\section{Even orthogonal descent and its final Fourier coefficient}
\label{fl:sec:odd-symplectic}

For an even orthogonal target the descent depends on a vector $v$
in the last row of the unipotent radical, and it is the stabilizer
of a nondegenerate $v$ that has the right type. The
Siegel--Whittaker coefficient of the residue only gives a
coefficient in which the corresponding frequency is zero. The
problem of this section is to pass from frequency zero to a nonzero
frequency $c\in F^\times$. In odd characteristic this is done by a
Fourier--Jacobi argument (Lemma~\ref{fl:lem:even-jacobi-nonzero});
the argument in characteristic two is different and is given in
Section~\ref{fl:sec:char2-d}. In both cases the vector is found
over $F$ without control of its norm, and the vanishing of the
local Jacquet module in the nonsplit case
(Proposition~\ref{fl:prop:uniform-b-jacquet}) then shows that the
stabilizer is split. The result is
Proposition~\ref{fl:prop:even-generic-nonzero}.

The argument has the following shape. A family of Fourier
coefficients $\Lambda_c$, $c\in F$, is defined on a unipotent
subgroup $U_{\rm corr}$ of the source group. For $c\ne0$ it is
equivalent to the Whittaker coefficient of the descent with
endpoint of norm $c/2$
(Lemma~\ref{fl:lem:even-period-to-generic}); for $c=0$ it is
implied by the Siegel--Whittaker coefficient
(Lemma~\ref{fl:lem:descent-vanishing-and-period}). The group
$U_{\rm rad}$ of the parabolic subgroup with Levi subgroup
$\operatorname{GL}_2^{n-1}\times\operatorname{SO}_5$ is normalized
by a Jacobi group which preserves $\lambda_{\rm rad}$. After
integration over $(U_{\rm rad},\psi\lambda_{\rm rad})$ one has a
function on the Jacobi group, and the $\Lambda_c$ are its Fourier
coefficients. A function on the Jacobi group with a nonzero
coefficient at $c=0$
has a nonzero coefficient at some $c\ne0$.

\subsection{The residual representation attached to a parameter}
\label{fl:sec:even-residual}

Let $\sigma$ take values in $\operatorname{GSp}_{2n}(k)$ or
$\operatorname{GSO}_{2n}(k)$, with $n\ge2$, and suppose it is
continuous, finitely ramified, semisimple, elliptic modulo scalars
and strongly pure, with multiplier $\chi$ of finite order and
$\omega_\sigma=\operatorname{CFT}(\chi)$. The descent goes
\[
 \operatorname{GSpin}_{4n}\longrightarrow\operatorname{GSpin}_{2n+1},
 \qquad
 \operatorname{GSpin}_{4n+1}\longrightarrow\operatorname{GSpin}_{2n}
\]
in the two cases. By Proposition~\ref{fl:prop:prelim-block},
$\operatorname{Std}\sigma=\bigoplus_i\varrho_i$ with distinct
irreducible $\varrho_i$ of dimensions $m_i$, $\sum m_i=2n$, each
carrying a form of the same kind with multiplier $\chi$. In the
alternating case the $m_i$ are even and
$\det\varrho_i=\chi^{m_i/2}$; in the symmetric case odd dimensions,
including one, are allowed, and $(\det\varrho_i)^2=\chi^{m_i}$. By
L.~Lafforgue's theorem, and class field theory in dimension one,
there are unitary cuspidal representations $\Pi_i$ with parameters
$\varrho_i$; they satisfy
$\Pi_i^\vee\otimes\omega_\sigma\simeq\Pi_i$ and are tempered at
every place \cite[Theorem VI.10(i)]{LLa02}. The hypotheses of
Proposition~\ref{fl:lem:residual-analytic} hold, by the remark after
Lemma~\ref{fl:lem:prelim-pole}, and we obtain an irreducible
residual representation $\pi$ of the source group, on which the
connected centre acts through $\omega_\sigma$, with nonzero
Siegel--Whittaker coefficient.

At a place where everything is unramified, let the standard
eigenvalues of $\sigma$ be $a_j$ and $c/a_j$, $1\le j\le n$, with
$c=\chi(\operatorname{Frob}_v)$. The Satake parameter of $\pi_v$ on
the Siegel Levi subgroup $\operatorname{GL}_{2n}$ consists of all
these numbers shifted by $q_v^{-1/2}$. Replacing $c/a_j$ by its
partner under the Weyl group of the source, we can write it as the
pairs $(a_jq_v^{-1/2},a_jq_v^{1/2})$, so that $\pi_v$ is a
spherical subquotient of the representation
$I_{P'}(\boldsymbol\mu;\omega_\sigma)$ of
Proposition~\ref{fl:prop:uniform-b-jacquet} with
$\mu_j(\varpi)=a_j$. In type $B_{2n}$ the Weyl group contains all
the sign changes needed. In type $D_{2n}$ it contains only an even
number of them; one uses $P'_+$ if $n$ is even and $P'_-$ if $n$ is
odd, which accounts for the remaining sign. Since only spherical
subquotients are involved, this holds also when eigenvalues
coincide.

