\section{Quadratic theta lifting and all spherical Hecke moments}
\label{fl:sec:odd-theta}

To pass from the even orthogonal group $\operatorname{SO}_{2m}$ to
the symplectic group $\operatorname{Sp}_{2m}$ we use the theta
lift for the dual pair of equal rank. On parameters it adds a
trivial line to the standard representation. Two things have to be
proved about the lift of a generic cusp form: that it is nonzero,
and that its Whittaker coefficient $\Lambda$ satisfies the
expected Hecke equations at all unramified places, that is,
$\Lambda(T\Theta)=\eta(T)\Lambda(\Theta)$ for the spherical Hecke
operators $T$. We obtain both from one computation, that of the
twisted Jacquet module of the Weil representation along the maximal
unipotent subgroup of the symplectic group
(Lemma~\ref{fl:lem:theta-jacquet-module}): it is the space of
compactly supported Whittaker functions on the orthogonal group.
Globally this gives a formula for the Whittaker coefficient of the
lift as an integral of the Whittaker function of the cusp form
against a test function at our disposal
(Proposition~\ref{fl:prop:odd-theta-Whittaker}), from which
nonvanishing is read off. At unramified places the test function
attached to a Hecke operator applied to the spherical vector is
computed explicitly
(Proposition~\ref{rw:prop:theta-spherical}); it is governed by the
branching rule from $\operatorname{SO}_{2m+1}$ to
$\operatorname{SO}_{2m}$, and this gives these equations. A
generalized Hecke eigenfunction with a nonzero Whittaker
coefficient is then extracted by
Lemma~\xref{I}{fl:lem:hecke-moment-extraction}
(Corollary~\ref{fl:cor:odd-theta-moment}). The
Weil representation is written in a model that uses the quadratic
form and not only its bilinear form, so that the argument applies
in characteristic two.

\subsection{A model of the Weil representation in every characteristic}

Let $V=V_+\oplus V_+^\vee$ with $\dim V_+=m\ge2$,
\[
 Q(x,\xi)=\xi(x),\qquad
 B((x,\xi),(x',\xi'))=\xi(x')+\xi'(x),
\]
and put $G_0=\operatorname{SO}(V)$ and $G_1=\operatorname{Sp}_{2m}$.
Over a local field, $G_1$ acts by right multiplication on
$\operatorname{Mat}_{m,2m}=V_+\otimes F^{2m}$, hence on its
Schwartz space. Writing a matrix as $(\mathbf x,\mathbf y)$ with
two blocks of size $m\times m$, we transport this action by the
partial Fourier transform
\[
 (\mathcal F\Phi)(\mathbf x,\boldsymbol\xi)
   =\int\Phi(\mathbf x,\mathbf y)
     \psi(-\operatorname{tr}(\mathbf y^t\boldsymbol\xi))\,d\mathbf y
\]
to a representation $\mathscr W_\psi$ of $G_1$ on
$\mathcal S(V^m)$, where $z=(z_1,\ldots,z_m)$ and
$z_i=(x_i,\xi_i)\in V$. Let
$m(A)=\operatorname{diag}(A,A^{-t})$,
$n(C)=\left(\begin{smallmatrix}1&C\\0&1\end{smallmatrix}\right)$
with $C=C^t$, and
$w=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$.
Then
\begin{align}
 \mathscr W_\psi(m(A))\phi(z)&=|\det A|^m\phi(zA),
   \label{fl:eq:quadratic-levi}\\
 \mathscr W_\psi(n(C))\phi(z)&=
 \psi\left(\sum_iC_{ii}Q(z_i)+\sum_{i<j}C_{ij}B(z_i,z_j)\right)\phi(z),
   \label{fl:eq:quadratic-unipotent}\\
 \mathscr W_\psi(w)\phi(z)&=\int\phi(z')
                    \psi\left(\sum_i B(z_i,z_i')\right)\,dz'.
   \label{fl:eq:quadratic-fourier}
\end{align}
The first follows from the substitution $\mathbf y\mapsto
\mathbf yA^{-t}$ in the Fourier transform. For the second,
substitute $\mathbf y+\mathbf xC$ for $\mathbf y$; the phase that
remains is $\psi(\operatorname{tr}(C^t\mathbf x^t\boldsymbol\xi))$,
and its diagonal and off-diagonal terms are the two sums
displayed. The third is the Fourier transform in all variables.
Since these operators come from an action of $G_1$ by pullback,
they define a representation of $G_1$ itself; no metaplectic cover
occurs, and no quadratic phase is divided by two. The group
$O(V)$ acts by $r(h)\phi(z)=\phi(h^{-1}z)$, and the two actions
commute.

