\section{Orthogonal descent and its local parameter}
\label{fl:sec:orthogonal-descent}

The residual representation of Proposition~\ref{fl:lem:residual-analytic}
lives on a special orthogonal group, or on the corresponding general
spin group $\operatorname{GSpin}$ (an extension of the special
orthogonal group by $\mathbb G_m$), about twice as large as the
group we want. The descent is a Fourier coefficient of it along a
unipotent subgroup $U_{\rm desc}$, for a character
$\psi_{\rm desc}$ whose stabilizer $H$ in the Levi subgroup is the
target group. This section does three things. It computes the
twisted Jacquet module of the local components for
$(U_{\rm desc},\psi_{\rm desc})$
(Proposition~\ref{fl:prop:uniform-b-jacquet}), which gives the
Hecke eigenvalues of the descent if it is nonzero. It proves two
lemmas on rearranging Fourier coefficients
(Lemmas~\ref{fl:lem:uniform-row-separation}
and~\ref{fl:lem:uniform-interleave}). And it proves that the
descent to an odd orthogonal group is nonzero, with a nonzero
Whittaker coefficient, in every characteristic
(Proposition~\ref{fl:prop:uniform-b-nonzero}). The descent to an
even orthogonal group is treated in
Sections~\ref{fl:sec:odd-symplectic} and~\ref{fl:sec:char2-d}.

Two features of the local computation are used later. In the even
orthogonal case the Jacquet module has two summands, exchanged by
the outer automorphism. And it vanishes when the stabilizer is not
split, so that the stabilizer of a nonzero global descent is split
at every unramified place.

\subsection{The descent datum}

Write a split even quadratic space as
\[
 V=\bigoplus_{i=1}^m(Fe_i\oplus Ff_i),\qquad
 Q\Bigl(\sum x_ie_i+y_if_i\Bigr)=\sum x_iy_i,
\]
with root vectors
\[
 E_{e_i-e_j}=E_{ij}-E_{f_jf_i},\qquad
 E_{e_i+e_j}=E_{if_j}-E_{jf_i},\qquad F_\alpha=E_\alpha^t.
\]
In odd dimension there is an additional basis vector $u$,
orthogonal to the others; its norm is $1$ in characteristic two,
and a convenient nonzero constant otherwise. Orthogonal groups are
the smooth special orthogonal groups. In even dimension this is the
kernel of the Dickson invariant on the orthogonal group of the
quadratic form; outside characteristic two it is the kernel of the
determinant. In odd dimension it is the kernel of the determinant
$\mathrm O(Q)\to\mu_2$ in every characteristic; in characteristic
two the bilinear form of an odd-dimensional space has the line of
$u$ as radical, and the quadratic form is not zero on it. The
rank of $g-1$ modulo two, which is the Dickson invariant in even
dimension, cannot be used in odd dimension: in characteristic two
a short root element of the odd orthogonal group differs from the
identity by a matrix of rank one. All unipotent
groups below are the same for the orthogonal group and for
$\operatorname{GSpin}$.

In both configurations $r\ge1$, and $U_{\rm desc}$ is the unipotent
radical of the parabolic subgroup fixing the flag
$\langle e_1\rangle\subset\cdots\subset\langle e_1,\ldots,e_r\rangle$,
whose Levi subgroup is the product of a torus of rank $r$ and the
orthogonal group of the space spanned by the remaining basis
vectors. The character $\psi_{\rm desc}$ is $\psi$ of the sum of
the coordinates on the simple roots $e_i-e_{i+1}$, $i<r$, plus a
linear form $\lambda$ of the last row, that is, of the component
of $U_{\rm desc}$ in the root subgroups from $e_r$ into the
remaining space.

\emph{First configuration.} Here $m=2r+2$ and the ambient group
has type $D_m$. The form $\lambda$ has coefficients $(-1,1)$ on
$(e_r-e_m,e_r+e_m)$ and zero elsewhere; it is the pairing with
$v=e_m-f_m$. Let $H$ be the subgroup of the Levi subgroup whose
torus component is trivial and whose orthogonal component fixes
$v$. It stabilizes $\psi_{\rm desc}$, and it is the special
orthogonal group of the orthogonal complement of $v$ in the
remaining space, a split group of type $B_{r+1}$; in characteristic
two this identification is explained below. The full stabilizer of
$\psi_{\rm desc}$ in the Levi subgroup is slightly larger: the
simple root coordinates force the $r$ torus coordinates to have a
common value $a$, the last row gives $gv=av$, and then $a^2=1$
because $Q(v)\ne0$; so the stabilizer is $\mu_2\times H$. Outside
characteristic two $H$ is its identity component. In characteristic
two $\mu_2$ is connected and not reduced, and $H$ is the reduced
identity component of the stabilizer.

\emph{Second configuration.} Here $m=2r$ and the ambient group has
type $B_m$. We assume
\[
 \lambda=B(v,\cdot)\text{ with }Q(v)\ne0\quad(p\ne2),
 \qquad \lambda(u)=1\quad(p=2).
\]
Then $V_\lambda=\ker\lambda$ is a nondegenerate even-dimensional
quadratic space and $H=\operatorname{SO}(V_\lambda)$ has type
$D_r$. It need not be split.

When the ambient group is $\operatorname{GSpin}$, we write $H$
also for the inverse image of this group.

In characteristic two the description of $H$ in the first
configuration requires an argument, because $v$ lies in its own
orthogonal complement. Put $v=e_m+f_m$ and $u_0=f_m$, and write the
remaining space as $Fv\oplus V_{\rm mid}\oplus Fu_0$, so that
\[
 Q(xv+w+yu_0)=x^2+xy+Q_{\rm mid}(w).
\]
An isometry of $v^\perp=Fv\oplus V_{\rm mid}$ fixing $v$ has the
form $w+xv\mapsto kw+(x+c(w))v$, with $k$ symplectic for the
bilinear form of $V_{\rm mid}$ and
$c(w)^2=Q_{\rm mid}(kw)+Q_{\rm mid}(w)$. Its extensions to the
whole space send
\[
 u_0\longmapsto u_0+b+Av,\qquad
 B(kw,b)=c(w),\qquad A^2+A=Q_{\rm mid}(b),
\]
over every base ring $R$. The equation for $A$ need not have a
solution in $R$; it has one in the $R$-algebra
$R'=R[A]/(A^2+A-Q_{\rm mid}(b))$, which is finite \'etale and
faithfully flat. Over $R'$ we thus have an extension. The
reflection $x\mapsto x+B(x,v)v$ changes $A$ into $A+1$ and has
Dickson invariant one; composing with it on the open and closed
part of $\operatorname{Spec}R'$ where the Dickson invariant of
the extension is one, we get an extension of Dickson invariant
zero. It is unique: the difference $T$ of two values of $A$
satisfies $T^2+T=0$, so it is an idempotent, and the two
extensions differ by the reflection on the part where $T=1$. By
uniqueness the two pullbacks of this extension to
$R'\otimes_RR'$ coincide, and it descends to $R$. Thus
restriction to $v^\perp$ identifies $H$ with the special orthogonal
group of the nondegenerate odd-dimensional quadratic space
$v^\perp$, the split group $\operatorname{SO}_{2r+3}$, as group
schemes. On a remaining
hyperbolic pair its short root subgroup is
\[
 h_j(s)=x_{e_j-e_m}(s)x_{e_j+e_m}(s),\qquad
 h_j(s)f_j=f_j+sv+s^2e_j,
\]
on which the torus acts through $z_j$.