\subsection{A nonzero frequency}
\label{fl:sec:even-jacobi}

For the rest of this section assume $p\ne2$. Let $\mathsf H$ be the
Heisenberg group with coordinates $(x,y,z)$ and multiplication
\[
 (x,y,z)(x',y',z')=(x+x',y+y',z+z'+(xy'-yx')/2),
\]
on which $\operatorname{SL}_2$ acts; its upper unipotent element
$n(b)$ acts by $(x,y,z)\mapsto(x,y-bx,z)$. We write $X,Y,Z$ for the
three coordinate subgroups. The subgroup generated by $Y$, $Z$ and
the $n(b)$ is abelian, and we consider its characters
$\psi(z+cb)$, $c\in F$.

\begin{lemma}[A nonzero frequency]
\label{fl:lem:even-jacobi-nonzero}
Let $\mathcal V$ be a space of automorphic functions on the Jacobi
group $\mathsf H\rtimes\operatorname{SL}_2$, stable under right
translation. If the Fourier coefficient for the character
$\psi(z)$, that is $c=0$, is not identically zero on $\mathcal V$,
then the coefficient for $\psi(z+cb)$ is not identically zero for
some $c\in F^\times$.
\end{lemma}
\begin{proof}
We pair with a theta series. This turns the coefficients into the
Fourier coefficients, along the upper unipotent subgroup, of a
genuine function on the metaplectic group, and a genuine function
cannot be invariant under both unipotent subgroups of
$\operatorname{SL}_2$.

\step{1}{unfolding} Let $\omega_\psi$ be the Schr\"odinger
representation,
\[
 \omega_\psi(x,y,z)\phi(t)=\psi(z+yt+xy/2)\phi(t+x),\qquad
 \omega_\psi(n(b))\phi(t)=\psi(bt^2/2)\phi(t).
\]
The functional $\phi\mapsto\sum_{\xi\in F}\phi(\xi)$ is invariant
under the rational points of the Jacobi group in its metaplectic
cover \cite[\S\S29,40--41, Proposition 5 and Theorem 6]{Weil64}.
For $f\in\mathcal V$ and a Schwartz function $\phi$, integrate $f$
against the theta series of $\omega_{\psi^{-1}}$ over
$[\mathsf H]$; the result is a genuine automorphic function $\Phi$
on the metaplectic group. Let $\mathcal F_cf$ be the coefficient
of $f$ for $\psi(z+cb)$. We claim that the Fourier coefficient of
$\Phi$ for the character $\psi(cb)$ of the upper unipotent
subgroup is
\[
 \int_{\mathbb A}\mathcal F_cf(X(x)g)
             (\omega_{\psi^{-1}}(g)\phi)(x)\,dx.
\]
Indeed, write the Heisenberg variable in the order $YZX$. The term
of the theta series with index $\xi$ has the phase
$\psi(-z-y\xi-b(\xi+x)^2/2)$; translating the variable by the
rational element $X(\xi)$ removes $\xi$ and combines the sum with
the integral over $[X]$ into an integral over $X(\mathbb A)$, and
the relation
\[
 X(x)n(b)=Y(bx)Z(bx^2/2)n(b)X(x)
\]
cancels the quadratic phase. All the quotients are compact, so
the manipulations are justified. Since the functions
$\omega_{\psi^{-1}}(g)\phi$ are arbitrary Schwartz functions of
$x$, the hypothesis $\mathcal F_0\ne0$ implies that $\Phi$ is not
identically zero for suitable $f$ and $\phi$, and it suffices to
show that some Fourier coefficient of some such $\Phi$ with
$c\ne0$ is nonzero.