\emph{Whittaker data.} Fix a hyperbolic basis $e_i,f_i$ of $V$
with $e_i\in V_+$. Let $U_1$ be the maximal unipotent subgroup of
$G_1$ consisting of the $m(u)n(C)$ with $u$ upper unipotent, and
for $c\in F^\times$ let
\[
 \chi_{1,c}(m(u)n(C))=\psi\left(\sum_{i<m}u_{i,i+1}+cC_{mm}\right).
\]
On the orthogonal side let $U_0$ be the maximal unipotent subgroup
of $G_0$ fixing the flag of the $e_i$, and
\[
 \mathbf v_c=(e_1,\ldots,e_{m-1},e_m+cf_m)\in V^m .
\]
The span of $\mathbf v_c$ is stable under $U_0$, which gives a
homomorphism $\operatorname{frame}_c$ from $U_0$ onto the group
$U_{\operatorname{GL}_m}$ of upper unipotent matrices, defined by
$n\mathbf v_c=\mathbf v_c\operatorname{frame}_c(n)$. Let
$\psi_{\rm GL}(u)=\psi(\sum_iu_{i,i+1})$ and
$\chi_{0,c}=\psi_{\rm GL}\circ\operatorname{frame}_c$. This is a
nondegenerate character of $U_0$, with coefficients
$1,\ldots,1,c$ on the simple roots, and every nondegenerate
character of $U_0$ is conjugate under the torus to one of these:
the first $m-1$ coefficients can be normalized to $1$, and the
ratio of the last two is then $c$.

\begin{lemma}[The Whittaker quotient of the Weil representation]
\label{fl:lem:theta-jacquet-module}
Over a nonarchimedean local field of any characteristic, the map
$\phi\mapsto\mathcal J_\phi$,
\begin{equation}\label{fl:eq:quadratic-whittaker-transform}
 \mathcal J_\phi(h)=\int_{U_{\operatorname{GL}_m}}
                 \phi(h^{-1}\mathbf v_cu)\psi_{\rm GL}(u)^{-1}\,du,
\end{equation}
induces an isomorphism of representations of $G_0$
\[
 J_{U_1,\chi_{1,c}}(\mathscr W_\psi)
       \simeq\operatorname{c\!-\!Ind}_{U_0}^{G_0}\chi_{0,c}^{-1}.
\]
In particular every smooth function on $G_0$ that transforms by
$\chi_{0,c}^{-1}$ on the left under $U_0$ and has compact support
modulo $U_0$ is of the form $\mathcal J_\phi$.
\end{lemma}
\begin{proof}
We take coinvariants in two stages, first along the subgroup of
the $n(C)$ and then along that of the $m(u)$. By
\eqref{fl:eq:quadratic-unipotent} the first stage restricts
functions to a fibre of the map which sends $z$ to the values
$Q(z_i)$ and $B(z_i,z_j)$; the second, by
\eqref{fl:eq:quadratic-levi}, is an integral over
$U_{\operatorname{GL}_m}$ acting on the right. The points of the
fibre where $z$ has rank less than $m$ do not contribute, and the
others form one orbit of $G_0$.

