#  Theta liftings of non-generic representations on double covers of
orthogonal groups

Yusheng Lei  Department of Mathematics, Boston College, Chestnut Hill,
MA-02467<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

We study the generalized theta lifting between the double covers of split
special orthogonal groups, which uses the non-minimal theta representations
constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of
non-generic representations and make a conjecture that gives an upper bound of
the first non-zero occurrence of the liftings, depending only on the unipotent
orbit. We prove both global and local results that support the conjecture.

###### Key words and phrases:

Unipotent Orbit, Automorphic Representation, Theta Correspondence, Theta Tower

###### 2021 Mathematics Subject Classification:

.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Split orthogonal groups
    2. 2.2 The double cover of the split orthogonal group $\operatorname{SO}_{m}$
  3. 3 Fourier coefficients associated to a unipotent orbit
    1. 3.1 Unipotent orbits
    2. 3.2 Generic characters
    3. 3.3 Global Fourier coefficients
    4. 3.4 Twisted Jacquet modules
    5. 3.5 Wave front sets
  4. 4 Theta representations and the tower of theta liftings
    1. 4.1 Local theta representations
    2. 4.2 Global theta representations
    3. 4.3 The tower of the theta liftings
  5. 5 Global theory
  6. 6 Local theory

##  1\. Introduction

The classical theta correspondence, based on the Weil representation, has been
much studied. Crucially, the Weil representation is a minimal representation.
This property plays a key role in establishing many properties of the
classical theta correspondence. In a small number of cases, there are also
non-minimal theta correspondences. Let $F$ be a number field containing the
group of fourth roots of unity, with the ring of adeles $\mathbb{A}$. In [4],
Bump-Friedberg-Ginzburg constructed the global theta representation
$\Theta_{m}$ on the double cover
$\widetilde{\operatorname{SO}}_{m}(\mathbb{A})$ of the split odd orthogonal
group $\operatorname{SO}_{m}(\mathbb{A})$ as the residues of certain
metaplectic Eisenstein series. In contrast to the Weil representation, such a
theta representation is not minimal. However, its Fourier coefficients
attached to most unipotent orbits vanish. This allows the construction of non-
minimal theta liftings [5]. Specifically, suppose $(\pi,\mathcal{V})$ is an
irreducible cuspidal genuine automorphic representation of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Let
$\operatorname{SO}_{2k^{\prime}}$ be a split even orthogonal group. The
embedding
$\operatorname{SO}_{2k^{\prime}}(\mathbb{A})\times\operatorname{SO}_{2k+1}(\mathbb{A})\hookrightarrow\operatorname{SO}_{2k+2k^{\prime}+1}(\mathbb{A})$
is covered by the embedding of
$\widetilde{\operatorname{SO}}_{2k^{\prime}}(\mathbb{A})\times\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$
into $\widetilde{\operatorname{SO}}_{2k+2k^{\prime}+1}(\mathbb{A})$. Consider
the theta representation $\Theta_{2k+2k^{\prime}+1}$ on
$\widetilde{\operatorname{SO}}_{2k+2k^{\prime}+1}(\mathbb{A})$. For any
$\varphi\in\mathcal{V}$ and $\theta_{2k+2k^{\prime}+1}$ a function in the
representation space of $\Theta_{2k+2k^{\prime}+1}$, Bump-Friedberg-Ginzburg
defined a function $f$ on
$\widetilde{\operatorname{SO}}_{2k^{\prime}}(\mathbb{A})$ (see equation (2) of
[5]) via the integral

(1.1)
$f(h)=\int_{\operatorname{SO}_{2k+1}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\varphi(g)\bar{\theta}_{2k+2k^{\prime}+1}(h,g)dg.$

Functions of the form $f(h)$ generate a genuine automorphic representation
$\Theta_{2k+2k^{\prime}+1}(\pi)$ on the cover
$\widetilde{\operatorname{SO}}_{2k^{\prime}}(\mathbb{A})$.

By fixing the representation $\pi$ and using the theta representations
$\widetilde{\operatorname{SO}}_{2k+2k^{\prime}+1}(\mathbb{A})$ with varying
$k^{\prime}$, one obtains a tower of liftings of the representation $\pi$ to
the groups $\widetilde{\operatorname{SO}}_{2k^{\prime}}(\mathbb{A})$.
According to [5], for a fixed genuine cuspidal automorphic representation
$\pi$ on $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$, one has the
following:

  1. (1)

As an automorphic representation of
$\widetilde{\operatorname{SO}}_{8k}(\mathbb{A})$, $\Theta_{10k+1}(\pi)\neq 0$.

  2. (2)

If $\Theta_{2k+2k^{\prime}+1}(\pi)=0$, then
$\Theta_{2k+2k^{\prime}-1}(\pi)=0$.

In view of these, it is natural to ask when the first non-zero lifting occurs
along the tower. In [5], Bump-Friedberg-Ginzburg show that if
$\Theta_{4k+5}(\pi)$ is generic as an automorphic representation of
$\widetilde{\operatorname{SO}}_{2k+4}(\mathbb{A})$, then the representation
$\pi$ of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ must be generic as
well. They also make a conjecture that a generic representation $\pi$ on
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ should lift to a generic
representation on $\widetilde{\operatorname{SO}}_{2k+4}(\mathbb{A})$.

However, little is known if the representations are not generic, i.e. not
supported on the maximal unipotent orbit. In this paper, we make a general
conjecture on when the lift of a given automorphic representation of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ is nonzero, depending only
on the unipotent orbit that the representation is supported on. Recall that
unipotent orbits are parametrized by partition of integers. We require that
the representation is supported on a unipotent orbit whose corresponding
partition consists of only odd integers. This condition implies that the
attached unipotent subgroup $V_{2,O}$ defined in Section 3 below is the
unipotent radical of a parabolic subgroup.

######  Conjecture 1.1.

Let $(\pi,\mathcal{V})$ be an irreducible cuspidal genuine automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Suppose
$\pi$ is supported on the unipotent orbit

$\mathcal{O}=\left((2n_{1}+1)^{r_{1}}(2n_{2}+1)^{r_{2}}\cdots(2n_{p}+1)^{r_{p}}\right)$

with $n_{1}>n_{2}>\cdots>n_{p}\geqslant 0$ and $r_{i}>0$ for all $i$. Let
$l=r_{1}+r_{2}+\cdots+r_{p}$ be the length of the partition corresponding to
$\mathcal{O}$. Then $\pi$ lifts nontrivially to an automorphic representation
$\Theta_{4k+2l+3}(\pi)$ of
$\widetilde{\operatorname{SO}}_{2k+2l+2}(\mathbb{A})$ which is supported on
the unipotent orbit

$\mathcal{O}^{\prime}=\left((2n_{1}+3)^{r_{1}}(2n_{2}+3)^{r_{2}}\cdots(2n_{p}+3)^{r_{p}}(1)\right).\\\
$

Conjecture 1.1 gives an upper bound of the first non-zero occurrence of the
theta lifting. In the generic case where $\mathcal{O}=(2k+1)$, Conjecture 1.1
agrees with the conjecture made in [5] and mentioned above. In Proposition
4.9, we show that this conjecture is consistent with the “dimension equation”
described in [8], [9] and [7], which proposes dimension constraints on when
the first non-zero lifting may occur. We remark that not every orbit of
$\operatorname{SO}_{2k+1}$ has all odd parts, but we do not know what to
expect when there are even parts in the partition.

In this paper, we prove the following theorem which gives evidence towards the
above conjecture.

######  Theorem 1.2.

Let $(\pi,\mathcal{V})$ be an irreducible cuspidal genuine automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Suppose
the theta lifting $\Theta_{4k+2l+3}(\pi)$, as a representation of
$\widetilde{\operatorname{SO}}_{2k+2l+2}(\mathbb{A})$, has a non-zero Fourier
coefficient associated with the unipotent orbit

$\mathcal{O}^{\prime}=\left((2n_{1}+3)^{r_{1}}(2n_{2}+3)^{r_{2}}\cdots(2n_{p}+3)^{r_{p}}(1)\right).$

Then the representation $\pi$ has a non-zero Fourier coefficient associated
with the unipotent orbit

$\mathcal{O}=\left((2n_{1}+1)^{r_{1}}(2n_{2}+1)^{r_{2}}\cdots(2n_{p}+1)^{r_{p}}\right).$

In the generic case, Theorem 1.2 agrees with the result proved in [5] and
mentioned above.