\step{2}{a genuine function has a nonzero frequency} Suppose all
the coefficients with $c\ne0$ vanished, for all $f$ and $\phi$.
Then every $\Phi$ would be invariant under the upper unipotent
subgroup, and, conjugating by the rational Weyl element, under the
lower one. But the canonical lifts of these two subgroups to the
metaplectic group generate the kernel of the covering at a place
$v$: with $w(t)=n_+(t)n_-(-t^{-1})n_+(t)$ and $h(t)=w(t)w(-1)$ one
has $h(s)h(t)=(s,t)_vh(st)$
\cite[Theorem 5.3, Corollary 5.5(2), Lemma 5.1(iii), Theorem A.4]{Rao1993},
and the Hilbert symbol of a uniformizer and a nonsquare unit is
$-1$, because norms from the unramified quadratic extension have
even valuation. So $\Phi$ would be invariant under the nontrivial
element of the kernel, on which a genuine function changes sign,
and $\Phi$ would vanish.
\end{proof}

The Jacobi group occurs in the source group as follows. Consider a
quadratic space of dimension five with basis
$(e_1,e_2,u,f_2,f_1)$, where $B(e_i,f_j)=\delta_{ij}$ and
$B(u,u)=1$, and let
\[
 x_{e_i}(t)=1+tE_{i,0}-tE_{0,-i}-\tfrac12t^2E_{i,-i}
\]
be its short root subgroups, the index $0$ referring to $u$ and
$-i$ to $f_i$.

\begin{lemma}[The Jacobi group in $\operatorname{SO}_5$]
\label{rw:lem:jacobi-embedding}
The assignments
\[
 X(x)\mapsto x_{e_2}(-2x),\quad
 Y(y)\mapsto x_{e_1}(y),\quad
 Z(z)\mapsto x_{e_1+e_2}(-2z),\quad
 n(b)\mapsto x_{e_1-e_2}(b/2)
\]
extend to an embedding of the Jacobi group in
$\operatorname{SO}_5$, in which $\operatorname{SL}_2$ is the
subgroup $\operatorname{SL}(\langle e_1,e_2\rangle)$ of the Levi
subgroup $\operatorname{GL}_2$. It lifts to
$\operatorname{GSpin}_5$.
\end{lemma}
\begin{proof}
One computes
\[
 X(x)Y(y)Z(z-xy/2)=
 \begin{pmatrix}
 1&0&y&xy-2z&-y^2/2\\
 0&1&-2x&-2x^2&xy+2z\\
 0&0&1&2x&-y\\
 0&0&0&1&0\\
 0&0&0&0&1
 \end{pmatrix}.
\]
In the product of two such matrices the entry $(1,4)$ is
$xy-2z+x'y'-2z'+2yx'$, which is the entry attached to
$(x+x',y+y',z+z'+(xy'-yx')/2)$; the other entries are checked in
the same way. Conjugation by $1+(b/2)(E_{1,2}-E_{-2,-1})$ replaces
$y$ by $y-bx$, and conjugation by $1+2a(E_{2,1}-E_{-1,-2})$
replaces $x$ by $x-ay$; both fix $z$. The group
$\operatorname{SL}_2$ and the unipotent elements lift canonically
to the spin group.
\end{proof}

Now let $m=2n$ and consider the source group of type $B_m$, with
the notation of Section~\ref{fl:sec:orthogonal-descent}. Let
$U_{\rm rad}$ be the unipotent radical of the parabolic subgroup
with Levi subgroup
$\operatorname{GL}_2^{n-1}\times\operatorname{SO}_5$, the last
factor acting on the span of
$e_{m-1},e_m,u,f_m,f_{m-1}$, with the character
\[
 \lambda_{\rm rad}(g)=\sum_{i=1}^{m-2}g_{i,i+2},
\]
the sum of the traces of the $n-1$ blocks of size two that
describe $U_{\rm rad}$ modulo its commutator subgroup. An element
$(g,\ldots,g,h)$ of the Levi subgroup preserves this character if
$hj=jg$, where $j$ is the inclusion of
$\langle e_{m-1},e_m\rangle$ in the five-dimensional space. So the
Jacobi group of Lemma~\ref{rw:lem:jacobi-embedding}, with
$\operatorname{SL}_2$ embedded diagonally in all the blocks,
normalizes $U_{\rm rad}$ and preserves $\lambda_{\rm rad}$, and
acts on the coefficients of automorphic functions along
$(U_{\rm rad},\psi\lambda_{\rm rad})$.