\step{1}{the fibre} The coinvariants for the characters
$C\mapsto\psi(cC_{mm})$ of the group of symmetric matrices are the
restrictions of Schwartz functions to the closed set
\[
 \mathcal Q_c=\{z:Q(z_i)=0\ (i<m),\ Q(z_m)=c,
                                      \ B(z_i,z_j)=0\ (i<j)\};
\]
a function vanishing on $\mathcal Q_c$ is killed by averaging
against the character, and restriction is surjective. In
characteristic two the conditions on the $Q(z_i)$ are independent
of those on the $B(z_i,z_j)$. At a point where $z$ has
rank $m$ the differentials of the equations are independent: in
terms of $\beta_{ij}=B(z_i,dz_j)$, which are independent linear
forms, they are $\beta_{ii}$ and $\beta_{ij}+\beta_{ji}$. So this
part of $\mathcal Q_c$ is smooth in every characteristic.

\step{2}{frames of rank less than $m$} Stratify the rest of
$\mathcal Q_c$ by the first index $j$ such that $z_j$ lies in the
span of $z_1,\ldots,z_{j-1}$; here $j<m$, because $Q(z_m)=c\ne0$
while the span of the others is isotropic. Acting by an element of
$U_{\operatorname{GL}_m}$, which preserves $\psi_{\rm GL}$ up to
conjugation within the stratum, we may assume $z_j=0$. Then the
root subgroup $u_{j,j+1}(a)$ fixes $z$ while $\psi_{\rm GL}$ is
nontrivial on it, so the fibre of the coinvariants at $z$ is
zero. This is the statement at one point; it extends to the stratum as
follows. Fix a compact subset of the stratum. On a neighbourhood
of one of its points where a given minor of
$(z_1,\ldots,z_{j-1})$ is invertible, the coefficients which
express $z_j$ in terms of $z_1,\ldots,z_{j-1}$ are continuous
functions, and so is the element of $U_{\operatorname{GL}_m}$
which they define. Conjugating by it a fixed element of
$u_{j,j+1}$ on which the character is not trivial, we get an
element of the stabilizer of the frame which depends continuously
on the frame and on which the character is not trivial. Cover the
compact subset by finitely many such neighbourhoods with compact
closure. All the elements obtained lie in one compact open
subgroup of $U_{\operatorname{GL}_m}$, and the average over this
subgroup against the character kills every function supported in
the compact subset: each orbit of the subgroup which meets the
support has in its stabilizer an element of the subgroup on which
the character is not trivial. Since coinvariants are exact, the
strata can be removed one after the other, in the order given by
the closed conditions that the first $j$ columns are dependent.

\step{3}{frames of rank $m$} These form one orbit of $G_0$. Indeed
the first $m-1$ vectors span an isotropic subspace and can be
moved to $e_1,\ldots,e_{m-1}$; then $z_m$ is, modulo their span,
of the form $xe_m+yf_m$ with $xy=c$, the torus makes $x=1$, and
the unipotent radical of the stabilizer of the span removes the
other coordinates. This uses the full orthogonal group; but the
reflection in $e_m-cf_m$ fixes $\mathbf v_c$ and has determinant
$-1$, or Dickson invariant one in characteristic two, so $G_0$ is
transitive as well. The stabilizer of $\mathbf v_c$ is
$S_c=\ker\operatorname{frame}_c$. Writing an element of $U_0$ as
\[
 n=\operatorname{diag}(u,u^{-t})
                 \begin{pmatrix}1&T\\0&1\end{pmatrix},
 \qquad T^t=-T,\quad T_{ii}=0,
\]
one finds $\operatorname{frame}_c(n)=u(1+c(Te_m)e_m^t)$, so that
$S_c$ is defined by $u=1-c(Te_m)e_m^t$; it is isomorphic to an
affine space, in every characteristic. Since the orbit and the
stabilizer are smooth, the orbit is $S_c\backslash G_0$.