\subsection{The local Jacquet module}

Let $P'$ be the parabolic subgroup of the ambient group whose Levi
subgroup consists of blocks $\operatorname{GL}_2$ on consecutive
pairs of basis vectors, $n$ in number, where $n=r+1$ in the first
configuration and $n=r$ in the second. In the first configuration
there are two such parabolic subgroups $P'_+$ and $P'_-$,
corresponding to the two families of maximal isotropic subspaces.
For characters $\mu_1,\ldots,\mu_n$ and $\omega_\sigma$ of the
multiplicative group of the local field put
\[
 I_{P'}(\boldsymbol\mu;\omega_\sigma)
 =i_{P'}^G\left(\mathop{\boxtimes}_j\mu_j\circ\det
                          \boxtimes\omega_\sigma\right),
\]
where the last factor is a character of the central $\mathbb G_m$
of $\operatorname{GSpin}$, which is a factor of the Levi subgroup,
and is omitted for orthogonal groups. Let
$z_1,\ldots,z_n$ be the coordinates of the maximal torus of $H$
and $z_0$ the coordinate of the central $\mathbb G_m$ of
$\operatorname{GSpin}$, that is, of the cocharacter $e_0^*$. In the
dual coordinates the similitude character of $\operatorname{GSpin}$
is $2e_0+\sum_ie_i$; the multiplier of the dual group
$\operatorname{GSp}$ or $\operatorname{GSO}$ is the character
$e_0^*$ of the dual torus.
Define characters of that torus by
\[
 \xi_\mu(z)=\omega_\sigma(z_0)\prod_{j=1}^n\mu_j(z_j),\qquad
 \xi_\mu^\iota(z)=\omega_\sigma(z_0)\prod_{j<n}\mu_j(z_j)
                              (\omega_\sigma\mu_n^{-1})(z_n).
\]

\begin{proposition}[Orthogonal Jacquet module]
\label{fl:prop:uniform-b-jacquet}
Over any nonarchimedean local field and for arbitrary smooth
characters $\mu_j$ and $\omega_\sigma$,
\[
 J_{U_{\rm desc},\psi_{\rm desc}}I_{P'}(\boldsymbol\mu;\omega_\sigma)
 \simeq\begin{cases}
 i_{B_H}^H\xi_\mu,&\text{ambient type }D_{2r+2},\\
 i_{B_H}^H\xi_\mu\oplus i_{B_H}^H\xi_\mu^\iota,
                     &\text{ambient type }B_{2r},\ V_\lambda\text{ split},\\
 0,&\text{ambient type }B_{2r},\ V_\lambda\text{ nonsplit}.
 \end{cases}
\]
In the first case the result is the same for $P'_+$ and for $P'_-$.
\end{proposition}
The proof occupies the rest of this subsection. It applies
Lemma~\ref{fl:lem:odd-averaging} with $P=MU_{\rm desc}$, the
parabolic subgroup of which $U_{\rm desc}$ is the radical, and it
has four parts. Only one double coset can contribute
(Lemma~\ref{rw:lem:jacquet-double-coset}). In this double coset
the intersection of $U_{\rm desc}$ with the conjugate of $P'$ is a
product of root subgroups, which we list
(Lemma~\ref{rw:lem:jacquet-intersection}). The locus on which the
character is trivial on this intersection is a flag variety of $H$
(Lemma~\ref{rw:lem:jacquet-locus}); this is the main point. The
characters given by Lemma~\ref{fl:lem:odd-averaging} are $\xi_\mu$
and $\xi_\mu^\iota$ (Lemma~\ref{rw:lem:jacquet-characters}). In the
first configuration we take $P'=P'_+$ in these four lemmas, and
compare with $P'_-$ afterwards
(Lemma~\ref{rw:lem:jacquet-two-parabolics}).

\medskip\noindent\emph{The character at a point of the Levi
subgroup.}\ The roots of $U_{\rm desc}$ are
\begin{equation}\label{rw:eq:descent-roots}
 e_i-e_j,\quad e_i+e_j\quad(1\le i\le r,\ i<j\le m),
 \qquad e_i\quad(1\le i\le r,\text{ ambient type }B).
\end{equation}
Let $W$ be the remaining space, spanned by the $e_h$, $f_h$ with
$h>r$ and by $u$ in odd dimension, so that
$M=\mathbb G_m^r\times G_W$ with $G_W$ the special orthogonal group
of $W$, or its inverse image in $\operatorname{GSpin}$. The elements
of the last row are the $n(y)$, $y\in W$, with
$n(y)f_r\equiv f_r+y$ modulo $e_r$; on them
$\psi_{\rm desc}(n(y))=\psi(\lambda(y))$, and the root subgroups are
\[
 x_{e_r-e_h}(t)=n(-tf_h),\qquad x_{e_r+e_h}(t)=n(-te_h),
\]
and in odd dimension the short root subgroup $U_{e_r}$ consists of
the $n(tu)$. Let $\gamma$ be an element of $M$, with components
$(a_1,\ldots,a_r)$ and $g\in G_W$. Put
$\psi^\gamma(n)=\psi_{\rm desc}(\gamma^{-1}n\gamma)$. Since
$\gamma^{-1}n(y)\gamma=n(a_r^{-1}g^{-1}y)$, this is the character of
$U_{\rm desc}$ of the same kind as $\psi_{\rm desc}$ in which the
coefficients on the simple roots $e_i-e_{i+1}$, $i<r$, are
multiplied by nonzero scalars and $\lambda$ is replaced by
$\lambda'=a_r^{-1}\lambda\circ g^{-1}$. It is trivial on every other
root subgroup of \eqref{rw:eq:descent-roots}. We shall use two
properties of $\lambda'$, valid for every $\gamma$.
\begin{enumerate}
\item[(a)] If $\lambda=B(v,\cdot)$, then $\lambda'=B(v',\cdot)$ with
$v'=a_r^{-1}gv$, and $Q(v')=a_r^{-2}Q(v)\ne0$. Writing
$v'=\sum_{h>r}(x_he_h+y_hf_h)+zu$, the coefficients of $\psi^\gamma$ on
$U_{e_r-e_h}$ and $U_{e_r+e_h}$ are $-x_h$ and $-y_h$, and
$Q(v')=\sum_hx_hy_h+Q(u)z^2$.
\item[(b)] In the second configuration in characteristic two,
$\lambda'(u)\ne0$: the line of $u$ is the radical of the bilinear
form of $W$ and $Q(u)\ne0$, so $g$ fixes $u$, and
$\lambda'(u)=a_r^{-1}\lambda(u)$.
\end{enumerate}
The condition of Lemma~\ref{fl:lem:odd-averaging} at the point $\gamma$
of the double coset of $w$ is that $\psi^\gamma$ be trivial on
$U_w=U_{\rm desc}\cap w^{-1}P'w$. This group is the product of the
root subgroups $U_\alpha$, $\alpha$ in \eqref{rw:eq:descent-roots},
for which $w\alpha$ is a root of $P'$, that is, a positive root or
a root $\pm(e_{2c-1}-e_{2c})$ of a block. So the condition is:
\begin{equation}\label{rw:eq:jacquet-criterion}
 \text{if }w\alpha\text{ is a root of }P',\text{ the coefficient of }
 \psi^\gamma\text{ on }U_\alpha\text{ is zero.}
\end{equation}
This is a condition on root subgroups and on the values of $\psi$
at their points, and the arguments below are the same in every
characteristic. The inducing character is trivial on all unipotent
elements of $P'$, including those of the blocks, so the $\mu_j$ do
not enter the condition.

\begin{lemma}[The contributing double coset]
\label{rw:lem:jacquet-double-coset}
Let $w$ be a signed permutation, and suppose that
\eqref{rw:eq:jacquet-criterion} holds at some point $\gamma$ of
$M$. Then $w$ lies in the double coset of the element $w_0$ with
\[
 w_0(e_i)=f_{2i-1},\qquad w_0(e_{r+k})=e_{2k}\quad(1\le i,k\le r),
\]
extended to the $f_i$ by duality. In the first configuration $w_0$
fixes $e_{m-1}$, and it exchanges $e_m$ and $f_m$ exactly when $r$
is odd; in the second configuration $w_0(u)=(-1)^ru$.
\end{lemma}
\begin{proof}
For $i\le r$ write
\[
 w(e_i)=e_{j_i}\ (\text{sign }+)\quad\text{or}\quad
 w(e_i)=f_{j_i}\ (\text{sign }-),\qquad A_i=\lceil j_i/2\rceil,
\]
so that $A_i$ is the block of $P'$ containing the image.

