j$ is a parabolic subgroup of a reductive group $H$, dividing
the character by $\delta_{B_j}^{1/2}$ writes it as a normalized
parabolic induction.
\end{proof}