Moveover, we establish a local counterpart of Theorem 1.2 (which is new even
in the generic case). In the local setting, we turn our attention to the
category of genuine admissible representations on the double covers of the
split special orthogonal groups over a non-archimedean local field $F$. In
[4], the local theta representation on the double cover of a split odd
orthogonal group $\widetilde{\operatorname{SO}}_{2k+1}(F)$ is constructed as
the image of an intertwining operator. Fourier coefficients as the global
analytic tool are replaced by the twisted Jacquet modules. We prove the
following result:

######  Theorem 1.3.

Let $(\pi,\mathcal{V})$ be an irreducible genuine admissible representation of
$\widetilde{\operatorname{SO}}_{2k+1}(F)$. Suppose there exists an irreducible
admissible representation $\Theta(\pi)$ of
$\widetilde{\operatorname{SO}}_{2k+2l+2}(F)$ such that, as representations of
the group
$\widetilde{\operatorname{SO}}_{2k+1}(F)\times\widetilde{\operatorname{SO}}_{2k+2l+2}(F)$,

(1.2)
$\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times\widetilde{\operatorname{SO}}_{2k+2l+2}}\left(\Theta_{4k+2l+3},\pi\otimes\Theta(\pi)\right)\neq
0.$

Furthermore, suppose there exists a non-trivial character
$\psi_{\mathcal{O}^{\prime}}$ (explicitly defined in Section 6) associated
with the unipotent orbit $\mathcal{O}^{\prime}$ such that the corresponding
twisted Jacquet module of $\Theta(\pi)$ is non-zero. Then there exists a non-
trivial character associated with the unipotent orbit $\mathcal{O}$ such that
the corresponding twisted Jacquet module of $\pi$ is also non-zero.

In the case of the classical symplectic-orthogonal theta liftings based on the
Weil representation, Ginzburg-Gurevich [11] give both upper and lower bounds
for the first non-zero occurrence in the theta tower. These bounds can be
parametrized by the partition corresponding to the unipotent orbit that
supports the cuspidal automorphic representation of
$\operatorname{Sp}_{2k}(\mathbb{A})$.

The liftings considered here are related to the extension of Langlands
functoriality to covering groups as follows. According to [19], [22] and [21],
one can define the dual group of a metaplectic group. In the case of the
metaplectic double cover of $\operatorname{SO}_{m}$, we have that

${}^{L}\widetilde{\operatorname{SO}}_{m}^{0}\cong\operatorname{SO}_{m}(\mathbb{C}).$

This suggests that there should be a lifting of genuine automorphic
representations from $\widetilde{\operatorname{SO}}_{2k+1}$ to
$\widetilde{\operatorname{SO}}_{2k^{\prime}}$ corresponding to the inclusion
of $\operatorname{SO}_{2k+1}(\mathbb{C})$ into
$\operatorname{SO}_{2k^{\prime}}(\mathbb{C})$ with $k^{\prime}>k$.

The study of non-generic cuspidal automorphic representations is an important
part of understanding the automorphic discrete spectrum. Jiang [13] proposed a
conjecture that relates Arthur parameters to the maximal unipotent orbit that
supports an automorphic representation. In Section 13 of [16], Leslie
conjectured an extension of Arthur parameters to the metapletic groups. In
view of these works, the results of this paper are conjecturally related to
the question of how Arthur parameters behave under the non-minimal theta
liftings introduced in [5].

This paper is organized as follows: After setting up the basic notations, we
briefly recall the construction of the metaplectic double cover of the split
orthogonal groups in Section 2. In Section 3, we review the definition of the
unipotent orbits and recall the construction of the Fourier coefficients and
the twisted Jacquet modules associated to a unipotent orbit. These are the
global and local tools for proving the respective main theorems. In Section 4,
we briefly recall the construction of both the local and global theta
representations of the double cover $\widetilde{\operatorname{SO}}_{2k+1}$. We
then prove an invariance property of the theta representations which is
crucial for the proof of the main theorem. We also establish the compatibility
of Conjecture 1.1 with the dimension equation. In Section 5, we prove the
global main theorem Theorem 1.2. Lastly, the local theory is treated in
Section 6.

Acknowledgement. I would like to express my gratitude to my advisor Solomon
Friedberg for suggesting this research topic, as well as for providing a
tremendous amount of advice and support. I would also like to thank Yuanqing
Cai, Stella Sue Gastineau, David Ginzburg and Hao Li for many helpful
discussions.

##  2\. Preliminaries

###  2.1. Split orthogonal groups

In this paper, we let $F$ be either a number field with ring of adeles
$\mathbb{A}$ or a non-archimedean local field of residue characteristic not
equal to $2$. Fix an algebraic closure $\bar{F}$ of $F$, and denote by

$\mu_{4}=\\{x\in\bar{F}:x^{4}=1\\}$

the group of all forth roots of unity in $\bar{F}$. Throughout this paper, we
assume that $F$ contains $\mu_{4}$. We also fix a choice of non-trivial
additive character by
$\psi:F\setminus\mathbb{A}\rightarrow\mathbb{C}^{\times}$ when $F$ is a number
field, or by $\psi:F\rightarrow\mathbb{C}^{\times}$ which is unramified when
$F$ is a non-archimedean local field.

For any positive integer $m$, let $\operatorname{SO}_{m}(F)$ denote the split
special orthogonal group consisting of $g\in\operatorname{SL}_{m}$ such that
$gJ_{m}g^{T}=J_{m}$, where

$J_{m}=\begin{pmatrix}&&&&1\\\ &&&1&\\\ &&\iddots&&\\\ &1&&&\\\
1&&&&\end{pmatrix}\in\operatorname{Mat}_{m\times m}(F).$

The maximal unipotent subgroup of $\operatorname{SO}_{m}$ contains $n=\lfloor
m/2\rfloor$ positive simple roots. Let $e_{i,j}$ be the $m\times m$ matrix
with value one on the $(i,j)$-th entry and zero elsewhere. We denote by
$\alpha_{i}$ ($1\leqslant i\leqslant n$) the positive simple roots with
respect to the usual order in the standard Borel subgroup of upper triangular
matrices, and we let each of the corresponding one-parameter subgroups be
$r\mapsto x_{\alpha_{i}}(r)$, where

$x_{\alpha_{i}}(r)=\exp\left(r(e_{i,i+1}-e_{n-i,n-i+1})\right)$

when $m$ is odd, and

$x_{\alpha_{i}}(r)=\begin{cases}\exp\left(r(e_{i,i+1}-e_{n-i,n-i+1})\right)&\text{if}\,1\leqslant
i<n\\\ \exp\left(r(e_{n-1,n+1}-e_{n,n+2})\right)&\text{if}\ i=n\end{cases}$

when $m$ is even.

We fix an embedding of any two orthogonal groups $\operatorname{SO}_{2k+1}$
and $\operatorname{SO}_{2k^{\prime}}$ into
$\operatorname{SO}_{2k+2k^{\prime}+1}$ by

(2.1) $\iota(h,g)=\begin{pmatrix}a&0&b\\\ 0&g&0\\\
c&0&d\end{pmatrix}\in\operatorname{SO}_{2k+2k^{\prime}+1},\quad
g\in\operatorname{SO}_{2k+1},h=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SO}_{2k^{\prime}}.$

###  2.2. The double cover of the split orthogonal group
$\operatorname{SO}_{m}$

Suppose $F$ is a non-archimedean local field. Let
$\widetilde{\operatorname{SL}}_{m}(F)$ be the metaplectic $4$-fold cover
defined in [17] and [15], with the corresponding cocycle denoted by $\sigma$.
This covering group satisfies the short exact sequence

(2.2)
$1\rightarrow\mu_{4}\rightarrow\widetilde{\operatorname{SL}}^{(\sigma)}_{m}(F)\rightarrow\operatorname{SL}_{m}(F)\rightarrow
1.$

Pulling back the image of $\operatorname{SO}_{m}(F)$ in
$\operatorname{SL}_{m}(F)$, we obtain a central extension
$\widetilde{\operatorname{SO}}_{m}(F)$ of $\operatorname{SO}_{m}(F)$.
According to [3] and [4], the square of the cocycle $\sigma$ restricted to
$\operatorname{SO}_{m}(F)$ is almost trivial. Hence, this gives a double cover
of the split orthogonal group $\operatorname{SO}_{m}(F)$.

The same construction goes through in the global situation as well. Suppose
$F$ is a number field with its ring of adeles $\mathbb{A}$. We denote the
double cover of $\operatorname{SO}_{m}(\mathbb{A})$ by
$\widetilde{\operatorname{SO}}_{m}(\mathbb{A})$.

######  Proposition 2.1.

The discrete subgroup $\operatorname{SO}_{m}(F)$ and the unipotent radical
$N(\mathbb{A})$ of the upper triangular Borel subgroup of
$\operatorname{SO}_{m}(\mathbb{A})$ split in the double cover
$\widetilde{\operatorname{SO}}_{m}(\mathbb{A})$.

###### Proof.

See Section 3 of [14]. ∎

##  3\. Fourier coefficients associated to a unipotent orbit

In this section, we recall the connection between Fourier coefficients and
unipotent orbits of the odd orthogonal group $\operatorname{SO}_{2k+1}$. See
[18], [8] and [9] for more details.