\step{4}{the integral} On functions on $S_c\backslash G_0$ the
remaining coinvariants, for $U_{\operatorname{GL}_m}=U_0/S_c$ and
the character $\psi_{\rm GL}$, are given by the integral
\eqref{fl:eq:quadratic-whittaker-transform}, which converges
because $u\mapsto h^{-1}\mathbf v_cu$ is a closed embedding. Its
image consists of all functions with compact support modulo
$U_0$, and its kernel is spanned by the differences
$\phi(\cdot u)-\psi_{\rm GL}(u)\phi$, as one sees with local
sections of $G_0\to U_0\backslash G_0$. A Schwartz function whose
support meets the strata of step~2 is, by that step, congruent
modulo the relations to one with compact support in the orbit of
frames of rank $m$, and the integral vanishes on the relations.
Finally $\mathcal J_{r(h_0)\phi}(h)=\mathcal J_\phi(hh_0)$.
\end{proof}

\subsection{The spherical vector}

Let $v$ be a place where the residue characteristic is arbitrary,
$\psi$ has conductor $\mathcal O_v$ and $c\in\mathcal O_v^\times$.
Let $K_0$ and $K_1$ be the hyperspecial subgroups of $G_0$ and
$G_1$ defined by the lattice spanned by the basis, and
$\phi^0=1_{V(\mathcal O_v)^m}$. By
\eqref{fl:eq:quadratic-levi}--\eqref{fl:eq:quadratic-fourier} it is
fixed by $K_1$, and it is fixed by $K_0$. For $J=G_0,G_1$ and a
dominant weight $\gamma$ of the dual group let $T^J_\gamma$ be the
Hecke operator corresponding under the normalized Satake
isomorphism to the character of the irreducible representation of
highest weight $\gamma$, and $t_\gamma$ the corresponding element
of the torus:
\[
 t_\lambda=\operatorname{diag}(\varpi^{\lambda_i},\varpi^{-\lambda_i})
      \in G_0,\qquad
 t_\mu=m(\operatorname{diag}(\varpi^{\mu_i}))\in G_1 .
\]

\begin{lemma}[Whittaker functionals and Hecke operators]
\label{rw:lem:whittaker-hecke}
Let $J$ be $G_0$ or $G_1$, let $\Lambda$ be a Whittaker functional
on a smooth representation of $J$, for a character of conductor
$\mathcal O_v$ on each simple root, and let $v_0$ be a vector fixed
by the hyperspecial subgroup. Then
\begin{equation}\label{fl:eq:universal-whittaker-hecke}
 \Lambda(T_\gamma^Jv_0)=\delta_{B_J}(t_\gamma)^{-1/2}
                                             \Lambda(t_\gamma v_0).
\end{equation}
\end{lemma}
\begin{proof}
No irreducibility is assumed, so this is a statement about the
module of compactly supported Whittaker functions invariant under
the hyperspecial subgroup, of which the function
$g\mapsto\Lambda(gv_0)$ is a linear functional. Precisely, pair
compactly supported functions which transform by $\psi^{-1}$ under
the maximal unipotent subgroup $U$ with Whittaker functions, which
transform by $\psi$, by integration over $U\backslash J$; this
pairing is bilinear, with no complex conjugation. The compact
module has a basis $b_\gamma$ indexed by the dominant $\gamma$:
$b_\gamma$ is supported on $Ut_\gamma K$ and normalized by
$b_\gamma(t_\gamma)=\delta_{B_J}(t_\gamma)$, so that
$\langle b_\gamma,W\rangle=W(t_\gamma)$ for every Whittaker
function $W$ invariant under $K$. For a Hecke operator $T$ let
$T^*(g)=T(g^{-1})$; then $\langle T^*b,W\rangle=\langle b,TW\rangle$.
With $W(g)=\Lambda(gv_0)$ the two sides of
\eqref{fl:eq:universal-whittaker-hecke} are
$\langle b_0,T_\gamma^JW\rangle
=\langle(T_\gamma^J)^*b_0,W\rangle$ and
$\delta_{B_J}(t_\gamma)^{-1/2}\langle b_\gamma,W\rangle$. So the
lemma follows from the identity
\[
 (T_\gamma^J)^*b_0=\delta_{B_J}(t_\gamma)^{-1/2}b_\gamma
\]
in the compact module; the transpose matters when
$-w_0\gamma\ne\gamma$. To verify the identity, pair both sides
with the Whittaker functions $W_t$ of the unramified principal
series, normalized by $W_t(1)=1$, for $t$ in a Zariski-dense set of
unramified parameters. By the formula of Casselman and
Shalika \cite[Theorem 5.4]{CS80},
$W_t(t_\gamma)=\delta_{B_J}(t_\gamma)^{1/2}\chi_\gamma(t)$, where
$\chi_\gamma$ is the character of highest weight $\gamma$; and
$T_\gamma^JW_t=\chi_\gamma(t)W_t$. So both sides pair with $W_t$ to
$\chi_\gamma(t)$. An element of the compact module is a finite
combination of the $b_\lambda$, and its pairing with $W_t$ is the
corresponding combination of the functions
$\delta_{B_J}(t_\lambda)^{1/2}\chi_\lambda(t)$ of $t$. These are
regular functions which are linearly independent, also on a
Zariski-dense set; so an element which pairs to zero with all these
$W_t$ is zero.
\end{proof}