\step{1}{the sign of $e_r$ is negative} Suppose $w(e_r)=e_j$. Let
$h>r$. If $w(e_h)=e_l$, then $w(e_r+e_h)=e_j+e_l$ is a positive
root, so $y_h=0$. If $w(e_h)=f_l$, then $w(e_r-e_h)=e_j+e_l$, so
$x_h=0$. Hence $x_hy_h=0$ for all $h$. In even dimension this gives
$Q(v')=0$, which contradicts (a). In odd dimension $w(e_r)=e_j$ is
a positive short root, so $\lambda'(u)=0$. In odd characteristic
$\lambda'(u)=2Q(u)z$, hence $z=0$ and again $Q(v')=0$. In
characteristic two $\lambda'(u)\ne0$ by (b), a contradiction.

\step{2}{all signs are negative} If the signs of $e_i$ and
$e_{i+1}$ are $(+,-)$, then $w(e_i-e_{i+1})=e_{j_i}+e_{j_{i+1}}$ is
a positive root, while the coefficient of $\psi^\gamma$ on the
simple root $e_i-e_{i+1}$ is not zero. So a positive sign cannot be
followed by a negative one; since the last sign is negative, all
are.

\step{3}{the blocks} For $i<r$ we now have
$w(e_i-e_{i+1})=e_{j_{i+1}}-e_{j_i}$. If $A_{i+1}<A_i$, this is a
positive root; if $A_{i+1}=A_i$, it is a root of a block. Both are
excluded by \eqref{rw:eq:jacquet-criterion}, the coefficient on a
simple root being nonzero. Hence
\begin{equation}\label{rw:eq:blocks-increase}
 A_1<A_2<\cdots<A_r.
\end{equation}
In the
second configuration there are $r$ blocks, so $A_i=i$. In the first
there are $r+1$, and \eqref{rw:eq:blocks-increase} leaves the
possibility $A_r=r+1$. Suppose this is the case, and let $h>r$,
$w(e_h)=e_l$ or $f_l$. Then $l\ne j_r$ and $\lceil l/2\rceil\le
r+1=A_r$, so $e_l-e_{j_r}$ is a root of $P'$: a positive root if
$l$ lies in an earlier block than $j_r$, a root of the last block
otherwise. It is the image of $e_r+e_h$ if $w(e_h)=e_l$ and of
$e_r-e_h$ if $w(e_h)=f_l$. So again $x_hy_h=0$ for every $h$ and
$Q(v')=0$, a contradiction. Therefore $A_i=i$ for all $i$ in both
configurations.

\step{4}{the representative} Multiplying $w$ on
the left by the Weyl group of the Levi subgroup of $P'$, which
exchanges the two indices of a block, we may assume
$j_i=2i-1$. The images of the $e_h$, $h>r$, are then, up to signs,
the $e_{2k}$ with $k\le r$ and, in the first configuration,
$e_{m-1}$ and $e_m$. Multiplying on the right by the Weyl group of
$G_W$, which consists of the signed permutations of the indices
$h>r$, with an even number of sign changes when $W$ has even
dimension, we arrive at $w(e_i)=f_{2i-1}$ and $w(e_{r+k})=e_{2k}$
for $1\le i,k\le r$. In the first configuration we arrive in
addition at $w(e_{m-1})=e_{m-1}$, and then $w$ exchanges $e_m$ and
$f_m$ exactly when $r$ is odd, since the number of sign changes of
an element of the Weyl group of type $D_m$ is even. In the second
configuration $w(u)=(-1)^ru$, the determinant being one. This is
$w_0$.
\end{proof}

\begin{lemma}[The intersection]
\label{rw:lem:jacquet-intersection}
For $w=w_0$ the group $U_w=U_{\rm desc}\cap w^{-1}P'w$ is
\[
 U_{\rm int}=\prod_{1\le k\le i\le r}U_{e_i+e_{r+k}} ,
\]
a product of commuting root subgroups.
\end{lemma}
\begin{proof}
We apply $w_0$ to the list \eqref{rw:eq:descent-roots}. In the
following table $1\le i\le r$, $i<j\le r$ and $1\le k\le r$.
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
root $\alpha$ of $U_{\rm desc}$ & $w_0\alpha$ & root of $P'$\\
\midrule
$e_i-e_j$ & $e_{2j-1}-e_{2i-1}$ & no\\
$e_i+e_j$ & $-e_{2i-1}-e_{2j-1}$ & no\\
$e_i-e_{r+k}$ & $-e_{2i-1}-e_{2k}$ & no\\
$e_i+e_{r+k}$, $k<i$ & $e_{2k}-e_{2i-1}$ & yes, a positive root\\
$e_i+e_{r+i}$ & $e_{2i}-e_{2i-1}$ & yes, a root of the $i$th block\\
$e_i+e_{r+k}$, $k>i$ & $e_{2k}-e_{2i-1}$ & no\\
$e_i\pm e_{m-1}$, $e_i\pm e_m$ (type $D$) &
 $-e_{2i-1}\pm e_{m-1}$, $-e_{2i-1}\pm e_m$ & no\\
$e_i$ (type $B$) & $-e_{2i-1}$ & no\\
\bottomrule
\end{tabular}
\end{center}
The roots marked ``no'' are negative, and none of them is a root
$\pm(e_{2c-1}-e_{2c})$ of a block: in $e_{2j-1}-e_{2i-1}$ both
indices are odd, in $e_{2k}-e_{2i-1}$ with $k>i$ the indices lie in
different blocks, and in type $D$ we have $2i-1<m-1$. This gives
$U_{\rm int}$. The sum of two roots $e_i+e_{r+k}$ is not a root, so
the root subgroups commute.
\end{proof}