###  3.1. Unipotent orbits

Let $F$ be either a number field or a non-archimedean local field, with a
fixed algebraic closure $\bar{F}$. Unipotent orbits of the group
$\operatorname{SO}_{2k+1}$ are parametrized by partitions of $2k+1$ with the
restriction that each even number occurs with even multiplicity. For an orbit
$\mathcal{O}$ corresponding to the partition
$(p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{s}^{r_{s}})$ where $p_{i}>p_{i+1}$ and
$r_{i}>0$ for all $i$, we write

$\mathcal{O}=(p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{s}^{r_{s}}).$

Define the length of the partition to be $l=r_{1}+r_{2}+\cdots+r_{s}$.

Suppose $\mathcal{O}_{1}=(p_{1}p_{2}\cdots p_{r})$ and
$\mathcal{O}_{2}=(q_{1}q_{2}\cdots q_{s})$. We impose a partial order by
$\mathcal{O}_{1}\geqslant\mathcal{O}_{2}$ if $p_{1}+\cdots+p_{i}\geqslant
q_{1}+\cdots+q_{i}$ for all $1\leqslant i\leqslant s$.

Let $\mathcal{O}=(p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{s}^{r_{s}})$. For each
$p_{i}$, we associate $r^{i}$ copies of the torus element

$h_{p_{i}}(t)=\operatorname{diag}(t^{p_{i}-1},t^{p_{i}-3},\cdots,t^{3-p_{i}},t^{1-p_{i}}).$

We obtain a one parameter torus element $h_{\mathcal{O}}(t)$ with non-
increasing powers of $t$ along the diagonal after combining and rearranging
all the $h_{p_{i}}(t)$’s. For example, if $\mathcal{O}=(3^{2}1)$, then

$h_{\mathcal{O}}(t)=\operatorname{diag}(t^{2},t^{2},1,1,1,t^{-2},t^{-2}).$

The conjugation action of $h_{\mathcal{O}}(t)$ on the unipotent radical $N$ of
the upper triangular Borel subgroup $B$ of $\operatorname{SO}_{2k+1}$ induces
a filtration on $N$

$I_{2k+1}\subset\cdots\subset V_{2,\mathcal{O}}\subset
V_{1,\mathcal{O}}\subset V_{0,\mathcal{O}}=N,$

where

$V_{i,\mathcal{O}}=\\{x_{\alpha}(r)\in
N:h_{\mathcal{O}}(t)x_{\alpha}(r)h_{\mathcal{O}}(t)^{-1}=x_{\alpha}(t^{j}r)\quad\text{for
some}\quad j\geqslant i\\}.$

Define

$M(\mathcal{O})=T\cdot\\{x_{\pm\alpha}(r):h_{\mathcal{O}}(t)x_{\alpha}(r)h_{\mathcal{O}}(t)^{-1}=x_{\alpha}(r)\\}.$

Then $P(\mathcal{O})=M(\mathcal{O})V_{1,\mathcal{O}}$ is a standard maximal
parabolic subgroup of $\operatorname{SO}_{2k+1}$.

We say that a unipotent orbit $\mathcal{O}=(p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots
p_{s}^{r_{s}})$ is odd if all the integers $p_{i}$ are odd. In Section 3.3, we
will see that the Fourier coefficients associated to the unipotent orbit
$\mathcal{O}$ are given by integration against $V_{2,\mathcal{O}}$. In
general, $V_{2,\mathcal{O}}$ is not the unipotent radical of a parabolic
subgroup. However, if $\mathcal{O}$ is odd, then
$V_{2,\mathcal{O}}=V_{1,\mathcal{O}}$ is the unipotent radical of the
parabolic subgroup $P(\mathcal{O})$.

Let $V_{2,\mathcal{O}}^{(1)}$ be the commutator subgroup of
$V_{2,\mathcal{O}}$. Over $\bar{F}$, the Levi subgroup $M(\mathcal{O})$ acts
by conjugation on the maximal abelian quotient
$V_{2,\mathcal{O}}/V_{2,\mathcal{O}}^{(1)}$ with a dense open orbit. Pick a
representative $u_{0}$ of this orbit, and set
$M^{u_{0}}(\mathcal{O})(\bar{F})$ to be its stabilizer. Although the group
$M^{u_{0}}(\mathcal{O})(\bar{F})$ depends on the choice of $u_{0}$, its Cartan
type is independent of the choice.

###  3.2. Generic characters

Suppose $F$ is a number field with ring of adeles $\mathbb{A}$. Let
$L_{2,\mathcal{O}}=V_{2,\mathcal{O}}/V_{2,\mathcal{O}}^{(1)}$ be the maximal
abelian quotient of $V_{2,\mathcal{O}}$. The action by conjugation of
$M(\mathcal{O})$ on $V_{2,\mathcal{O}}$ induces an action of
$M(\mathcal{O})(F)$ on the character group

$\widehat{L_{2,\mathcal{O}}(F)\setminus L_{2,\mathcal{O}}(\mathbb{A})}\cong
L_{2,\mathcal{O}}(F).$

We call $\psi_{\mathcal{O}}:L_{2,\mathcal{O}}(F)\setminus
L_{2,\mathcal{O}}(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ a generic
character if the connected component of its stabilizer in $M(\mathcal{O})(F)$
is of the same type as $M^{u_{0}}(\mathcal{O})(\bar{F})$. We extend any such
character trivially to $V_{2,\mathcal{O}}(F)\setminus
V_{2,\mathcal{O}}(\mathbb{A})$. There may exist infinitely many
$M(\mathcal{O})$-conjugacy classes of such generic characters for a specific
unipotent orbit $\mathcal{O}$.

######  Example 3.1.

Let $\mathcal{O}$ be the unipotent orbit corresponding to the partition
$(3^{2}1)$ in $\operatorname{SO}_{7}$. We have

$V_{2,\mathcal{O}}=\left\\{\begin{pmatrix}I_{2}&X&Y\\\ &I_{3}&X^{\ast}\\\
&&I_{2}\end{pmatrix}\in\operatorname{SO}_{7}:X\in\operatorname{Mat}_{2\times
3},\,X^{\ast}=-J_{3}X^{T}J_{2},\,Y^{T}J_{2}+J_{2}Y=0\right\\}.$

A choice of generic character is
$\psi_{\mathcal{O}}:V_{2,\mathcal{O}}(F)\setminus
V_{2,\mathcal{O}}(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ given by

$\psi_{\mathcal{O}}(v)=\psi(v_{1,3}+v_{2,5}).$

By Pontryagin dualilty, we may identify each generic character with an element
in $L_{2,\mathcal{O}}(F)\cong\operatorname{Mat}_{2\times 3}(F)$. The above
character corresponds to the matrix

$\begin{pmatrix}1&0&0\\\ 0&0&1\end{pmatrix}.$

The rank of the matrix and the fact that its row space is not totally
isotropic are invariant under the action of
$M(\mathcal{O})\cong\operatorname{GL}_{2}(F)\times\operatorname{SO}_{3}(F)$ on
$\operatorname{Mat}_{2\times 3}(F)$. Any choice with full rank and non-
totally-isotropic row space corresponds to a generic character.

###  3.3. Global Fourier coefficients

Let $(\pi,\mathcal{V})$ be an automorphic representation of
$\operatorname{SO}_{2k+1}(\mathbb{A})$. We define the Fourier coefficients of
$\pi$ associated with a unipotent orbit $\mathcal{O}$ by the following:

######  Definition 3.2.

Let $\psi_{\mathcal{O}}:V_{2,\mathcal{O}}(F)\setminus
V_{2,\mathcal{O}}(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ be a generic
character associated with a unipotent orbit $\mathcal{O}$ in
$\operatorname{SO}_{2k+1}$. For an automorphic function $\varphi\in\pi$, the
Fourier coefficient of $\varphi$ with respect to $\psi_{\mathcal{O}}$ is

(3.1)
$F_{\psi_{\mathcal{O}}}(\varphi_{\pi})(g)=\int_{[V_{2,\mathcal{O}}]}\varphi(ug)\psi_{\mathcal{O}}(u)\,du.$

Henceforth, we use $[K]$ to denote $K(F)\setminus K(\mathbb{A})$ for any group
$K$. We say that the orbit $\mathcal{O}$ supports $\pi$ if there exists some
$\varphi\in\mathcal{V}$ and $\psi_{\mathcal{O}}$ generic such that the above
integral is non-zero. Otherwise, we say that $\mathcal{O}$ does not support
the representation $\pi$.