A weight $\lambda=(\lambda_1,\ldots,\lambda_m)$ of
$\operatorname{SO}_{2m}$ is said to interlace a dominant weight
$\mu$ of $\operatorname{SO}_{2m+1}$, written $\lambda\prec\mu$, if
\begin{equation}\label{fl:eq:quadratic-interlacing}
 \mu_i\ge\lambda_i\ge\mu_{i+1}\ (i<m),\qquad
 \mu_m\ge|\lambda_m|.
\end{equation}
Such $\lambda$ are dominant. Put
$d_0(\lambda)=\sum_i(m-i)\lambda_i$ and
$d_1(\mu)=\sum_i(m-i+1)\mu_i$, so that
$\delta_{B_0}(t_\lambda)^{1/2}=q_v^{-d_0(\lambda)}$ and
$\delta_{B_1}(t_\mu)^{1/2}=q_v^{-d_1(\mu)}$.

\begin{proposition}[Hecke operators on the spherical vector]
\label{rw:prop:theta-spherical}
For dominant $\mu$ and $\lambda$,
\begin{equation}\label{fl:eq:quadratic-lattice-value}
 \mathcal J_{\mathscr W_\psi(t_\mu)\phi^0}(t_\lambda)
 =\begin{cases}q_v^{-d_0(\lambda)-d_1(\mu)},&\lambda\prec\mu,\\
                 0,&\text{otherwise}.
   \end{cases}
\end{equation}
Consequently, if $W_\lambda^0$ denotes the function in
$\operatorname{c\!-\!Ind}_{U_0}^{G_0}\chi_{0,c}^{-1}$ supported on
$U_0t_\lambda K_0$ with value $q_v^{-d_0(\lambda)}$ at
$t_\lambda$, then
\[
 \mathcal J_{\mathscr W_\psi(T_\mu^{G_1})\phi^0}
                  =\sum_{\lambda\prec\mu}W_\lambda^0,
\]
and, if $\Lambda$ is a Whittaker functional for $\chi_{0,c}$ on a
smooth representation of $G_0$ and $v_0$ is a vector fixed by
$K_0$,
\[
 \int_{U_0\backslash G_0}\Lambda(hv_0)\,
 \mathcal J_{\mathscr W_\psi(T_\mu^{G_1})\phi^0}(h)\,dh
 =\sum_{\lambda\prec\mu}\Lambda(T_\lambda^{G_0}v_0).
\]
The operator $\sum_{\lambda\prec\mu}T_\lambda^{G_0}$ corresponds
under the Satake isomorphism to the restriction of the character
of highest weight $\mu$ of $\operatorname{SO}_{2m+1}$ along
$s\mapsto\operatorname{diag}(s,1)$.
\end{proposition}
\begin{proof}
By \eqref{fl:eq:quadratic-levi},
$\mathscr W_\psi(t_\mu)\phi^0$ is $q_v^{-m\sum\mu_j}$ times the
characteristic function of the $z$ with
$z_j\in\varpi^{-\mu_j}V(\mathcal O_v)$. In the integral
\eqref{fl:eq:quadratic-whittaker-transform} at $h=t_\lambda$ the
variable is the upper unipotent matrix $u$, and the $j$th column
of $t_\lambda^{-1}\mathbf v_cu$ has coordinates
$\varpi^{-\lambda_i}u_{ij}$ on $e_i$, $i\le j$, with in addition
the coordinate $c\varpi^{\lambda_m}$ on $f_m$ for $j=m$. The
integrand is therefore nonzero only if $\lambda_j\le\mu_j$ for