\begin{lemma}[The locus]
\label{rw:lem:jacquet-locus}
Let $w=w_0$, and put
$Z_k^0=\langle e_{r+1},\ldots,e_{r+k}\rangle$ for $k\le r$.
\begin{enumerate}
\item The quotient $M_w\backslash M$ is the variety of isotropic
flags $Z_1\subset\cdots\subset Z_r$ in $W$ with $\dim Z_k=k$,
completed in the first configuration by a maximal isotropic
subspace $Z_{r+2}\supset Z_r$ in a fixed family. The class of
$\gamma$ corresponds to the flag $(\gamma^{-1}Z_k^0)$.
\item The condition \eqref{rw:eq:jacquet-criterion} at $\gamma$ is
$\lambda(Z_r)=0$.
\item In the first configuration the locus defined by this
condition is one orbit of $H$, isomorphic to $B_H\backslash H$ for
a Borel subgroup $B_H$ of $H$.
\item In the second configuration the locus is empty if $V_\lambda$
is not split. If $V_\lambda$ is split, it consists of two orbits of
$H$, each isomorphic to $B_H\backslash H$ with the same Borel
subgroup $B_H$.
\end{enumerate}
In all cases the locus is closed and the orbit maps have local
sections. As base points we take $\gamma_1=1$, with the coordinates
of $W$ chosen so that $Z^0_r\subset\ker\lambda$, and for the second
orbit in (4) an element $\gamma_2\in G_W$ which normalizes the
torus, exchanges $e_{2r}$ and $f_{2r}$ and fixes the other basis
vectors of $V_\lambda$.
\end{lemma}
\begin{proof}
\step{1}{the flag variety} The group
$M_w=M\cap w^{-1}P'w$ contains the torus $\mathbb G_m^r$, and its
component in $G_W$ is the parabolic subgroup whose roots $\beta$
are those with $w\beta$ a root of $P'$. From
$w(e_{r+k}-e_{r+k'})=e_{2k}-e_{2k'}$ and
$w(e_{r+k}+e_{r+k'})=e_{2k}+e_{2k'}$, and the corresponding
formulas for $e_{m-1}$, $e_m$ and $u$, one finds that it is the
stabilizer of the flag
\[
 Z_1^0\subset\cdots\subset Z_r^0,
\]
completed in the first configuration by the maximal isotropic
subspace $Z^0_{r+2}$ spanned by $Z_r^0$, $e_{m-1}$ and $w^{-1}e_m$.
So $M_w\backslash M$ is the variety of isotropic flags
$(Z_k)=\gamma^{-1}(Z^0_k)$ in $W$ of dimensions $1,\ldots,r$, followed in
the first configuration by a maximal isotropic subspace $Z_{r+2}$
in the family of $Z^0_{r+2}$. This is (1).

\step{2}{the condition} By
Lemma~\ref{rw:lem:jacquet-intersection} the condition is that
$\psi^\gamma$ be trivial on the subgroups $U_{e_i+e_{r+k}}$ with
$k\le i\le r$. The character $\psi^\gamma$ is trivial on
$U_{e_i+e_{r+k}}$
for $i<r$, and on $x_{e_r+e_{r+k}}(t)=n(-te_{r+k})$ it is
$\psi(-t\lambda'(e_{r+k}))$. So the condition is
$\lambda'(Z^0_r)=0$, that is, $\lambda(Z_r)=0$. This is (2).

\step{3}{the first configuration} The condition says
$Z_r\subset v^\perp$. Then
$Y=Z_{r+2}\cap v^\perp$ has dimension $r+1$: it contains $Z_r$, and
it is not $Z_{r+2}$ because $v$, having nonzero norm, is not in
$Z_{r+2}^\perp=Z_{r+2}$. So $(Z_1,\ldots,Z_r,Y)$ is a complete
isotropic flag of the space $v^\perp$ of dimension $2r+3$.
Conversely, for such a flag the space $Y^\perp/Y$ is a split
quadratic plane; its two isotropic lines give the two maximal
isotropic subspaces of $W$ containing $Y$, one in each family, and
$Z_{r+2}$ is the one in the prescribed family. So the locus is the
variety of complete isotropic flags of $v^\perp$, on whose rational
points $H$ acts transitively with stabilizer a Borel subgroup: it
is one orbit $B_H\backslash H$. These statements hold in
characteristic two, where $v\in v^\perp$ and isotropic means that
the quadratic form vanishes.

\step{4}{the second configuration} The locus is the variety of
complete
isotropic flags of length $r$ in $W$ contained in
$V_\lambda=\ker\lambda$, a space of dimension $2r$. It has no
rational point if $V_\lambda$ is not split. If $V_\lambda$ is
split, the stabilizer in $H$ of
$(Z_1,\ldots,Z_{r-1})$ is a Borel subgroup $B_H$, and there are two
choices of $Z_r$, one in each family of maximal isotropic
subspaces. So the locus consists of two orbits of $H$, each
isomorphic to $B_H\backslash H$ with the same $B_H$. In
characteristic two the decomposition $W=Fu\oplus V_\lambda$, which
exists because $\lambda(u)=1$, gives the same description.

\step{5}{local sections} The locus is defined by the closed
condition $\lambda(Z_r)=0$, and the orbit maps have local sections,
given by the big cells. The elements $\gamma_1$ and $\gamma_2$ are
points of the two orbits.
\end{proof}

\begin{lemma}[The characters]
\label{rw:lem:jacquet-characters}
Suppose that the locus of Lemma~\ref{rw:lem:jacquet-locus} is not
empty, and let $\chi$ be the inducing character of
$I_{P'}(\boldsymbol\mu;\omega_\sigma)$. For the base point
$\gamma_j$ put
$U_j=U_{\rm desc}\cap(w_0\gamma_j)^{-1}P'w_0\gamma_j$. Then, on the
maximal torus of $H$,
\[
 (\chi\delta_{P'}^{1/2})\circ\operatorname{Ad}(w_0\gamma_j)\cdot
 \left|\det\operatorname{Ad}(\cdot)|_{\Lie U_{\rm desc}/\Lie U_j}\right|
 =\begin{cases}
   \xi_\mu\,\delta_{B_H}^{1/2},&j=1,\\
   \xi_\mu^\iota\,\delta_{B_H}^{1/2},&j=2.
  \end{cases}
\]
\end{lemma}
\begin{proof}
\step{1}{the modulus characters} Let $\epsilon=0$ in ambient type
$D$ and $\epsilon=1$ in ambient type $B$. On the $k$th remaining
pair $(e_{r+k},f_{r+k})$ the exponents of $|z_k|$ in the half
modulus of $P'$, in the modulus of $U_{\rm int}$ and in the half
modulus of $B_H$ are
\[
 m-2k+\frac{1+\epsilon}{2},\qquad r-k+1,\qquad
 m-r-k+\frac{\epsilon-1}{2}.
\]
The first is the coefficient $m-j+\epsilon/2$ of the half sum of
positive roots of the ambient group at $j=2k$, minus the
contribution $-1/2$ of the block. The second counts the $r-k+1$
roots $e_i+e_{r+k}$ in $U_{\rm int}$; the modulus of the whole of
$U_{\rm desc}$ is trivial on $H$ because the weights $z_k$ and
$z_k^{-1}$ occur in pairs. The first minus the second is the
third. For the last
coordinate in the first configuration the three exponents are
$1/2,0,1/2$. So at $\gamma_1=1$, where $U_1=U_{\rm int}$, the two
modulus factors on the left combine to $\delta_{B_H}^{1/2}$.