###  3.4. Twisted Jacquet modules

Suppose now $F$ is a non-archimedean local field. Let $U$ be a unipotent
subgroup of $\operatorname{SO}_{2k+1}(F)$, with
$\psi_{U}:U\rightarrow\mathbb{C}^{\times}$ a character on $U$. Let
$(\pi,\mathcal{V})$ be a smooth representaion of
$\operatorname{SO}_{2k+1}(F)$. Suppose there exists a subgroup
$M\subset\operatorname{SO}_{2k+1}(F)$ which normalizes $U$ and stablizes the
character $\psi_{U}$. Consider the subspace $\mathcal{V}(U,\psi_{U})$ of
$\mathcal{V}$ generated by vectors of the form
$\\{\pi(u)v-\psi_{U}(u)v\,|\,v\in\mathcal{V},u\in U\\}$. The twisted Jacquet
module of $\pi$ with respect to $\psi_{U}$ is defined by
$J_{U,\psi_{U}}(\pi)=\mathcal{V}/\mathcal{V}(U,\psi_{U})$. If $\psi_{U}$ is
trivial, we denote it by $J_{U}(\pi)$ and call it the Jacquet module of $\pi$
with respect to $U$. This defines an exact functor between the categories of
smooth representations on the two groups

$J_{U,\psi_{U}}:Rep(\operatorname{SO}_{2k+1}(F))\rightarrow Rep(M).$

The Levi subgroup $M(\mathcal{O})(F)$ acts on $L_{2,\mathcal{O}}(F)$, and
hence on the character group

$\widehat{L_{2,\mathcal{O}}(F)}\cong L_{2,\mathcal{O}}(F).$

Again, we only look at those generic characters whose stablizer under this
action is of the same Cartan type as $M^{\mu_{0}}(\mathcal{O})(\bar{F})$.

######  Definition 3.3.

Let $(\pi,\mathcal{V})$ be an admissible representation of
$\operatorname{SO}_{2k+1}(F)$. The twisted Jacquet module of $\pi$ associated
to a unipotent orbit $\mathcal{O}$ and a generic character
$\psi_{\mathcal{O}}:V_{2,\mathcal{O}}(F)\rightarrow\mathbb{C}^{\times}$ is
given by

(3.2) $J_{V_{2,\mathcal{O}},\psi_{\mathcal{O}}}(\pi).$

We say that the unipotent orbit $\mathcal{O}$ supports $\pi$ if there exists
some generic character $\psi_{\mathcal{O}}$ such that (3.2) is non-zero.

###  3.5. Wave front sets

We have the following definition that applies to both global and local
situations.

######  Definition 3.4.

Let $(\pi,\mathcal{V})$ be either an automorphic representation of
$\operatorname{SO}_{2k+1}(\mathbb{A})$ (where $\mathbb{A}$ is the ring of
adeles of a number field $F$) or an admissible representation of
$\operatorname{SO}_{2k+1}(F)$ (where $F$ is a non-archimedean local field).
The wave front set $\mathcal{O}(\pi)$ is the set of maximal unipotent orbits
in $\operatorname{SO}_{2k+1}$ such that $\mathcal{O}\in\mathcal{O}(\pi)$.

##  4\. Theta representations and the tower of theta liftings

###  4.1. Local theta representations

Let $F$ be a non-archimedean local field. In [4], Bump-Friedberg-Ginzburg
constructed the local theta representation $\Theta_{2k+1}$ of the double cover
$\widetilde{\operatorname{SO}}_{2k+1}(F)$ as the irreducible image of an
intertwining operator

The theta representation $\Theta_{2k+1}$ of the double cover
$\widetilde{\operatorname{SO}}_{2k+1}(F)$ is a small representation in the
terminology of [4]. It agrees with the minimal representation when $k=2$ or
$3$. When $k=4$ or $5$, $\mathcal{O}(\Theta_{2k+1})$ is the singleton set
containing the next smallest unipotent orbit. Bump-Friedberg-Ginzburg [4]
proved the following result:

######  Proposition 4.1.

Let $\Theta_{2k+1}$ be the theta representation of the double cover
$\widetilde{\operatorname{SO}}_{2k+1}(F)$. Let $n$ be a positive integer. Then

(4.1) $\mathcal{O}(\Theta_{2k+1})=\begin{cases}(2^{2n}1)&\text{if}\;k=2n,\\\
(2^{2n}1^{3})&\text{if}\;k=2n+1.\end{cases}$

###### Proof.

See either [4] or Section 2 of [14]. ∎

Let $r$ be an integer such that $1\leqslant r\leqslant k$. Suppose
$P_{r}=(\operatorname{GL}_{r}\times\operatorname{SO}_{2k-2r+1})U_{r}$ is a
maximal parabolic subgroup of $\operatorname{SO}_{2k+1}$ with the indicated
Levi decomposition, where $U_{r}$ is the unipotent radical. Recall from [15]
that the two-fold metapletic cover $\widetilde{\operatorname{GL}}_{r}(F)$ of
the general linear group $\operatorname{GL}_{r}(F)$ affords a theta
representation $\Theta_{\operatorname{GL}_{r}}$. Note that the two-fold cover
$\widetilde{\operatorname{GL}}_{r}(F)$ embeds into
$\widetilde{\operatorname{SO}}_{2k+1}(F)$ as the Levi factor of the inverse
image of the parabolic subgroup $P_{r}(F)$. The reader may refer to [15] for
more details. We have the following:

######  Proposition 4.2.

Let $\Theta_{2k+1}$ be the local theta representation of
$\widetilde{\operatorname{SO}}_{2k+1}(F)$. Considered as a representation of
$\widetilde{\operatorname{GL}}_{r}\times\widetilde{\operatorname{SO}}_{2k-2r+1}(F)$,
the Jacquet module of $\Theta_{2k+1}$ with respect to $U_{r}$ is ismorphic to
$\Theta_{\operatorname{GL}_{r}}\otimes\Theta_{2k-2r+1}$. When $r=k$,
$\Theta_{1}$ is the trivial representation.

###### Proof.

This is Theorem 2.3 of [4], or Proposition 1 of [5]. ∎

###  4.2. Global theta representations

Suppose now $F$ is a number field, with its ring of adeles $\mathbb{A}$. The
global theta representation of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ is given by the residues of
an Eisenstein series constructed in [4]. In the terminology of [4], let
$\chi_{s}:T(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ be the character on the
split torus attached to the complex parameter
$s=(s_{1},s_{2},\cdots,s_{k})\in\mathbb{C}^{k}$. Let
$\operatorname{Ind}(\chi_{\theta})$ be the representation of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ parabolically induced from
the character $\chi_{\theta}:Z(\widetilde{T}(F))\setminus
Z(\widetilde{T}(\mathbb{A}))\rightarrow\mathbb{C}^{\times}$ attached to the
complex parameter $s_{\theta}=(k/2,(k-1)/2,\cdots,1/2)$. For any
$f\in\operatorname{Ind}(\chi_{\theta})$, there is a unique global smooth
section sending any $s\in\mathbb{C}^{k}$ to a function
$f_{s}\in\operatorname{Ind}(\chi_{s})$ such that $f_{s_{\theta}}=f$. Bump-
Friedberg-Ginzburg [4] define the Eisenstein series associated to this smooth
section by

(4.2) $E(g,f_{s})=\sum_{\gamma\in
B(F)\setminus\operatorname{SO}_{2k+1}(F)}f_{s}(\gamma g),\quad
g\in\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A}).$

By taking the residue of this Eisenstein series at $s=s_{\theta}$, they obtain
an automorphic form

$\theta_{f}(g)=\operatorname{res}_{s_{1}=k/2}\cdots\operatorname{res}_{s_{k-1}=1}\operatorname{res}_{s_{k}=1/2}E(g,f_{s}).$

Then, the global theta representation $\Theta_{2k+1}$ of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ is the representation on
the subspace of $L^{2}([\widetilde{\operatorname{SO}}_{2k+1}])$ spanned by all
$\theta_{f}$ with $f\in\operatorname{Ind}(\chi_{\theta})$.

######  Proposition 4.3.

Let $\theta$ be a function in the theta representation $\Theta_{2k+1}$ of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Considered as a function
of
$(g,h)\in\widetilde{\operatorname{GL}}_{r}\times\widetilde{\operatorname{SO}}_{2k-2r+1}$,
the integral

$\int_{U_{r}(F)\setminus U_{r}(\mathbb{A})}\theta(u(g,h))\,du$

is in the space of the automorphic representation
$\Theta_{\operatorname{GL}_{r}}\otimes\Theta_{2k-2r+1}$. Here,
$\Theta_{\operatorname{GL}_{r}}$ is the global exceptional representation on
the double cover $\widetilde{\operatorname{GL}}_{r}(\mathbb{A})$ in the sense
of [15].

This is the global version of Proposition 4.2. The case $r=1$ is proved in
[4]. Similar statement for the n-fold metaplectic cover
$\widetilde{\operatorname{Sp}}^{(n)}_{2k}$ can be found in [10]. The proof is
similar to these two sources. For the convenience of the readers, we outline
the proof here.