\subsection{The coefficients \texorpdfstring{$\Lambda_c$}{Lambda-c}}

Let $U$ be the standard maximal unipotent subgroup of the source
group and $U_{\rm last}^0=\{g\in U:g_{m,0}=0\}$. Let
$U_{\rm corr}\subset U_{\rm last}^0$ be the subgroup on which the
entries $g_{1,2},g_{3,4},\ldots,g_{m-1,m}$ have a common value,
denoted $a$. In terms of the preceding paragraph,
\[
 U_{\rm corr}=U_{\rm rad}\rtimes(YZn),\qquad a=b/2 ,
\]
where $YZn$ is the abelian subgroup of the Jacobi group. For
$c\in F$ define on $U_{\rm corr}$
\begin{equation}\label{fl:eq:even-corrected-period}
 \Lambda_c(g)=\sum_{i=1}^{m-2}g_{i,i+2}
       -a\sum_{j=1}^{n-1}g_{2j,2j+1}
       +g_{m-1,-m}+\frac c2a.
\end{equation}
The first sum is not $\lambda_{\rm rad}$ of the component of $g$
in $U_{\rm rad}$: a block $C_i$ of $g$ is the corresponding block
of that component times
$\left(\begin{smallmatrix}1&a\\0&1\end{smallmatrix}\right)$, and
the trace of $C_i\left(\begin{smallmatrix}1&a\\0&1
\end{smallmatrix}\right)^{-1}$ is the trace of $C_i$ minus
$ag_{2i,2i+1}$. This is the second sum. The entry $g_{m-1,-m}$ is
$-2z$. Thus
\[
 \Lambda_c=\lambda_{\rm rad}-2z+cb/4 ,
\]
and $\psi\Lambda_c$ is the character $\lambda_{\rm rad}$ on
$U_{\rm rad}$ times the character of $YZn$ which, for the additive
character $\psi(-2\,\cdot)$, is the one with frequency $-c/8$ in
Lemma~\ref{fl:lem:even-jacobi-nonzero}.

The rest of this section passes between the coefficients along
five families of subgroups of $U_{\rm last}^0$.
Table~\ref{rw:tab:even-descent-route} lists these subgroups with
their characters, and then the order in which the coefficients are
reached in the proof of
Proposition~\ref{fl:prop:even-generic-nonzero}. The groups
$U_{{\rm Wh},c}$ and $U_{\rm alt}$ are defined in
Lemma~\ref{fl:lem:even-period-to-generic} and its proof, the groups
$U_{{\rm chain},r}$ in the proof of
Lemma~\ref{fl:lem:descent-vanishing-and-period}.
Figure~\ref{rw:fig:even-groups} shows three of them for $n=3$.

\begin{table}[tp]
\centering
\small
\renewcommand{\arraystretch}{1.3}
\setlength{\tabcolsep}{5pt}
\begin{tabular}{@{}lll@{}}
\toprule
subgroup & defined in $U_{\rm last}^0$ by & character, composed with $\psi$\\
\midrule
$U_{{\rm chain},m-1}$ & no condition &
 $\lambda_{m-1}=\sum_{i<m}g_{i,i+1}$\\
$U_{{\rm chain},r}$, $r\le m-2$ & $g_{i,m}=0$ for $r<i<m$ &
 $\lambda_r=\sum_{i<r}g_{i,i+1}+g_{r,m}
   +\sum_{i=r+1}^{m-2}g_{i,i+1}+g_{m-1,-m}$\\
$U_{{\rm Wh},c}$ & $L_i=0$ for $n<i<m$ &
 $\lambda_{{\rm Wh},c}=\sum_{i<n}g_{i,i+1}+L_n
   +\sum_{i=n+1}^{m-2}g_{i,i+1}+g_{m-1,-m}$\\
$U_{\rm alt}$ & $g_{2j,2j+1}=0$ for $j<n$ &
 $\lambda_{{\rm alt},c}=\sum_{i=1}^{m-2}g_{i,i+2}
   +g_{m-1,-m}+\tfrac c2g_{m-1,m}$\\
$U_{\rm corr}$ & $g_{2j-1,2j}=a$ for $j\le n$ &
 $\Lambda_c=\sum_{i=1}^{m-2}g_{i,i+2}
   -a\sum_{j<n}g_{2j,2j+1}+g_{m-1,-m}+\tfrac c2a$\\
\bottomrule
\end{tabular}