all $j$, from the diagonal entries $u_{jj}=1$, and
$\lambda_m\ge-\mu_m$, from the coordinate on $f_m$; and then the
domain of integration is
$u_{ij}\in\varpi^{\lambda_i-\mu_j}\mathcal O_v$ for $i<j$. The
integral over the variable $u_{i,i+1}$
against $\psi$ is the volume $q_v^{\mu_{i+1}-\lambda_i}$ of its
domain if this domain is contained in $\mathcal O_v$, that is if
$\lambda_i\ge\mu_{i+1}$, and zero otherwise; the other variables
contribute the volumes $q_v^{\mu_j-\lambda_i}$. This gives the
conditions \eqref{fl:eq:quadratic-interlacing}, and the product of
the volumes and of the factor $q_v^{-m\sum\mu_j}$ is
$q_v^{-d_0(\lambda)-d_1(\mu)}$.

The function $\mathcal J_{\mathscr W_\psi(t_\mu)\phi^0}$ is
invariant under $K_0$ on the right, so by the Iwasawa
decomposition it is determined by its values at the $t_\lambda$,
which vanish unless $\lambda$ is dominant because of the
character $\chi_{0,c}$. Applying
Lemma~\ref{rw:lem:whittaker-hecke} for $G_1$ to the functionals
$\phi\mapsto\mathcal J_\phi(h)$, which are Whittaker functionals
on $\mathscr W_\psi$ by
Lemma~\ref{fl:lem:theta-jacquet-module}, we get
$\mathcal J_{\mathscr W_\psi(T_\mu^{G_1})\phi^0}
=q_v^{d_1(\mu)}\mathcal J_{\mathscr W_\psi(t_\mu)\phi^0}$, which is
the sum of the $W_\lambda^0$. The integral of $\Lambda(hv_0)$
against $W_\lambda^0$ is, by the Iwasawa decomposition,
$\delta_{B_0}(t_\lambda)^{-1/2}\Lambda(t_\lambda v_0)$, and this is
$\Lambda(T_\lambda^{G_0}v_0)$ by
Lemma~\ref{rw:lem:whittaker-hecke} for $G_0$. The last assertion
is the branching rule
\[
 \operatorname{tr}_{B_m,\mu}(\operatorname{diag}(s,1))
               =\sum_{\lambda\prec\mu}\operatorname{tr}_{D_m,\lambda}(s)
 \quad\text{\cite[\S4.1, (4.4)]{Molev06}},
\]
in which each $\lambda$ occurs with multiplicity one.
\end{proof}

\begin{remark}
For $m=2$ and $\mu=(1,0)$ the interlacing weights are
$\lambda=(0,0)$ and $(1,0)$, with values $q_v^{-2}$ and
$q_v^{-3}$ in \eqref{fl:eq:quadratic-lattice-value}; multiplied by
$q_v^{d_1(\mu)}=q_v^2$ they are $1$ and $q_v^{-1}$, the values of
$W^0_\lambda$. The branching rule is
$\operatorname{Std}_{\operatorname{SO}_5}|_{\operatorname{SO}_4}
=\operatorname{Std}_{\operatorname{SO}_4}\oplus1$: the lift adds a
trivial line to the standard parameter.
\end{remark}