###### Proof.

Consider the exceptional quasicharacter
$\chi_{\theta}:T(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ defined by

$\chi_{\theta}(\operatorname{diag}(t_{1},t_{2},\cdots,t_{k},1,t_{k},\cdots,t_{2},t_{1}))=|t_{1}|^{\frac{k}{2}}|t_{2}|^{\frac{k-1}{2}}\cdots|t_{k}|^{\frac{1}{2}}.$

Denote $\delta_{B}$ and $\delta_{P_{r}}$ the modular characters of the
standard Borel subgroup of upper triangular matrices and $P_{r}$ respectively.
By induction in stages, we deduce that
$\operatorname{Ind}_{\widetilde{B}}^{\widetilde{\operatorname{SO}}_{2k+1}}(\delta_{B}^{1/2}\chi_{\theta})$
is equal to
$\operatorname{Ind}_{\widetilde{P}_{r}}^{\widetilde{\operatorname{SO}}_{2k+1}}(\Theta_{\operatorname{GL}_{r}}\otimes\Theta_{2k-2r+1})\delta_{P_{r}}^{\frac{3n-2r+1}{4n-2r}}$.

Let $E_{P_{r}}(g,s,f_{s})$ be the Eisenstein series of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ associated to

$f_{s}\in\operatorname{Ind}_{\widetilde{P}_{r}}^{\widetilde{\operatorname{SO}}_{2k+1}}(\Theta_{\operatorname{GL}_{r}}\otimes\Theta_{2k-2r+1})\delta_{P_{r}}^{s}.$

It follows from [4] that any function in the representation $\Theta_{2k+1}$ is
the residue of $E_{P_{r}}(g,s,f_{s})$ at the point $s=\frac{3n-2r+1}{4n-2r}$.
Consider the constant term

(4.3) $\int_{U_{r}(F)\setminus
U_{r}(\mathbb{A})}E_{P_{r}}(u(g,h),s,f_{s})\,du.$

For $\Re(s)$ large enough we can unfold the Eisenstein series and obtain that
(4.3) equals

(4.4) $\sum_{\omega\in
P_{r}(F)\setminus\operatorname{SO}_{2k+1}(F)/P_{r}(F)}\int_{U_{r}^{\omega}(F)\setminus
U_{r}(\mathbb{A})}\sum_{\gamma\in(\omega^{-1}P_{r}(F)\omega\cap
P_{r}(F))\setminus P_{r}(F)}f_{s}(\omega^{-1}\gamma u(g,h))\,du,$

where
$U_{r}^{\omega}=\omega^{-1}U_{r}\omega\cap\omega_{0}^{-1}U_{r}\omega_{0}$ with
$\omega_{0}$ being the long Weyl group element.

The double cosets $P_{r}(F)\setminus\operatorname{SO}_{2k+1}(F)/P_{r}(F)$ may
be represented by Weyl group elements. By Section II.1.7 of [20], any of the
inner summations is either an Eisenstein series or a product of such series.
All such Eisenstein series are holomorphic at $s=\frac{3n-2r+1}{4n-2r}$ except
the one corresponding to the long Weyl group element. Following the same
argument as in Proposition 1 of [10], by taking the residue of (4.3) at
$s=\frac{3n-2r+1}{4n-2r}$, we obtain

(4.5) $\int_{U_{r}(F)\setminus
U_{r}(\mathbb{A})}\theta(u(g,h))\,du=\operatorname{Res}_{s=\frac{3n-2r+1}{4n-2r}}M_{\omega_{0}}f_{s}(g,h),$

where

(4.6)
$M_{\omega_{0}}f_{s}(g,h)=\int_{\omega_{0}^{-1}U_{r}(F)\omega_{0}\setminus
U_{r}(\mathbb{A})}f_{s}(\omega_{0}^{-1}u(g,h))\,du.$

As $\operatorname{Res}_{s=\frac{3n-2r+1}{4n-2r}}M_{\omega_{0}}f_{s}(g,h)$ is
an element of $\Theta_{\operatorname{GL}_{r}}\otimes\Theta_{2k-2r+1}$, the
proof is complete. ∎

The global theta representation has the same smallness property given by 4.7.
We have the following statement:

######  Proposition 4.4.

Let $\Theta_{2k+1}$ be the global theta representation of the double cover
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Let $n$ be a positive
integer. Then

(4.7) $\mathcal{O}(\Theta_{2k+1})=\begin{cases}(2^{2n}1)&\text{if}\;k=2n,\\\
(2^{2n}1^{3})&\text{if}\;k=2n+1.\end{cases}$

###### Proof.

See Theorem 4.2(i) of [4], or Proposition 2 of [5]. ∎

There is another important property we need for the theta representations
$\Theta_{2k+1}$. If $r<k/4$ is a positive integer, we denote $H_{r,2k+1}$ the
unipotent radical of the maximal parabolic subgroup of
$\operatorname{SO}_{2k+1}$ whose Levi part is
$\operatorname{GL}_{r}\times\operatorname{SO}_{2k-2r+1}$. In other words,
$H_{r,2k+1}$ consists of upper triangular matrices of the form

$\left\\{\begin{pmatrix}I_{r}&x&\ast\\\ &I_{2k-2r+1}&x^{\ast}\\\
&&I_{r}\end{pmatrix}\in\operatorname{SO}_{2k+1}:x^{\ast}=-J_{2k-2r+1}x^{T}J_{r}\right\\},$

where $\ast$ denotes whatever is needed for the matrix to be orthogonal.
Similarly, let $H_{r,2k-2r+1}$ be the unipotent of the standard maximal
parabolic subgroup of $\operatorname{SO}_{2k-2r+1}$ with Levi subgroup
$\operatorname{GL}_{r}\times\operatorname{SO}_{2k-4r+1}$. Via the embedding
(2.1), we identify $H_{r,2k-2r+1}$ with its image in
$\operatorname{SO}_{2k+1}$. Hence, $H_{r,2k-2r+1}$ consists of matrices of the
form

$\left\\{\begin{pmatrix}I_{r}&&&&\\\ &I_{r}&y&\ast&\\\
&&I_{2k-4r+1}&y^{\ast}&\\\ &&&I_{r}&\\\
&&&&I_{r}\end{pmatrix}\in\operatorname{SO}_{2k+1}(\mathbb{A})\right\\}.$

For any $u=(u_{i,j})\in H_{r,2k+1}(\mathbb{A})$, we define the character
$\psi_{1}:H_{r,2k+1}(F)\setminus
H_{r,2k+1}(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ by

$\psi_{1}(u)=\psi(\sum_{j=1}^{r}u_{j,j+r}).$

######  Proposition 4.5.

Fix a function $\theta\in\Theta_{2k+1}$. The integral

(4.8) $f(g)=\int_{H_{r,2k+1}(F)\setminus
H_{r,2k+1}(\mathbb{A})}\theta(ug)\psi_{1}(u)\,du$

is left invariant by $H_{r,2k-2r+1}(\mathbb{A})$. That is, $f(g)=f(vg)$ for
any $v\in H_{r,2k-2r+1}(\mathbb{A})$.

###### Proof.

Via the embedding (2.1), the center $Z(H_{r,2k-2r+1})$ consists of matrices of
the form

$\left\\{\begin{pmatrix}I_{r}&&&&\\\ &I_{r}&&z&\\\ &&I_{2k-4r+1}&&\\\
&&&I_{r}&\\\ &&&&I_{r}\end{pmatrix}:\quad z\in\operatorname{Mat}_{r\times
r},\quad z^{T}J_{r}+J_{r}z=0\right\\}.$

We first expand (4.8) against $Z(R_{r,2k-2r+1})(F)\setminus
Z(R_{r,2k-2r+1})(\mathbb{A})$. Embed the group $\operatorname{GL}_{r}(F)$ into
$\operatorname{SO}_{2k+1}(F)$ via

$h\hookrightarrow\operatorname{diag}(I_{r},h,I_{2k-4r+1},h^{\ast},I_{r}),\quad\forall
h\in\operatorname{GL}_{r}(F).$

The $\operatorname{GL}_{r}(F)$-action on $Z(H_{r,2k-2r+1})(F)\setminus
Z(H_{r,2k-2r+1})(\mathbb{A})$ induces an action on its character group, which
may be identified with $Z(H_{r,2k-2r+1})(F)$. This action preserves the rank
of the matrices in
$Z(H_{r,2k-2r+1})(F)\cong\left\\{z\in\operatorname{Mat}_{r\times
r},\,z^{T}J_{r}+J_{r}z=0\right\\}$. On the other hand, the rank of any $z\in
Z(H_{r,2k-2r+1})(F)$ must be even. Suppose $\operatorname{Rank}(z)=2q$, where
$0<2q\leqslant r$. We may choose a representative

(4.9) $\begin{pmatrix}0&z_{q}\\\
0&0\end{pmatrix}\in\operatorname{Mat}_{r\times r}(F),$

where

$z_{q}=\operatorname{diag}(\lambda_{1},\lambda_{2},\cdots,\lambda_{q},-\lambda_{q},\cdots,-\lambda_{2},-\lambda_{1})\in\operatorname{Mat}_{2q\times
2q}(F),\lambda_{i}\in F^{\times}\quad\forall i=1,2,\cdots,q.$

This gives the corresponding character on $[Z(H_{r,2k-2r+1})]$ by:

$\psi_{Z,q}(v)=\psi(\lambda_{1}v_{1,r-2q+1}+\lambda_{2}v_{2,r-2q+2}+\cdots+\lambda_{q}v_{q,r-q}),$

where

$v\in Z(H_{s,2k-2s+1})(\mathbb{A})\cong\operatorname{Mat}_{r\times
r}(\mathbb{A}).$

For any character in the same orbit as $\psi_{Z,q}$, its contribution in the
expansion is given by the integral

(4.10)
$\int_{[H_{r,2k+1}]}\int_{[Z(H_{r,2k-2r+1})]}\theta(uvg)\psi_{1}(u)\psi_{Z,q}(v)\,dvdu.$

Notice that the character $\psi_{1}\psi_{Z,q}$ coincides with a generic
character associated to the unipotent orbit
$\mathcal{O}_{q}=(4^{2q}3^{r-2q}1^{2k-2q-3r+1})$ (or $(4^{r}1^{2k-4r+1})$ in
the case $r=2q$). Moreover, we have $V_{2,\mathcal{O}_{q}}\subseteq
H_{r,2k+1}Z(H_{r,2k-2r+1})$. As a result, the integral (4.10) contains an
inner integral that is a Fourier coefficient of $\theta$ with respect to the
unipotent orbit $\mathcal{O}_{q}$. The integral (4.10) is then zero because
any such Fourier coefficient of $\theta$ is zero by Proposition 4.4. Thus,
only the integral corresponding to the trivial character contributes, and
(4.8) is equal to

(4.11) $f(g)=\int_{[H_{r,2k+1}Z(H_{r,2k-2r+1})]}\theta(ug)\psi_{1}(u)\,du.$

As the center $Z(H_{r,2k-2r+1})$ is now contained in the domain of
integration, we can further expand (4.11) against

$[H_{r,2k-2r+1}/Z(H_{r,2k-2r+1})]\cong\operatorname{Mat}_{r\times(2k-4r+1)}(F)\setminus\operatorname{Mat}_{r\times(2k-4r+1)}(\mathbb{A}).$

Embed the group $\operatorname{GL}_{r}(F)\times\operatorname{SO}_{2k-4r+1}(F)$
into $\operatorname{SO}_{2k+1}(F)$ via

$(h_{1},h_{2})\hookrightarrow\operatorname{diag}(I_{r},h_{1},h_{2},h_{1}^{\ast},I_{r}),\quad(h_{1},h_{2})\in\operatorname{GL}_{r}(F)\times\operatorname{SO}_{2k-4r+1}(F).$

The conjugation action of
$\operatorname{GL}_{r}(F)\times\operatorname{SO}_{2k-4r+1}(F)$ on the quotient
$[H_{r,2k-2r+1}/Z(H_{r,2k-2r+1})]$ induces an action on the character group of
the latter, which may be identified with
$\operatorname{Mat}_{r\times(2k-4r+1)}(F)$. For any
$\xi\in\operatorname{Mat}_{r\times(2k-4r+1)}(F)$ with
$\operatorname{Rank}(\xi)=q,\,1\leqslant q\leqslant r$, if any of its row
vectors is non-isotropic with respect to the bilinear form on $F^{2k-4r+1}$
given by

$(x,y)\mapsto xJ_{2k-4r+1}y^{T},$

then the corresponding contribution is given by the integral

(4.12)
$\int_{[H_{r,2k+1}]}\int_{[H_{r,2k-2r+1}]}\theta(uvg)\psi_{1}(u)\psi_{\xi}(v)\,dvdu.$

The product of the characters $\psi_{1}$ and $\psi_{\xi}$ is a generic
character associated to the unipotent orbit $(5^{q}3^{r-q}1^{2k-3r-2q+1})$.
Thus, the integral (4.12) contains a Fourier coefficient of $\Theta_{2k+1}$
associated to the unipotent orbit $(5^{q}3^{r-q}1^{2k-3r-2q+1})$, which is
zero by Proposition 4.4. Under the conjugation action by
$\operatorname{GL}_{r}(F)\times\operatorname{SO}_{2k-4r+1}(F)$, any
$\xi\in\operatorname{Mat}_{r\times(2k-4r+1)}(F)$ whose row space is not
totally isotropic lies in the same orbit as the non-isotropic ones with the
same rank. The same argument shows that the corresponding contribution is
zero. Therefore, the only possible non-zero contributions are from those
$\xi\in\operatorname{Mat}_{r\times(2k-4r+1)}(F)$ whose row space is totally
isotropic. We may pick the representatives of these orbits to be

$z_{q}=\begin{pmatrix}I_{q}&0\\\
0&0\end{pmatrix}\in\operatorname{Mat}_{r\times(2k-4r+1)}(F),q=0,1,\cdots,r.$

For any $z_{q}\in\operatorname{Mat}_{r\times(2k-4r+1)}(F)$, the correponding
contribution in the expansion is

(4.13)
$\int_{[H_{r,2k+1}]}\int_{[H_{r,2k-2r+1}]}\theta(uvzg)\psi_{1}(u)\psi_{2,q}(v)\,dvdu,$

where the character $\psi_{2,q}$ corresponding to $z_{q}$ is the trivial
character when $q=0$ and is given by

$\psi_{2,q}(u)=\psi(u_{1,1}+u_{2,2}+\cdots+u_{q,q}),\quad\forall u\in
H_{r,2k-2r+1}/Z(H_{r,2k-2r+1})(\mathbb{A})\cong\operatorname{Mat}_{r\times(2k-4r+1)}(\mathbb{A})$

when $q>0$. We claim that the only non-zero contribution is when $q=0$.
Suppose on the contrary that $q>0$. We can further expand (4.13) against
$[Z(H_{r,2k-4r+1})]$ and follow by another expansion against the abelian
quotient $[H_{r,2k-4r+1}/Z(H_{r,2k-4r+1})]$. Here, $H_{r,2k-4r+1}$ is
similarly defined as the unipotent subgroup of $\operatorname{SO}_{2k+1}$
consisting of matrices of the form

$\left\\{\begin{pmatrix}I_{2r}&&&&\\\ &I_{r}&y&\ast&\\\
&&I_{2k-6r+1}&y^{\ast}&\\\ &&&I_{r}&\\\ &&&&I_{2r}\end{pmatrix},\quad
y\in\operatorname{Mat}_{r\times(2k-6r+1)}\right\\}.$

For any character $\psi^{\ast}$ on $[H_{r,2k-4r+1}/Z(H_{r,2k-4r+1})]$, the
corresponding contribution is

(4.14)
$\int_{[H_{r,2k+1}]}\int_{[H_{r,2k-2r+1}]}\int_{[H_{r,2k-4r+1}]}\theta(uvwg)\psi_{1}(u)\psi_{2,q}(v)\psi^{\ast}(w)\,dwdvdu.$

Any character $\psi^{\ast}$ corresponding to a matrix in
$\operatorname{Mat}_{r\times(2k-6r+1)}(F)$ that contains non-isotropic row
vectors in $F^{2k-6r+1}$ again contributes zero.

We claim that the constant term is also zero. If we denote

$H_{r}=H_{r,2k+1}H_{r,2k-2r+1}H_{r,2k-4r+1},$

then the constant term is of the form

(4.15) $\int_{[H_{r}]}\theta(ug)\psi_{1,q}(u)\,du.$

Here, if $u=u_{1}u_{2}u_{3}$ with $u_{1}\in H_{r,2k+1}(\mathbb{A}),u_{2}\in
H_{r,2k-2r+1}(\mathbb{A})$ and $u_{3}\in H_{r,2k-4r+1}(\mathbb{A})$, then

$\psi_{1,q}(u)=\psi_{1}(u_{1})\psi_{2,q}(u_{2}).$

This integral is a Fourier coefficient of the constant term of $\Theta_{2k+1}$
with respect to the parabolic subgroup whose Levi part is
$\operatorname{GL}_{3r}\times\operatorname{SO}_{2k-6r+1}$. By Proposition 4.3,
we may regard this integral as a function in the representation
$\Theta_{\operatorname{GL}_{3r}}\otimes\Theta_{2k-6r+1}$ of
$\widetilde{\operatorname{GL}}_{3r}(F)\times\widetilde{\operatorname{SO}}_{2k-6r+1}(F)$.
According to [6], the corresponding Fourier coefficient of
$\Theta_{\operatorname{GL}_{3r}}$ is the semi-Whittaker coefficient associated
with the partition $\Lambda=(3^{q}2^{r-q}1^{r-q})$. Following [6], let
$P_{\Lambda}=M_{\Lambda}U_{\Lambda}$ be the standard parabolic subgroup of
$\operatorname{GL}_{3r}$ with the Levi subgroup
$M_{\Lambda}\cong\operatorname{GL}^{q}_{3}\times\operatorname{GL}_{2}^{r-q}\times\operatorname{GL}_{1}^{r-qr}$
and $U_{\Lambda}$ the unipotent radical. Let $U$ be the unipotent radical of
the standard Borel subgroup of $\operatorname{GL}_{3r}$ and
$\psi_{\Lambda}:U(F)\setminus U(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ be
the character such that it acts non-trivially as $\psi$ on the one-parameter
subgroup corresponding to simple positive root contained in $M_{\Lambda}$ and
act trivially otherwise. Then the $\Lambda$-semi-Whittaker coefficient of any
function $\theta_{\operatorname{GL}_{3r}}$ in
$\Theta_{\operatorname{GL}_{3r}}$ is given by

(4.16) $\int_{U(F)\setminus
U(\mathbb{A})}\theta_{\operatorname{GL}_{3r}}(ug)\psi_{\Lambda}(u)\,du.$

By Proposition 4.1 of [6], any such semi-Whittaker coefficient is zero.