\step{2}{the inducing characters} Under $w_0$ the determinant of the
$k$th block restricts to $z_k$ on the torus of $H$. For
$\operatorname{GSpin}$, with coroots $e_i^*-e_j^*$ and
$e_i^*+e_j^*-e_0^*$, the coordinates made negative by $w_0$ are the
first $r$, and in the first configuration for odd $r$ also the last
one, since $w_0$ then exchanges $e_m$ and $f_m$. All of them are
trivial on $H$, so the central coordinate restricts to $z_0$. This gives $\xi_\mu$ at $\gamma_1$. At
$\gamma_2$ the left side is evaluated at
$\gamma_2z\gamma_2^{-1}$, which is $z$ with $z_n$ inverted and, for
$\operatorname{GSpin}$, with $z_0$ multiplied by $z_n$. The half
modulus of $B_H$ does not involve $z_n$, the exponent
$m-r-k+(\epsilon-1)/2$ being zero for $k=r=n$; so step~1 applies
without change, and the inducing character becomes
$\xi_\mu^\iota$.
\end{proof}

\begin{lemma}[The two parabolic subgroups]
\label{rw:lem:jacquet-two-parabolics}
In the first configuration there is an automorphism of the ambient
group which preserves $U_{\rm desc}$ and $\psi_{\rm desc}$, is the
identity on $H$, and carries $P'_+$ and its inducing character to
$P'_-$ and its inducing character.
\end{lemma}
\begin{proof}
Consider the isometry which exchanges $e_m$ and $f_m$ and acts by
$-1$ on the first $r$ hyperbolic pairs. The exchange of $e_m$ and
$f_m$ is the reflection in $v$. It fixes the vectors of $v^\perp$,
so it commutes with $H$, and it changes the sign of $\lambda$. The
sign $-1$ on the first $r$ pairs commutes with $H$ and restores the
sign of $\lambda$ without affecting the
coordinates on the roots $e_i-e_{i+1}$, $i<r$. So the isometry
preserves $\psi_{\rm desc}$ and commutes with $H$. The exchange of
$e_m$ and $f_m$ also exchanges the two families of
maximal isotropic subspaces, hence $P'_+$ and $P'_-$, and it
carries the last block of one to the last block of the other
compatibly with the determinants; the second factor lies in the
maximal torus and changes neither the parabolic subgroups nor
their characters. So conjugation by this isometry is the required
automorphism of the orthogonal group.

For $\operatorname{GSpin}$ we lift it. In the Clifford algebra,
normalized by $e_i^2=f_i^2=0$ and
$e_if_i+f_ie_i=1$, put
\[
 \tilde t_r=\prod_{i=1}^r(e_i+f_i)(e_i-f_i).
\]
This is an even element of the Clifford group whose image in the
orthogonal group is $-1$ on the first $r$ hyperbolic pairs and the
identity on their complement, and conjugation by the vector $v$
induces conjugation by the reflection in $v$. So the automorphism
of $\operatorname{GSpin}$ which lifts our isometry is conjugation
by $\tilde t_rv$; it preserves the even Clifford group although
$v$ is odd. It is trivial on $H$: the element $\tilde t_r$ lies in
the even Clifford algebra of the first $r$ hyperbolic pairs and
commutes with the even Clifford algebra of the remaining space,
and $v$ anticommutes with the vectors of $v^\perp$, hence commutes
with their even products. In characteristic two each factor
$(e_i+f_i)(e_i-f_i)$ is $(e_i+f_i)^2=1$, the automorphism is
conjugation by $v$, and the isometry has Dickson invariant one.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{fl:prop:uniform-b-jacquet}]
Take $P'=P'_+$ in the first configuration. The condition
\eqref{rw:eq:jacquet-criterion} is closed in $\gamma$. By
Lemma~\ref{fl:lem:odd-averaging}(1) a double coset contributes to
the Jacquet module only through the points at which this condition
holds, and by Lemma~\ref{rw:lem:jacquet-double-coset} there are
such points only in the double coset of $w_0$. If $V_\lambda$ is
not split, there are none in this double coset either
(Lemma~\ref{rw:lem:jacquet-locus}), and the Jacquet module is zero.
Otherwise the locus is a finite disjoint union of closed orbits
$B_H\backslash H$ with local sections, and
Lemma~\ref{fl:lem:odd-averaging}(2) gives the Jacquet module as the
sum over these orbits of the compact inductions from $B_H$ of the
characters of Lemma~\ref{rw:lem:jacquet-characters}. Since
$B_H\backslash H$ is compact, these are the normalized inductions
$i_{B_H}^H\xi_\mu$ and $i_{B_H}^H\xi_\mu^\iota$.

In the first configuration, composing
$I_{P'_+}(\boldsymbol\mu;\omega_\sigma)$ with the automorphism of
Lemma~\ref{rw:lem:jacquet-two-parabolics} gives
$I_{P'_-}(\boldsymbol\mu;\omega_\sigma)$, and it does not change the
twisted Jacquet module as a representation of $H$.
\end{proof}

Let $a_j=\mu_j(\varpi)$ and $c=\omega_\sigma(\varpi)$ for
unramified characters. Since Jacquet modules and invariants under
a compact open subgroup are exact, the proposition shows that
every spherical subquotient of the twisted Jacquet module of a
subquotient of $I_{P'}(\boldsymbol\mu;\omega_\sigma)$ has multiplier
$c$ and standard eigenvalues
\[
 a_1,\ldots,a_n,c/a_n,\ldots,c/a_1.
\]

\begin{remark}
The vanishing in the nonsplit case holds because the flag variety
of Lemma~\ref{rw:lem:jacquet-locus} has no rational point. It allows us to construct a global
descent without controlling the discriminant of $V_\lambda$ in
advance, and to conclude afterwards that $V_\lambda$ is split. In
characteristic two the discriminant is replaced by an
Artin--Schreier class, the Arf invariant.
\end{remark}

\subsection{The smallest case}
\label{rw:sec:smallest-case}

It may help to see the proof of
Proposition~\ref{fl:prop:uniform-b-jacquet} in the smallest case,
the second configuration with $r=1$. The ambient group is
$\operatorname{SO}_5$, with basis $e_1,e_2,f_2,f_1$ and $u$, and
$W=\langle e_2,f_2,u\rangle$. The group $U_{\rm desc}$ is the
unipotent radical of the stabilizer of the line of $e_1$. It is
abelian, $y\mapsto n(y)$ identifies it with $W$, and its root
subgroups are $U_{e_1-e_2}$, $U_{e_1+e_2}$ and $U_{e_1}$. There is
no simple root $e_i-e_{i+1}$ with $i<r$, so
$\psi_{\rm desc}=\psi\circ\lambda$ for a linear form $\lambda$ on
$W$ whose kernel $V_\lambda$ is a nondegenerate plane, and
$H=\operatorname{SO}(V_\lambda)$ is a torus of dimension one, split
if and only if $V_\lambda$ is a hyperbolic plane. The parabolic
subgroup $P'$ is the stabilizer of the isotropic plane
$\langle e_1,e_2\rangle$, its Levi subgroup is $\operatorname{GL}_2$,
and $I_{P'}(\mu)$ is induced from $\mu\circ\det$.

There are two double cosets, according to the sign of $w(e_1)$
(Lemma~\ref{rw:lem:jacquet-double-coset}). If the sign is positive
we may take $w=1$. All
three roots of $U_{\rm desc}$ are then roots of $P'$, so
\eqref{rw:eq:jacquet-criterion} asks that $\lambda'=0$, which never
happens: this double coset does not contribute. If the sign is
negative, $w(e_1)=f_1$, $w(e_2)=e_2$ and $w(u)=-u$. The images of
$e_1-e_2$, $e_1+e_2$ and $e_1$ are $-e_1-e_2$, $e_2-e_1$ and
$-e_1$, of which only the second is a root of $P'$, a root of the
block. So $U_{\rm int}=U_{e_1+e_2}$
(Lemma~\ref{rw:lem:jacquet-intersection}), and the condition at
the point $\gamma=(a_1,g)$ is $\lambda(g^{-1}e_2)=0$.

The quotient $M_w\backslash M$ is the variety of isotropic lines
$Z_1=g^{-1}\langle e_2\rangle$ in the quadratic space $W$ of
dimension three, a conic, and the condition says that $Z_1$ lies in
the plane $V_\lambda$ (Lemma~\ref{rw:lem:jacquet-locus}). So the
locus is the intersection of
a conic with a line. It consists of two rational points if
$V_\lambda$ is split and has no rational point otherwise; in the
second case the Jacquet module is zero. In the split case choose
the coordinates so that $V_\lambda=\langle e_2,f_2\rangle$; the two
points are the lines of $e_2$ and of $f_2$, both fixed by $H$, whose
element $z$ acts by $z$ on $e_2$ and by $z^{-1}$ on $f_2$. Here
$B_H=H$.