\bigskip
\begin{tabular}{@{}rlcl>{\raggedright\arraybackslash}p{0.35\textwidth}@{}}
\toprule
 & from & & to & by\\
\midrule
1 & Siegel--Whittaker coefficient & $\Rightarrow$ &
 $(U_{{\rm chain},m-1},\lambda_{m-1})$ &
 the first is the integral of the second over $U_{e_m}$
 (Lemma~\ref{fl:lem:descent-vanishing-and-period}, step~1)\\
2 & $(U_{{\rm chain},r},\lambda_r)$ & $\Rightarrow$ &
 $(U_{{\rm chain},r-1},\lambda_{r-1})$ &
 for $r=m-1,\ldots,n+1$: row separation, removal of
 $U_{e_r+e_m}$, exchange of $U_{e_r}$ for $U_{-e_m}$, conjugation
 by $w_r$
 (Lemma~\ref{fl:lem:descent-vanishing-and-period}, steps~2 and~3)\\
3 & $(U_{{\rm chain},n},\lambda_n)$ & $=$ &
 $(U_{{\rm Wh},0},\lambda_{{\rm Wh},0})$ & the definitions agree\\
4 & $(U_{{\rm Wh},0},\lambda_{{\rm Wh},0})$ & $\sim$ &
 $(U_{\rm alt},\lambda_{{\rm alt},0})$ &
 row separation, interleaving of the chains $e_1,\ldots,e_n$ and
 $e_{n+1},\ldots,e_{m-1}$, an isometry
 (Lemma~\ref{fl:lem:even-period-to-generic}, step~2)\\
5 & $(U_{\rm alt},\lambda_{{\rm alt},0})$ & $\sim$ &
 $(U_{\rm corr},\Lambda_0)$ &
 exchange of $\prod_{j<n}U_{e_{2j-1}-e_{2j}}$ for
 $\prod_{j<n}U_{e_{2j}-e_{2j+1}}$
 (Lemma~\ref{fl:lem:even-period-to-generic}, step~1)\\
6 & $(U_{\rm corr},\Lambda_0)$ & $\Rightarrow$ &
 $(U_{\rm corr},\Lambda_c)$, some $c\ne0$ &
 Fourier expansion on the Jacobi group, after integration over
 $(U_{\rm rad},\psi\lambda_{\rm rad})$
 (Lemma~\ref{fl:lem:even-jacobi-nonzero};
 Proposition~\ref{fl:prop:even-generic-nonzero}, step~1)\\
7 & $(U_{\rm corr},\Lambda_c)$ & $\sim$ &
 $(U_{{\rm Wh},c},\lambda_{{\rm Wh},c})$ &
 lines 5 and 4 for this $c$, through
 $(U_{\rm alt},\lambda_{{\rm alt},c})$\\
8 & $(U_{{\rm Wh},c},\lambda_{{\rm Wh},c})$ & $=$ &
 descent, then Whittaker &
 $U_{{\rm Wh},c}=U_{{\rm desc},n}\rtimes U_{H_c}$: the descent with
 $r=n$ and endpoint $-v_c$, then a Whittaker coefficient of $H_c$
 (Lemma~\ref{fl:lem:even-period-to-generic}, step~3)\\
\bottomrule
\end{tabular}
\caption{The coefficients of Section~\ref{fl:sec:odd-symplectic}, for
$m=2n$ and $p\ne2$. Above: the subgroups and their characters; here
$n\le r\le m-1$ and $c\in F$, $L_i=g_{i,m}+(c/2)g_{i,-m}$, and $a$
denotes the common value of the entries
$g_{1,2},g_{3,4},\ldots,g_{m-1,m}$. Below: the route from the
Siegel--Whittaker coefficient to the Whittaker coefficient of the
descent. The sign $\Rightarrow$ means that if the coefficient on the
left is not identically zero on the space of functions considered,
neither is the one on the right; $\sim$ is the relation of
Section~\ref{fl:sec:odd-tools}. The frequency $c$ changes only in
line~6. That $c$ is then a square, so that $H_c$ is split, is step~2
of the proof of Proposition~\ref{fl:prop:even-generic-nonzero}.}
\label{rw:tab:even-descent-route}
\end{table}

\input{sections/fig-even}

\begin{lemma}[The coefficient $\Lambda_c$ and the descent]
\label{fl:lem:even-period-to-generic}
For $c\in F$ put $L_i=g_{i,m}+(c/2)g_{i,-m}$, let
$U_{{\rm Wh},c}\subset U_{\rm last}^0$ be defined by $L_i=0$ for
$n<i<m$, and let
\[
 \lambda_{{\rm Wh},c}=\sum_{i<n}g_{i,i+1}+L_n
                  +\sum_{i=n+1}^{m-2}g_{i,i+1}+g_{m-1,-m}
\]
on $U_{{\rm Wh},c}$. Then
$(U_{{\rm Wh},c},\psi\lambda_{{\rm Wh},c})\sim
(U_{\rm corr},\psi\Lambda_c)$.
For $c\ne0$ the coefficient along
$(U_{{\rm Wh},c},\psi\lambda_{{\rm Wh},c})$ is the descent of
Section~\ref{fl:sec:orthogonal-descent}, in the second
configuration with $r=n$ and endpoint $-v_c$, where
$v_c=e_m+(c/2)f_m$, followed
by a Whittaker coefficient of its stabilizer.
\end{lemma}
\begin{proof}
Both coefficients are compared with a third. Let
$U_{\rm alt}\subset U_{\rm last}^0$ be defined by the vanishing of
the entries $g_{2j,2j+1}$, with the character
$\psi\lambda_{{\rm alt},c}$, where
\[
 \lambda_{{\rm alt},c}(g)
   =\sum_{i=1}^{m-2}g_{i,i+2}+g_{m-1,-m}+(c/2)g_{m-1,m}.
\]