Therefore, we only need to consider those terms corresponding to characters on
$[H_{r,2k-4r+1}]$ represented by

(4.17)
$\psi_{3,q^{\prime}}(u)=\psi(y_{1,1}+y_{2,2}+\cdots+y_{q^{\prime},q^{\prime}}),$

with $1<q^{\prime}\leqslant r$ and

(4.18) $u=\begin{pmatrix}I_{2r}&&&&\\\ &I_{r}&y&\ast&\\\
&&I_{2k-6r+1}&y^{\ast}&\\\ &&&I_{r}&\\\ &&&&I_{2r}\end{pmatrix}\in
H_{r,2k-4r+1}(\mathbb{A}),y=(y_{i,j})\in\operatorname{Mat}_{r\times(2k-6r+1)}(\mathbb{A}).$

We continue this argument by further expanding the integral (4.17) against
relevant unipotent subgroups. For each step, we either obtain a Fourier
coefficient of $\Theta_{2k+1}$ associated to unipotent orbit that is
incomparable to $\mathcal{O}(\Theta_{2k+1})$, or we obtain some semi-Whittaker
coefficient on the double cover $\widetilde{\operatorname{GL}}_{nr}$
corresponding to a partition of the integer $nr$ (with $n>3$) that contains an
integer greater than $2$. The former is zero by Proposition 4.4, while the
latter is also zero by Proposition 4.1 of [6]. This shows that the only
contribution of the expansion of (4.11) against
$[H_{r,2k-2r+1}/Z(H_{r,2k-2r+1})]$ is the constant term, which completes the
proof. ∎

For intergers $r_{1}\leqslant r_{2}\leqslant\cdots\leqslant r_{n}$, we further
denote $H(r_{1},r_{2},\cdots,r_{n-1};r_{n})$ the unipotent radical of the
parabolic subgroup of $\operatorname{SO}_{2k+1}$ with Levi subgroup

$L(r_{1},r_{2},\cdots,r_{n-1};r_{n})=\operatorname{GL}^{2}_{r_{1}}\times\operatorname{GL}^{2}_{r_{2}}\times\cdots\times\operatorname{GL}^{2}_{r_{n-1}}\times\operatorname{GL}_{r_{n}}\times\operatorname{SO}_{2k-4(r_{1}+\cdots+r_{n-1})-2r_{n}+1}.$

We also let

$L=\operatorname{GL}_{2r_{1}}\times\operatorname{GL}_{2r_{2}}\times\cdots\times\operatorname{GL}_{2r_{n-1}}\times\operatorname{SO}_{2k-4(r_{1}+\cdots+r_{n-1})+1}.$

Denote the diagonal embedding
$\iota^{\ast}:L(r_{1},r_{2},\cdots,r_{n-1};r_{n})\subset
L\hookrightarrow\operatorname{SO}_{2k+1}$.

For each $i=1,2,\cdots,n-1$, let $H_{r_{i}}$ be the unipotent radical of the
standard Siegel parabolic subgroup of $\operatorname{GL}_{2r_{i}}$ whose Levi
part is $\operatorname{GL}_{r_{i}}\times\operatorname{GL}_{r_{i}}$. Also, let
$H_{r_{n}}$ be the unipotent radical of the maximal parabolic subgroup of
$\operatorname{SO}_{2k-4(r_{1}+\cdots+r_{n-1})+1}$ with Levi subgroup
$\operatorname{GL}_{r_{n}}\times\operatorname{SO}_{2k-4(r_{1}+\cdots+r_{n-1})-2r_{n}+1}$.
Then we define

$H^{0}:=H_{r_{1}}\times H_{r_{2}}\times\cdots\times H_{r_{n-1}}\times
H_{r_{n}}\subseteq H(r_{1},r_{2},\cdots,r_{n-1};r_{n}).$

We will define a character on $[H^{0}]$ and then extend it trivially to
$[H(r_{1},r_{2},\cdots,r_{n-1};r_{n})]$. For any $u_{i}\in
H_{r_{i}}(\mathbb{A})$ with $i=1,2,\cdots,n-1$, we may write

$u_{i}=\ \begin{pmatrix}I_{r_{i}}&x_{i}\\\ &I_{r_{i}}\end{pmatrix},\quad
x_{i}\in\operatorname{Mat}_{r_{i}\times r_{i}}(\mathbb{A}).$

Define $\psi_{r_{i}}:H_{r_{i}}(F)\setminus
H_{r_{i}}(\mathbb{A})\rightarrow\mathbb{C}^{\times}$ by

$\psi_{r_{i}}(u_{i})=\psi(\operatorname{Tr}(x_{i})).$

For the factor $H_{r_{n}}$, define the character by

$\psi_{r_{n}}(u)=\psi(\sum_{j=1}^{r_{n}}u_{j,j+r_{n}}),\quad u\in
H_{r_{n}}(\mathbb{A}).$

Pulling back each $\psi_{r_{i}}$ via the projection of $H^{0}$ onto the
corresponding factor and taking the product afterwards, we obtain
$\psi_{n}:=\prod_{i=1}^{n}\psi_{r_{i}}:[H^{0}]\rightarrow\mathbb{C}^{\times}$.
Extend $\psi_{n}$ trivially to $[H(r_{1},r_{2},\cdots,r_{n-1};r_{n})]$.
Furthermore, let $H_{n}$ be the unipotent radical of the standard maximal
parabolic subgroup of
$\operatorname{SO}_{2k-4(r_{1}+\cdots+r_{n-1})-2r_{n}+1}$ with Levi subgroup
given by

$\operatorname{GL}_{r_{n}}\times\operatorname{SO}_{2k-4(r_{1}+\cdots+r_{n-1})-2r_{n}+1}.$

Embed $H_{n}$ into $\operatorname{SO}_{2k+1}$ via (2.1) and still denote its
image by $H_{n}$. Proposition 4.5 admits a straightforward corollary.

######  Corollary 4.6.

The function

(4.19)
$f(g)=\int_{[H(r_{1},r_{2},\cdots,r_{n-1};r_{n})]}\theta(ug)\psi_{n}(u)\,du$

is left invariant under $H_{n}(\mathbb{A})$.

###### Proof.

Apply Proposition 4.3 with $r=2(r_{1}+r_{2}+\cdots+r_{n-1})$. Then apply
Proposition 4.5 on the theta representation of the smaller orthogonal group. ∎

###  4.3. The tower of the theta liftings

Let $(\pi,\mathcal{V})$ be an irreducible cuspidal genuine automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Suppose
$\operatorname{SO}_{2k^{\prime}}$ is a split even orthogonal group. By
identifying $\operatorname{SO}_{2k^{\prime}}\times\operatorname{SO}_{2k+1}$
with its embedded image in $\operatorname{SO}_{2k+2k^{\prime}+1}$ via (2.1),
we consider functions on
$\widetilde{\operatorname{SO}}_{2k^{\prime}}(\mathbb{A})$ of the form

(4.20)
$f(h)=\int_{[\operatorname{SO}_{2k+1}]}\varphi(g)\bar{\theta}_{2k+2k^{\prime}+1}(h,g)\,dg,$

where $\varphi$ is any vector in $\mathcal{V}$ and $\theta_{2k+2k^{\prime}+1}$
is any function in the representation space of $\Theta_{2k+2k^{\prime}+1}$.
This integral defines a map from the irreducible cuspidal genuine automorphic
representations on $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ to a
genuine automorphic representation $\Theta_{2k+2k^{\prime}+1}(\pi)$ on
$\widetilde{\operatorname{SO}}_{2k^{\prime}}(\mathbb{A})$.