For the characters (Lemma~\ref{rw:lem:jacquet-characters}): at the
first point the element
$z$ is sent by $w$ to the element of the block with determinant $z$,
so the inducing character gives $\mu(z)$. The half modulus of $P'$
contributes $|z|$, because the half sum of the positive roots of
$\operatorname{SO}_5$ is $(3/2,1/2)$ and that of the block is
$(1/2,-1/2)$. The roots of $U_{\rm desc}$ have weights $z^{-1}$,
$z$ and $1$ on $H$, and removing $U_{\rm int}$, of weight $z$,
leaves $|z|^{-1}$. The two cancel. At the
second point $z$ is replaced by $z^{-1}$. Hence
\[
 J_{U_{\rm desc},\psi_{\rm desc}}I_{P'}(\mu)\simeq
 \begin{cases}\mu\oplus\mu^{-1},&V_\lambda\text{ split},\\
              0,&V_\lambda\text{ not split},\end{cases}
\]
as a representation of $H$, which in the split case is the
multiplicative group of the local field. The
two summands are the two points in which the line meets the conic,
and in general the two families of maximal isotropic subspaces of
$V_\lambda$ play the same role.

\subsection{Two rearrangements of Fourier coefficients}

The global argument uses the following two operations on root
subgroups. All the commutator pairings in them have coefficient
$\pm1$, so they are valid in characteristic two.

\begin{lemma}[Separating a row]
\label{fl:lem:uniform-row-separation}
In the odd orthogonal group, let $U$ be the standard maximal
unipotent subgroup, $U_{\rm last}^0=\{g\in U:g_{u,f_m}=0\}$, and
$L_i(g)=g_{i,m}+bg_{i,-m}$ for a constant $b$, where the indices
$m$ and $-m$ refer to $e_m$ and $f_m$. Let
$\nu:U_{\rm last}^0\to F$ be an additive homomorphism, so that
$\psi\circ\nu$ is a character, which is trivial on $U_{e_r-e_j}$
and on $U_{e_j-e_m}$ for $r<j<m$. Then
$(U_{\rm tail},\psi(\nu+L_r))\sim(U_{\rm row},\psi(\nu+L_r))$
for the subgroups of $U_{\rm last}^0$
\[
 U_{\rm tail}=\{L_i=0:r<i<m\},\qquad
 U_{\rm row}=\{g_{r,i}=0:r<i<m\}.
\]
\end{lemma}
\begin{proof}
Write the form as $Q=a_0z^2+\sum x_iy_i$. On $U_{\rm last}^0$ the
entry $g_{m,-m}=-a_0g_{u,f_m}^2$ vanishes. Hence in the entry
$(i,-m)$ of a product $gg'$ the terms passing through $u$ and
$e_m$ vanish, and those passing through the $f_j$ vanish because
the matrices are triangular. Adding $b$ times this entry to the
entry $(i,m)$ gives
\[
 L_i(gg')=L_i(g)+L_i(g')+\sum_{i<j<m}g_{ij}L_j(g').
\]
For $j=r+1,\ldots,m-1$ in turn, let the common subgroup be the one
defined by $g_{ri}=0$ for $r<i\le j$ and $L_i=0$ for $j\le i<m$, and
exchange
$U_{e_r-e_j}$ for $U_{e_j-e_m}$ by
Lemma~\ref{fl:lem:odd-exchange}. Before the exchange the entry
$g_{rj}$ is free and $L_j=0$; after it $g_{rj}=0$ and $L_j$ is
free. So the first step starts from $U_{\rm tail}$ and the last
ends at $U_{\rm row}$. The displayed formula shows that
the common subgroup is a group and that its character is
invariant. The commutator of the two root subgroups is
$U_{e_r-e_m}$, on which the character is nontrivial, and the
pairing is $\pm st$. In odd characteristic the condition defining
$U_{\rm last}^0$ is equivalent to $g_{m,u}=0$.
\end{proof}

\begin{lemma}[Interleaving two chains]
\label{fl:lem:uniform-interleave}
Let $A_1,\ldots,A_r$ and $B_1,\ldots,B_b$ be two sequences of
weights, with $b=r-d\ge1$ and $d\in\{-1,0,1\}$. Consider the
unipotent group generated by the root subgroups of the positive
roots $A_i-A_{i'}$ and $B_j-B_{j'}$ within each sequence and of
the cross roots $A_i-B_j$ with $i<r$, with a character which is
$\psi$ of the sum of the coordinates on the adjacent roots
$A_i-A_{i+1}$ and $B_j-B_{j+1}$, and possibly of further
coordinates on an auxiliary subgroup which is normalized by all
these root subgroups and by $U_{B_j-A_{i+1}}$ for $j\le i-d$, the
character being invariant. Then the coefficient is equivalent, in
the sense of $\sim$, to the one in which the cross roots are
replaced by
\[
 A_i-B_j\ (j>i-d),\qquad B_j-A_i\ (i\ge2,\ j<i-d).
\]
\end{lemma}
\begin{proof}
The roots involved are those of the general linear group on basis
vectors indexed by $A_1,\ldots,A_r,B_1,\ldots,B_b$, where
$[1+sE_{ab},1+tE_{bc}]=1+stE_{ac}$ for distinct $a,b,c$. Put
$X_{ij}=U_{A_i-B_j}$ and $Y_{ij}=U_{B_j-A_{i+1}}$. Their commutator
is $U_{A_i-A_{i+1}}$, on which the character is $\psi$, so the
pairing between $X_{ij}$ and $Y_{ij}$ is $\psi(st)$. We exchange
$X_{ij}$ for $Y_{ij}$ by Lemma~\ref{fl:lem:odd-exchange}, for
$i=r-1,r-2,\ldots,1$ in this order and, for each $i$, for
$j=1,\ldots,i-d$ in this order; here $i-d\le b-1$. At the end
the cross roots $A_i-B_j$ that remain are those with $j>i-d$, and
the root $B_j-A_h$ has been added, at the step $(h-1,j)$, exactly
when $h\ge2$ and $j<h-d$. This is the statement. It remains to
check the hypotheses of Lemma~\ref{fl:lem:odd-exchange} at the
step $(i,j)$.

\emph{The common subgroup.} Remove $X_{ij}$. The root $A_i-B_j$
could be a sum of two remaining roots only as
$(A_i-A_h)+(A_h-B_j)$ with $h>i$ or as $(A_i-B_h)+(B_h-B_j)$ with
$h<j$. But $A_h-B_j$ with $h>i$ was removed at an earlier value of
$i$, and $A_i-B_h$ with $h<j$ at an earlier value of $j$. So the
remaining roots form a group.

\emph{Conjugation by $Y_{ij}$.} The root $B_j-A_{i+1}$ can be added
to a remaining root in four ways:
\[
\begin{array}{c|c}
\text{remaining root}&\text{sum}\\ \hline
 A_h-B_j&A_h-A_{i+1}\\
 A_{i+1}-A_h&B_j-A_h\\
 B_h-B_j&B_h-A_{i+1}\\
 A_{i+1}-B_h&B_j-B_h.
\end{array}
\]
In the first row $h<i$, since the root with $h=i$ has just been
removed and those with $h>i$ earlier; the sum is a nonadjacent
root of the first chain. In the second row $h>i+1$, and the sum
was added at the step $(h-1,j)$. In the third row $h<j$, and the
sum was added at the step $(i,h)$. In the fourth row
$A_{i+1}-B_h$ remains only if $h>i+1-d\ge j+1$, and the sum is a
nonadjacent root of the second chain. In all cases the sum belongs
to the common subgroup and the character is trivial on it.