\step{1}{$U_{\rm corr}\sim U_{\rm alt}$} The two groups contain
$U_{\rm corr}\cap U_{\rm alt}$, and are obtained from it by adding
$\prod_{j<n}U_{e_{2j}-e_{2j+1}}$, respectively
$\prod_{j<n}U_{e_{2j-1}-e_{2j}}$. Conjugation by $U_{e_{2j}-e_{2j+1}}$ changes two
neighbouring entries on the second superdiagonal by opposite
multiples of $a$, which cancel in
\eqref{fl:eq:even-corrected-period}; conjugation by a root
subgroup $U_{e_{2j-1}-e_{2j}}$ preserves the character of
$U_{\rm alt}$ because the entries $g_{2j,2j+1}$ vanish there. The
commutator pairing, in coordinates $s_j$ on the first group and
$t_j$ on the second, with $t_n=0$, is
$\sum_{j<n}(t_j-t_{j+1})s_j$, a perfect pairing. So
Lemma~\ref{fl:lem:odd-exchange} applies.

\step{2}{$U_{{\rm Wh},c}\sim U_{\rm alt}$} By
Lemma~\ref{fl:lem:uniform-row-separation} with $r=n$ and $b=c/2$
the conditions $L_i=0$ can be replaced by $g_{n,j}=0$ for
$n<j<m$. Then apply Lemma~\ref{fl:lem:uniform-interleave} to the
chains $A_i=e_i$, $i\le n$, and $B_j=e_{n+j}$, $j<n$, with $d=1$.
The auxiliary subgroup is the unipotent radical of the parabolic
subgroup with Levi subgroup
$\operatorname{GL}_{m-1}\times\operatorname{SO}_3$; the character
on it only involves the rows of $A_n$ and $B_{n-1}$, paired with
$e_m+(c/2)f_m$ and with $f_m$, and the commutators that occur in
the exchanges lie in other rows. The isometry over $F$
\[
 A_i\mapsto e_{2i-1},\quad B_j\mapsto e_{2j},\quad
 e_m\leftrightarrow f_m,\quad u\mapsto-u ,
\]
of determinant one, carries the resulting group and character to
$U_{\rm alt}$ and $\psi\lambda_{{\rm alt},c}$.

\step{3}{the case $c\ne0$} The equations $L_i=0$ say that the rows
$n<i<m$ are orthogonal to $v_c$, so that
$U_{{\rm Wh},c}=U_{{\rm desc},n}\rtimes U_{H_c}$, where
$U_{{\rm desc},n}$ is the descent radical with $r=n$ and $U_{H_c}$
is a maximal unipotent subgroup of
$H_c=\operatorname{GSpin}(v_c^\perp)$, the orthogonal complement
being taken in the space spanned by the last $n$ hyperbolic pairs
and $u$. In the convention $n(y)f_i\equiv f_i+y\bmod Fe_i$
of \eqref{rw:eq:descent-roots} and the following formulas,
$g_{i,m}=-B(y,e_m)$ and $g_{i,-m}=-B(y,f_m)$. Thus $L_n$ is the
pairing with $-v_c$, and $\psi\lambda_{{\rm Wh},c}$ restricts to
$\psi_{\rm desc}$ with endpoint $-v_c$. Its restriction to
$U_{H_c}$ has coefficient one on the simple roots $e_i-e_{i+1}$,
$n<i<m-1$. The last term $g_{m-1,-m}$ pairs the last row with
$-f_m$, whose component in $v_c^\perp$ is
$z_c=-f_m+c^{-1}v_c$. This vector is orthogonal to
the other basis vectors of $v_c^\perp$ and $B(z_c,z_c)=-c^{-1}$;
over a field where $H_c$ splits, it has nonzero components along
both of the last two simple root vectors. So the character is a
nondegenerate character of the maximal unipotent subgroup of the
quasi-split group $H_c$.
\end{proof}

\begin{lemma}[From the Siegel--Whittaker coefficient to frequency zero]
\label{fl:lem:descent-vanishing-and-period}
Let $\mathcal V$ be a space of automorphic functions on
$\operatorname{GSpin}_{4n+1}$, stable under right translation. If
the Siegel--Whittaker coefficient is not identically zero on
$\mathcal V$, neither is the coefficient along
$(U_{\rm corr},\psi\Lambda_0)$.
\end{lemma}
\begin{proof}
By Lemma~\ref{fl:lem:even-period-to-generic} with $c=0$ it
suffices to treat the coefficient along
$(U_{{\rm Wh},0},\psi\lambda_{{\rm Wh},0})$. We reach it through a
chain of
coefficients indexed by $r=m-1,m-2,\ldots,n$, each implied by the
preceding one.