\subsection{The global lift}

Let $f$ be a cusp form on $G_0(\mathbb A)$ in a generic cuspidal
representation, with Satake parameters $s_v^0$ at the unramified
places, and choose $c\in F^\times$ such that its Whittaker
coefficient $\operatorname{Wh}_{0,c}f$ for $\chi_{0,c}$ is not
zero. For a Schwartz function $\phi$ on $V(\mathbb A)^m$ define
\[
 \theta(h,g;\phi)=\sum_{z\in V(F)^m}(r(h)\mathscr W_\psi(g)\phi)(z),
 \qquad
 \Theta(f,\phi)(g)=\int_{[G_0]}f(h)\theta(h,g;\phi)\,dh.
\]
The theta function is invariant under $G_0(F)\times G_1(F)$: for
the three kinds of generators of $G_1(F)$ this follows from the
product formula, from $\psi(F)=1$, and from the Poisson summation
formula \cite[\S\S29--30,40--41]{Weil64}. The support of a
cusp form of given level is compact modulo the centre
\cite[Chapter I, Proposition~5.3]{Morris1982}, and here the centre
is finite, so that its group of adelic points is compact. Choose a
compact subset of $G_0(\mathbb A)$ which maps onto this support.
For $h$ in this subset and $g$ in a compact set, the supports of
the functions $r(h)\mathscr W_\psi(g)\phi$ lie in one compact
subset of $V(\mathbb A)^m$, which contains only finitely many
points of $V(F)^m$. So the integral defining the lift, its
integrals over compact unipotent quotients and the unfolding below
are finite sums. With Schwartz functions taking algebraic values
they make sense over $k$ as well, after
Lemma~\ref{fl:lem:prelim-coeff}.

\begin{proposition}[Whittaker coefficient of the theta lift]
\label{fl:prop:odd-theta-Whittaker}
The Whittaker coefficient of the lift for $\chi_{1,c}$ is
\begin{equation}\label{fl:eq:quadratic-global-unfolding}
 \operatorname{Wh}_{1,c}\Theta(f,\phi)(g)
   =\int_{U_0(\mathbb A)\backslash G_0(\mathbb A)}
       \operatorname{Wh}_{0,c}f(h)\,
                 \mathcal J_{\mathscr W_\psi(g)\phi}(h)\,dh.
\end{equation}
Moreover there is $\phi$, equal to $\phi_v^0$ outside finitely many
places, such that $v=\Theta(f,\phi)$ and the functional
$\Lambda=\operatorname{Wh}_{1,c}(-)(1)$ satisfy
\[
 \Lambda(Tv)=\eta(T)\Lambda(v),\qquad \Lambda(v)\ne0
\]
for every $T$ in the tensor product of the spherical Hecke algebras
at the places outside a finite set. This set contains the places
$v$ where $f$ is not spherical, where the conductor of $\psi$ is
not $\mathcal O_v$, or where $c$ is not a unit; and $\eta$ is the
character whose
standard parameter at $v$ is $\operatorname{Std}(s_v^0)\oplus1$.
\end{proposition}
\begin{proof}
\step{1}{unfolding} The global computation is parallel to the
proof of Lemma~\ref{fl:lem:theta-jacquet-module}. Integrating
against the character of the $n(C)$ restricts the sum over
$V(F)^m$ to $\mathcal Q_c(F)$. In an orbit of
$U_{\operatorname{GL}_m}(F)$ consisting of frames of rank less
than $m$, choose the frame in which the first dependent column is
zero, $z_j=0$, as in Lemma~\ref{fl:lem:theta-jacquet-module}.
Unfolding the sum over this orbit leaves an integral over the
quotient by the stabilizer of the frame. The stabilizer contains
$u_{j,j+1}$, on which the character is not trivial, so the
integral over $u_{j,j+1}(F)\backslash u_{j,j+1}(\mathbb A)$
vanishes. The points of rank
$m$ form the orbit $S_c(F)\backslash G_0(F)$, and unfolding the
sum against the integral over $[G_0]$, then combining the integral
over $[S_c]$ with the integral over $[U_{\operatorname{GL}_m}]$
into an integral of $f$ over $[U_0]$, gives
\eqref{fl:eq:quadratic-global-unfolding}.