\emph{Conjugation by $X_{ij}$.} The possible sums are $A_i-B_h$
with $h>j$, $A_h-B_j$ with $h<i$, and $A_i-A_h$ with $h\ge i+2$.
The first two have not been removed yet and the third is not
adjacent. An adjacent root of the second chain could only arise
from a root $B_h-A_i$, and such roots are added only at the later
value $i-1$. So $X_{ij}$ normalizes the common subgroup and
preserves its character. The auxiliary subgroup is covered by the
hypothesis.
\end{proof}

Figure~\ref{rw:fig:interleave} shows the cross roots before and after
the exchanges, in the two cases in which the lemma is applied below.

\input{sections/fig-interleave}

\subsection{Nonvanishing for an odd orthogonal target}

The next proposition shows that the descent in the first
configuration is nonzero. The Siegel--Whittaker coefficient is the
coefficient $\mathcal{SW}$ of Section~\ref{fl:sec:common-descent}.

\begin{proposition}[Nonvanishing of the odd orthogonal descent]
\label{fl:prop:uniform-b-nonzero}
Let $\mathcal V$ be a space of automorphic functions on
$\operatorname{SO}_{4n}$ or $\operatorname{GSpin}_{4n}$, $n\ge2$,
stable under right translation. If the Siegel--Whittaker
coefficient is not identically zero on $\mathcal V$, then the
descent of $\mathcal V$ to the split group of type $B_n$ has a
nonzero principal Whittaker coefficient.
\end{proposition}
\begin{proof}
The Whittaker coefficient of the descent is a Fourier coefficient
of $\mathcal V$ along a unipotent subgroup of the ambient group.
The coefficients to be compared are taken along subgroups
generated by the root subgroups of positive weight for a
cocharacter $D$. In the gradings used below the smallest positive
weight is $2$. The quotient of such a subgroup by the subgroup of
the weights greater than $2$ is a vector group, with the root
subgroups of weight $2$ as coordinates, and an element $f$ of
weight $-2$ defines a character of it, by the pairing over the
integers explained in step~3. We transform the pair for the
descent, by exchanges of root subgroups and by one conjugation
over $F$, into the pair $(D_{\rm SW},f_{\rm SW})$ of a principal
nilpotent element of the Siegel Levi subgroup, and then relate the
latter to the Siegel--Whittaker coefficient. The conjugation in
step~3 is the essential point.

\step{1}{combining the two coefficients} Put $m=2n$ and $r=n-1$.
The composite of the descent and of the Whittaker coefficient of
$H$ is the coefficient along the product of $U_{\rm desc}$ and of
the maximal unipotent subgroup of $H$.
Let $N$ be the unipotent radical of the parabolic subgroup with
Levi subgroup $\operatorname{GL}_{m-1}\times\operatorname{SO}_2$,
and $N_c\subset N$ the subgroup of the roots $e_i+e_j$ with
$i<j<m$. The quotient $N/N_c$ has coordinates $x_i$ and $y_i$,
for the roots $e_i-e_m$ and $e_i+e_m$, and the upper triangular
unipotent subgroup of $\operatorname{GL}_{m-1}$ acts on the two
columns $x$ and $y$ in the usual way. We write a point, modulo
$N_c$, as $(A,x,y)$, with $A$ in that unipotent subgroup. In the
initial group $x_j=y_j$ for $n\le j<m$, because its short root
subgroup in the row $j$ is $x_{e_j-e_m}(s)x_{e_j+e_m}(s)$. The
character is the sum of the adjacent coordinates of the two chains
of $A$, together with $L_r+y_{m-1}$, where $L_i=-x_i+y_i$.

For $j=n,\ldots,m-1$ in turn we exchange $U_{e_r-e_j}$ for
$U_{e_j-e_m}$. The common subgroup of the exchange is defined by
$A_{ri}=0$ for $n\le i\le j$ and $L_i=0$ for $j\le i<m$; it is a
subgroup, by the rule for the multiplication of the columns. The
root subgroup of $e_r-e_j$ preserves $L_r$ because $L_j=0$, and
that of $e_j-e_m$ preserves it because $A_{rj}=0$. Both normalize
the common subgroup and preserve the other coordinates of the
character, and the commutator of two of their elements changes
$L_r$ by $\pm st$. The subgroup $N_c$ is central in $N$ and is
normalized by the upper triangular unipotent subgroup of
$\operatorname{GL}_{m-1}$, hence by all the groups in question; and
it is contained in the common subgroup and in the kernel of the
character. So it suffices to check all this in the quotient. After the exchanges the conditions
$L_j=0$ have been replaced by $A_{rj}=0$. The resulting character
is the pairing with
\[
 f=\sum_{i<r}F_{e_i-e_{i+1}}-F_{e_r-e_m}+F_{e_r+e_m}
       +\sum_{j=n}^{m-2}F_{e_j-e_{j+1}}+F_{e_{m-1}+e_m}.
\]

\step{2}{interleaving} Apply
Lemma~\ref{fl:lem:uniform-interleave} with $A_i=e_i$,
$B_j=e_{n+j-1}$ and $d=-1$; the auxiliary subgroup is $N$.
The root elements of $\operatorname{GL}_{m-1}$ which are exchanged
act on the columns only in rows other than $r$ and $m-1$: the
root $A_i-B_j$ has row $i<r$, and the root $B_j-A_{i+1}$ has row
$n+j-1\le m-2$, since $j\le i+1\le r$. So they preserve the
auxiliary character $L_r+y_{m-1}$. The group
obtained is the subgroup $\{\delta=0\}$ of the group of positive
degree for the grading
\[
 D(e_i)=2(n-i)\ (i<n),\qquad
 D(e_j)=2(m-j)\ (j\ge n),\qquad D(f_i)=-D(e_i),
\]
where $\delta$ is the coordinate on the root $e_n-e_1$, of degree
$2$; all other positive degrees are at least $2$ as well. The
characters of the whole group of positive degree that extend ours
are the pairings with
\[
 f_t=f+tE_{e_1-e_n},\qquad t\in F .
\]

\step{3}{conjugation to a principal nilpotent element of the
Siegel Levi subgroup} For $t\in\{0,-1\}$ define vectors
\begin{align*}
 v_1&=e_n,\qquad w_1=f_n,\\
 v_k&=t e_{k-1}+e_{n+k-1},\qquad
 w_k=(2t+1)f_{k-1}+(t+1)f_{n+k-1}\quad(2\le k\le n),\\
 v_{n+1}&=-t e_m+(t+1)f_m,\qquad
 w_{n+1}=(t+1)e_m-tf_m,\\
 v_{n+2+h}&=(-1)^h\bigl((2t+1)f_{r-h}+t f_{m-1-h}\bigr),\\
 w_{n+2+h}&=(-1)^h\bigl((t+1)e_{r-h}-e_{m-1-h}\bigr)
                 \quad(0\le h\le n-2).
\end{align*}
For these two values of $t$ they form a hyperbolic basis:
$Q(v_i)=Q(w_i)=0$, $B(v_i,w_j)=\delta_{ij}$ and
$B(v_i,v_j)=B(w_i,w_j)=0$. So $P_t:e_i\mapsto v_i$, $f_i\mapsto w_i$
is an isometry, and
\[
 P_t^{-1}f_tP_t=\sum_{i<m}F_{e_i-e_{i+1}}=:f_{\rm SW},\qquad
 P_t^{-1}DP_t=D_{\rm SW},\quad D_{\rm SW}(e_i)=m+2-2i.
\]
These are identities between matrices with integer entries; the
defect of each for general $t$ is divisible by $t(t+1)$, for
instance $f_t^m=(-1)^{n-1}2t(t+1)E_{f_n,e_n}$ whereas
$f_{\rm SW}^m=0$. The span of the $v_i$ meets that of the $e_i$ in
dimension $n$ for $t=0$ and $n+1$ for $t=-1$. Hence $P_t$ lies in
the special orthogonal group, of Dickson invariant zero, for
exactly one of the two values:
\[
 t_*=0\ \text{ if $n$ is even},\qquad t_*=-1\ \text{ if $n$ is odd}.
\]
In characteristic two the pairing between the root vectors
$F_\alpha$ and $E_\alpha$ that defines the characters is the
reduction of the integral pairing $\operatorname{tr}(AB)/2$ on the
Lie algebra of the even orthogonal group over $\mathbb Z$, and the
identities above hold over $\mathbb Z$.