\step{1}{the chain} For $n\le r\le m-2$ let
$U_{{\rm chain},r}\subset U_{\rm last}^0$ be defined by
$g_{i,m}=0$ for $r<i<m$, with the character
\[
 \lambda_r=\sum_{i<r}g_{i,i+1}+g_{r,m}
                   +\sum_{i=r+1}^{m-2}g_{i,i+1}+g_{m-1,-m}.
\]
For $r=m-1$ let $U_{{\rm chain},m-1}=U_{\rm last}^0$ and
$\lambda_{m-1}=\sum_{i<m}g_{i,i+1}$. For $r=n$ this is
$(U_{{\rm Wh},0},\lambda_{{\rm Wh},0})$. The Siegel--Whittaker
coefficient is
the integral of the coefficient for $r=m-1$ over the root subgroup
$U_{e_m}$, so the latter is not identically zero.

\step{2}{from $r$ to $r-1$: an exchange} Let $n<r\le m-1$. By
Lemma~\ref{fl:lem:uniform-row-separation} with $b=0$ we may
replace the conditions $g_{i,m}=0$ by $g_{r,j}=0$ for $r<j<m$.
Remove the root subgroup $U_{e_r+e_m}$, on which the character is
trivial; this preserves nonvanishing. What remains is
$U_{{\rm com},r}\rtimes Z_r$ with $Z_r=U_{e_r}$, where
$U_{{\rm com},r}$ is the group with roots
\[
 \Phi^+\setminus\{e_m,\ e_r,\ e_r+e_m,\ e_r-e_j\ (r<j<m)\}.
\]
Let $Y_r=U_{-e_m}$, with
\[
 x_{-e_m}(s)=1+s(E_{f_m,u}-E_{u,e_m})-\tfrac12s^2E_{f_m,e_m}.
\]
We exchange $Z_r$ for $Y_r$ by Lemma~\ref{fl:lem:odd-exchange}.
Both normalize $U_{{\rm com},r}$ and preserve its character:
conjugation by $Y_r$ produces from $e_i+e_m$ the roots $e_i$ and
$e_i-e_m$, and from $e_i$ the root $e_i-e_m$, with $i\ne r$;
conjugation by $Z_r$ produces from $e_i-e_r$ the roots $e_i$ and
$e_i+e_r$, and from $e_i$ the root $e_i+e_r$. All of these belong
to $U_{{\rm com},r}$ and the character is trivial on them; the
root $e_{m-1}+e_m$, on which it is not trivial, does not occur
because $r<m$. Finally
\[
 [x_{-e_m}(s),x_{e_r}(t)]=x_{e_r-e_m}(st),
\]
as one checks in the five-dimensional space with basis
$(e_r,e_m,u,f_m,f_r)$, and the character on $U_{e_r-e_m}$ is
$\psi$. So the pairing is perfect.

\step{3}{from $r$ to $r-1$: a conjugation} Let $w_r$ be the
isometry which fixes the first $r-1$ hyperbolic pairs, sends
$(e_i,f_i)$ to $(-e_{i+1},-f_{i+1})$ for $r\le i\le m-2$, sends
$(e_{m-1},f_{m-1})$ to $(-f_m,-e_m)$ and $(e_m,f_m)$ to
$(e_r,f_r)$, and sends $u$ to $-u$. It has determinant one, the
exchange of $e$ and $f$ in one pair being compensated by the sign
on $u$. For $i\le m-2$ put $i'=i$ if $i<r$ and $i'=i+1$
otherwise. The roots supported on the indices at most $m-2$ are
mapped to all the positive roots which involve neither $r$ nor
$m$. The roots $e_i\pm e_{m-1}$ are mapped to $e_{i'}\mp e_m$, and
$e_{m-1}$ to $-e_m$. The sums $e_i+e_m$ with $i\le m-2$ are mapped
to the sums which involve $e_r$ and not $e_m$, and
$e_{m-1}+e_m$ to $e_r-e_m$. The differences $e_i-e_m$ which are
allowed, those with $i<r$, are mapped to $e_i-e_r$. Altogether
these are the roots of $U_{{\rm com},r}$ and $-e_m$, so
\[
 w_rU_{{\rm chain},r-1}w_r^{-1}=U_{{\rm com},r}Y_r,
\]
and on the roots where $\lambda_{r-1}$ is nontrivial it acts as
follows, for $r\le m-2$:
\[
\begin{array}{c|c}
\text{root}&\text{image}\\ \hline
e_i-e_{i+1}\ (i<r-1)&e_i-e_{i+1}\\
e_{r-1}-e_m&e_{r-1}-e_r\\
e_i-e_{i+1}\ (r\le i\le m-3)&e_{i+1}-e_{i+2}\\
e_{m-2}-e_{m-1}&e_{m-1}+e_m\\
e_{m-1}+e_m&e_r-e_m.
\end{array}
\]
The coefficient is $+1$ in each row: in a root $e_i-e_j$ the two
signs cancel, and in the last two rows the exchange of $e_m$ and
$f_m$ turns the sign of $E_{e_i+e_j}$ into that of
$E_{e_i-e_j}$. For $r=m-1$ the last two rows are replaced by
$e_{m-2}-e_{m-1}$ and $e_{m-1}-e_m$. So conjugation by
$w_r\in G(F)$ carries the coefficient for $r-1$ to the coefficient
obtained in step~2.
\end{proof}