\step{2}{Hecke operators} Let $v$ be a place which is not in the
finite set of the statement, and let
$\phi=\phi^0_v\otimes\phi^v$. We apply
Proposition~\ref{rw:prop:theta-spherical} to the representation of
$G_0(F_v)$ generated by $f$ and to its Whittaker functional
$f'\mapsto\operatorname{Wh}_{0,c}f'(h^v)$. With
\eqref{fl:eq:quadratic-global-unfolding} we get
\[
 \Lambda\bigl(T_\mu^{G_1}\Theta(f,\phi)\bigr)
 =\sum_{\lambda\prec\mu}\Lambda\bigl(\Theta(T_\lambda^{G_0}f,\phi)\bigr)
 =\eta(T^{G_1}_\mu)\Lambda\bigl(\Theta(f,\phi)\bigr),
\]
since $f$ is an eigenvector of the operators $T_\lambda^{G_0}$ and
the sum of the eigenvalues is the character of highest weight
$\mu$ at $\operatorname{diag}(s_v^0,1)$. The components at the
other places were arbitrary, so the argument can be repeated at
other unramified places, and products of operators at one place
are linear combinations of the $T^{G_1}_\mu$.

\step{3}{nonvanishing} Translate $f$ so that
$\operatorname{Wh}_{0,c}f(1)\ne0$, and let $K=\prod_vK_v$ be a
compact open subgroup of $G_0(\mathbb A)$ fixing $f$, hyperspecial
at almost all places, such that $\chi_{0,c}$ is trivial on
$U_0(\mathbb A)\cap K$. By
Lemma~\ref{fl:lem:theta-jacquet-module} at the remaining places,
and with $\phi_v^0$ at the others, there is $\phi$ such that
$\mathcal J_\phi$ is the function supported on $U_0(\mathbb A)K$
which is $\chi_{0,c}^{-1}$ on $U_0(\mathbb A)$. Then the right
side of \eqref{fl:eq:quadratic-global-unfolding} at $g=1$ is
$\operatorname{Wh}_{0,c}f(1)$ times the volume of
$(U_0(\mathbb A)\cap K)\backslash K$.
\end{proof}

The proposition gives the values of one functional $\Lambda$ on
the vectors $Tv$. A generalized eigenfunction is obtained from it
as follows. The functional
$\Lambda$ at the fixed level is a pairing with a function of
compact support. The identities $\Lambda(Tv)=\eta(T)\Lambda(v)$
with $\Lambda(v)\ne0$ show that this pairing does not vanish after
localization at the maximal ideal of the transposed character: an
operator $T$ that is invertible there cannot kill it, because
$\Lambda(Tv)=\eta(T)\Lambda(v)\ne0$.
Lemma~\xref{I}{fl:lem:hecke-moment-extraction} then gives a finite
quotient of the Hecke module, killed by a power of that maximal
ideal, in which the pairing is nonzero, and a functional on that
quotient which is nonzero on it is the required function.

\begin{corollary}[A generic function with the lifted Hecke character]
\label{fl:cor:odd-theta-moment}
There is an automorphic function of finite level on
$G_1(\mathbb A)$ on which the spherical Hecke algebras at the
unramified places act through $\eta$, as a generalized character,
and whose principal Whittaker coefficient is nonzero.
\end{corollary}