\input{sections/fig-siegel}

\step{4}{from $(D_{\rm SW},f_{\rm SW})$ to the Siegel--Whittaker
coefficient} The group of positive degree for $D_{\rm SW}$
contains the roots $e_i+e_j$ with $i+j<m+2$ and $-e_i-e_j$ with
$i+j>m+2$, whereas the Siegel--Whittaker coefficient is along the
group of all positive roots. We pass from the second to the first;
Figure~\ref{rw:fig:siegel-exchange} shows the case $m=6$.
For $i=m-2,\ldots,2$ in decreasing order: if $i\ge n$, first
remove the root $e_{i+1}+e_{i+2}$, on which the character is
trivial; then, for $j=m,\ldots,\max(i+2,m+2-i)$ in decreasing
order, exchange
\[
 U_{e_i+e_j}\quad\hbox{for}\quad U_{-e_{i+1}-e_j}.
\]
At the step $(i,j)$ the positive roots $e_k+e_l$ present are those
with $k<i$, or $k=i$ and $l\le j$, or $k>i$ and $k+l<m+2$, and the
negative roots $-e_k-e_l$ present are those with $k\ge i+2$ and
$k+l>m+2$, or $k=i+1$ and $l>j$; in both lists $k<l$. Call these
sets $\mathsf P$ and $\mathsf N$. Together with the roots
$e_k-e_l$, $k<l$, of the upper triangular group of
$\operatorname{GL}_m$ they form a closed set of roots which
contains no pair of opposite roots, and therefore the product of
their root subgroups is a unipotent group. We check this directly.
Recall that
$j\ge\max(i+2,m+2-i)$.
\begin{itemize}
\item \emph{$\mathsf P$ is stable under lowering an index.} If
$e_k+e_l\in\mathsf P$ and one of the two indices is replaced by a
smaller one, the result is in $\mathsf P$. The only case which is
not immediate is that of $k,l>i$ with $k+l<m+2$ when the new index
is $i$; then the other index is at most $m-i<j$. This gives the
sums of a root $e_a-e_b$ with a root of $\mathsf P$.
\item \emph{$\mathsf N$ is stable under raising an index.} If
$-e_k-e_l\in\mathsf N$ and an index is replaced by a larger one,
the result is in $\mathsf N$: in the case $k=i+1$, $l>j$, raising
$k$ gives two indices which are at least $i+2$, with sum larger
than $m+2$ because $l>m+2-i$. This gives the sums of a root
$e_a-e_b$ with a root of $\mathsf N$.
\item \emph{A root of $\mathsf P$ plus a root of $\mathsf N$.} If
$e_a+e_b\in\mathsf P$ and $-e_b-e_c\in\mathsf N$ with $a\ne c$,
the sum is $e_a-e_c$, and we must show $a<c$. Both $b$ and $c$ are
at least $i+1$. If $a\le i$ there is nothing to prove. If $a>i$,
then $a+b<m+2$. When $b,c\ge i+2$ and $b+c>m+2$ we get
$c>m+2-b>a$. When $b=i+1$ and $c>j$ we get
$a<m+1-i\le j-1<c$. The case $c=i+1$, $b>j$ does not occur, since
it would give $a<m+2-b<i$.
\item \emph{No opposite roots.} A pair $\{k,l\}$ cannot give a
root of $\mathsf P$ and one of $\mathsf N$: for $k\ge i+2$ the
conditions $k+l<m+2$ and $k+l>m+2$ exclude each other, and for
$k=i+1$ the conditions $l<m+1-i$ and $l>j$ do.
\end{itemize}
The same verification applies with the root $e_i+e_j$ removed, so
the common subgroup of the two groups of an exchange is a group.
The only commutator on which the character is nontrivial is the
root $e_i-e_{i+1}$, with coefficient $\pm1$: it is the commutator
of the two exchanged root subgroups. The other sums of an
exchanged root with a present root which could be simple roots do
not occur, because $e_{i+1}+e_{j-1}\notin\mathsf P$, as
$i+j\ge m+2$, and because no root of $\mathsf N$ involves the
index $i$. Consider the root removed at the beginning of the step
for $i\ge n$. Among the positive roots which are sums and are
still present, $e_{i+1}+e_{i+2}$ is the only one with both indices
greater than $i$; lowering one of its indices gives a root which
is still present, and the removed root is not obtained by lowering
an index of a root which is kept. Among the negative roots which
are sums, none involves the index $i+1$, and with a root
$-e_{i+2}-e_l$ one obtains $e_{i+1}-e_l$ with $l\ge i+3$, on which
the character is trivial. So the root subgroup which is removed
normalizes the subgroup which is kept, and its character. The
coefficient before a removal is an
integral of the coefficient after it, so nonvanishing is
preserved, and each exchange is an equivalence by
Lemma~\ref{fl:lem:odd-exchange}. At the end the
roots are those of positive degree for $D_{\rm SW}$.

\step{5}{conclusion} If the Siegel--Whittaker coefficient is not
identically zero on $\mathcal V$, then by step~4 neither is the
coefficient for $(D_{\rm SW},f_{\rm SW})$, and conjugating by
$P_{t_*}\in G(F)$ neither is the coefficient for $(D,f_{t_*})$.
The latter is a Fourier coefficient, along the root subgroup of
$e_n-e_1$, of the coefficient on $\{\delta=0\}$ of step~2, which
is therefore not identically zero. By steps~2 and~1 the Whittaker
coefficient of the descent is not identically zero.
\end{proof}

\begin{remark}\label{rw:rem:odd-descent-computer}
Of the two values $t=0,-1$ exactly one gives an isometry in the
identity component of the orthogonal group; which one depends on
the parity of $n$. The argument works in characteristic two because
it uses the quadratic form and not only its bilinear form: the
latter does not see the component or the action of the short root
subgroup. The identities of step~3, the determinant of $P_t$ and
the dimensions of the intersections have been verified by computer
for $n\le5$. The sequences of exchanges in steps~1, 2 and~4 and in
Lemma~\ref{fl:lem:uniform-interleave}, and the statements of
Lemmas~\ref{rw:lem:jacquet-double-coset}
and~\ref{rw:lem:jacquet-intersection}, have been verified in
small rank by a program. At each exchange it checks the conditions
of Lemma~\ref{fl:lem:odd-exchange} on the sets of roots and on the
supports of the characters, in the form in which the program
encodes them. The same was done for Lemmas~\ref{fl:lem:even-period-to-generic}
and~\ref{fl:lem:descent-vanishing-and-period}.
\end{remark}