\subsection{Nonvanishing}

\begin{proposition}[Nonvanishing of the even orthogonal descent, $p\ne2$]
\label{fl:prop:even-generic-nonzero}
Assume $p\ne2$. The descent of the residual representation of
Section~\ref{fl:sec:even-residual} on $\operatorname{GSpin}_{4n+1}$
to the split group $\operatorname{GSpin}_{2n}$ is nonzero and has a
nonzero Whittaker coefficient.
\end{proposition}
\begin{proof}
\step{1}{a nonzero descent for some $c$} The residual
representation has a nonzero Siegel--Whittaker coefficient. By
Lemma~\ref{fl:lem:descent-vanishing-and-period} the coefficient
$\Lambda_0$ is not identically zero on it. The coefficients
$\Lambda_c$ are the Fourier coefficients, for the characters of
Lemma~\ref{fl:lem:even-jacobi-nonzero}, of the functions on the
Jacobi group obtained by integrating over
$(U_{\rm rad},\psi\lambda_{\rm rad})$. So $\Lambda_c$ is not
identically zero for some $c\in F^\times$, and by
Lemma~\ref{fl:lem:even-period-to-generic} the descent with
endpoint $-v_c$ has a nonzero Whittaker coefficient on $H_c$. The
conjugations by isometries used in these lemmas lift to elements
of $\operatorname{GSpin}(F)$, because $H^1(F,\Gm)=0$, and two
lifts differ by a rational point of the centre, which acts
trivially on automorphic functions; so all the coefficients are
coefficients of $\pi$.

\step{2}{$c$ is a square} By Witt cancellation the space
$v_c^\perp$ is split if and only if $Q(v_c)/Q(u)$ is a square;
since $Q(v_c)=c/2$ and $Q(u)=1/2$, this means that $c$ is a
square. If it were not, by Chebotarev there would be a place $v$ where
$\pi$ and $\psi$ are unramified and $H_c$ is not split. There
$\pi_v$ is a subquotient of the representation $I_{P'}$ of
Section~\ref{fl:sec:even-residual}, and its twisted Jacquet module
vanishes by Proposition~\ref{fl:prop:uniform-b-jacquet} and
exactness. On the other hand a nonzero descent gives a nonzero
functional on this Jacquet module. Translate a vector with nonzero
descent on the right so that the integral which defines its
descent is not zero at the identity; write it as a sum of pure
tensors, one of which has a nonzero integral; fix the components
of this tensor at the places other than $v$, and let the component
at $v$ vary. By a change of variable in the adelic integral the
functional obtained is equivariant for $U_{{\rm desc},n}(F_v)$ and
$\psi_{{\rm desc},v}$, so it factors through the Jacquet module,
which is zero. This contradiction shows that $H_c$ is split. By
Witt's theorem and a suitable element of
the torus the endpoint $-v_c$ and the character are conjugate over
$F$ to the split datum.
\end{proof}

\begin{remark}
The Fourier--Jacobi argument only shows that some frequency
$c\ne0$ occurs. That the stabilizer is split follows from the
unramified local components together with the global nonvanishing.
Because these two steps are separate, the first can be replaced by
another argument in characteristic two without changing the
second. The identities of Lemma~\ref{rw:lem:jacobi-embedding} and
the commutator in step~2 of the proof of
Lemma~\ref{fl:lem:descent-vanishing-and-period} have been verified
by computer.
\end{remark}