By fixing the representation $\pi$ of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ and varying the theta
representations $\Theta_{2k+2k^{\prime}+1}$ of
$\widetilde{\operatorname{SO}}_{2k+2k^{\prime}+1}(\mathbb{A})$ with increasing
$k^{\prime}$, we obtain a tower of liftings of representations on
$\widetilde{\operatorname{SO}}_{2k^{\prime}}(\mathbb{A})$:

${\Theta_{2k+2k^{\prime}+3}(\pi)}$${\Theta_{2k+2k^{\prime}+1}(\pi)}$${\vdots}$${\pi}$${\Theta_{2k+3}(\pi).}$$\scriptstyle{\vdots}$

In [5], Bump-Friedberg-Ginzburg show that if
$\Theta_{2k+2k^{\prime}+1}(\pi)=0$, then $\Theta_{2k+2k^{\prime}-1}(\pi)=0$.
It is also proved in [5] that any genuine cuspidal automorphic representation
$\pi$ of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ lifts nontrivially
to an automorphic representation on
$\widetilde{\operatorname{SO}}_{8k}(\mathbb{A})$. This raises the question of
when the first non-zero theta lifting occurs along the tower for a fixed
$\pi$. In the case when $\pi$ is generic, a conjecture in [5] states that
$\pi$ should lift non-trivially to an automorphic representation of
$\widetilde{\operatorname{SO}}_{2k+4}(\mathbb{A})$. Those authors also proved
the following result:

######  Theorem 4.7.

Let $\pi$ be an irreducible cuspidal genuine automorphic representation of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. If the representation
$\Theta_{4k+5}(\pi)$ of $\widetilde{\operatorname{SO}}_{2k+4}(\mathbb{A})$ is
generic, then the representation $\pi$ is also generic.

On the other hand, there is not much known yet for the theta liftings when
$\pi$ is non-generic. Motivated by the generic case, we make the following
more general conjecture:

######  Conjecture 4.8.

Let $(\pi,\mathcal{V})$ be an irreducible cuspidal genuine automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Suppose

$\mathcal{O}=\left((2n_{1}+1)^{r_{1}}(2n_{2}+1)^{r_{2}}\cdots(2n_{p}+1)^{r_{p}}\right)\in\mathcal{O}(\pi)$

with $n_{1}>n_{2}>\cdots>n_{p}\geqslant 0$ and $r_{i}>0$ for all $i$. Then
$\pi$ lifts nontrivially to an automorphic representation $\Theta(\pi)$ of
$\widetilde{\operatorname{SO}}_{2k+2l+2}(\mathbb{A})$ such that

$\mathcal{O}^{\prime}=\left((2n_{1}+3)^{r_{1}}(2n_{2}+3)^{r_{2}}\cdots(2n_{p}+3)^{r_{p}}(1)\right)\in\mathcal{O}\left(\Theta(\pi)\right).\\\
$

Recall that $l=r_{1}+r_{2}+\cdots+r_{p}$ is the length of the partition
corresponding to $\mathcal{O}$. If $\pi$ is an irreducible cuspidal generic
automorphic representation of
$\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$, then
$\mathcal{O}(\pi)=(2k+1)$. Conjecture 4.8 predicts that it lifts to an
automorphic representation $\Theta(\pi)$ on
$\widetilde{\operatorname{SO}}_{2k+4}(\mathbb{A})$ with
$\mathcal{O}(\Theta(\pi))=((2k+3)(1))$, which agrees with the conjecture
proposed in [5].

For the remaining of this section, assume that the wave front set
$\mathcal{O}(\rho)$ is a singleton set for an automorphic representation
$\rho$. Recall that the Gelfand-Kirillov dimension of the representation
$\rho$ (see [7]) is given by

$\dim(\rho)=\frac{1}{2}\dim(\mathcal{O}(\rho))=\dim(V_{2,\mathcal{O}(\rho)})+\frac{1}{2}\dim(V_{1,\mathcal{O}(\rho)}/V_{2,\mathcal{O}(\rho)}).$

######  Proposition 4.9.

Suppose $(\pi,\mathcal{V})$ is an irreducible cuspidal genuine automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$ with
$\mathcal{O}(\pi)=\mathcal{O}$ such that its theta lifting $\Theta(\pi)$ on
$\widetilde{\operatorname{SO}}_{2k+2l+2}(\mathbb{A})$ has
$\mathcal{O}\left(\Theta(\pi)\right)=\mathcal{O}^{\prime}$. Then

(4.21)
$\dim(\operatorname{SO}_{2k+1})+\dim(\Theta(\pi))=\dim(\pi)+\dim(\Theta_{4k+2l+3}).$

Before we verify (4.21), we need to fix some notations. Suppose, as in
Conjecture 4.8,

$\mathcal{O}=\left((2n_{1}+1)^{r_{1}}(2n_{2}+1)^{r_{2}}\cdots(2n_{p}+1)^{r_{p}}\right),\quad
n_{1}>n_{2}>\cdots>n_{p}\geqslant 0.$

Denote

(4.22) $s_{i}:=\sum_{j=1}^{i}r_{j},\quad i=1,2,\cdots,p.$

In particular, $l=s_{p}$ is the length of the partition corresponding to
$\mathcal{O}$. The fact that $\mathcal{O}$ is odd implies that
$V_{1,\mathcal{O}}=V_{2,\mathcal{O}}$, and both of these are the unipotent
radical of the parabolic subgroup $P_{\mathcal{O}}$ whose Levi part is

$M(\mathcal{O})=\operatorname{GL}^{n_{1}-n_{2}}_{s_{1}}\times\operatorname{GL}^{n_{2}-n_{3}}_{s_{2}}\times\cdots\times\operatorname{GL}^{n_{p-1}-n_{p}}_{s_{p-1}}\times\operatorname{GL}_{l}^{n_{p}}\times\operatorname{SO}_{l}.$

For simplicity, we denote
$U_{\mathcal{O}}=V_{1,\mathcal{O}}=V_{2,\mathcal{O}}$.

Likewise, we denote by $V_{\mathcal{O}^{\prime}}$ the unipotent subgroup
$V_{1,\mathcal{O}^{\prime}}=V_{2,\mathcal{O}^{\prime}}$ associated to the
unipotent orbit

$\mathcal{O}^{\prime}=\left((2n_{1}+3)^{r_{1}}(2n_{2}+3)^{r_{2}}\cdots(2n_{p}+3)^{r_{p}}(1)\right)$

in $\operatorname{SO}_{2k+2l+2}$. The corresponding Levi subgroup is

$M(\mathcal{O}^{\prime})=\operatorname{GL}^{n_{1}-n_{2}}_{s_{1}}\times\operatorname{GL}^{n_{2}-n_{3}}_{s_{2}}\times\cdots\times\operatorname{GL}^{n_{p-1}-n_{p}}_{s_{p-1}}\times\operatorname{GL}^{n_{p}+1}_{l}\times\operatorname{SO}_{l+1}.$

###### Proof.

Observe that $\mathcal{O}(\Theta_{4k+2l+3})=(2^{2k+l+1}1)$ since $l$ is odd,
and $\dim(\Theta_{4k+2l+3})=\frac{(2k+l+1)^{2}}{2}$. Also,
$\dim(\operatorname{SO}_{2k+1})=2k^{2}+k$. Although the dimension of the
representations $\Theta_{4k+2l+3}(\pi)$ and $\pi$ may vary, it suffices to
check that the difference between the dimensions agrees with that between
$\dim(\Theta_{4k+2l+3})$ and $\dim(\operatorname{SO}_{2k+1})$.

Notice that the difference between $\dim(\pi)$ and the dimension of the
unipotent radical of the Borel subgroup of $\operatorname{SO}_{2k+1}$ is
related to that between $\dim(\Theta_{4k+2l+3}(\pi))$ and the dimension of the
unipotent radical of the Borel subgroup of $\operatorname{SO}_{2k+2l+2}$. The
former is precisely the dimension of the unipotent radical of the Borel
subgroup of $M(\mathcal{O})$. We denote this dimension by
$t+\frac{(l-1)^{2}}{4}$, where $\frac{(l-1)^{2}}{4}$ and $t$ are the
dimensions of the maximal unipotent subgroup of the factor
$\operatorname{SO}_{l}$ and the remaining Levi factors respectively. Hence, we
obtain that $\dim(\pi)=k^{2}-t-\frac{(l-1)^{2}}{4}$. Similarly, we have
$\dim(\Theta_{4k+2l+3}(\pi))=(k+l+1)(k+l)-t-\frac{l^{2}-1}{4}-\frac{(l-1)(l)}{2}$.
Therefore, the difference is exactly

(4.23)
$\frac{l^{2}+1}{2}+2kl+k+l=\frac{(2k+l+1)^{2}}{2}-(2k^{2}+k)=\dim(\Theta_{4k+2l+3})-\dim(\operatorname{SO}_{2k+1}).$

∎

Proposition 4.9 shows that Conjecture 4.8 agrees with the dimension equation
proposed in [8]. The general philosophy of the dimension equation is that the
sum of the dimensions of the representations is equal to the sum of the
dimensions of the groups in the domain of integration in a global unipotent
integral. In our case, this is given by equation (4.21). Refer to [8], [9] and
[7] for more details on dimension equations.

##  5\. Global theory

We follow the notations in and after Conjecture 4.8. Suppose
$(\pi,\mathcal{V})$ is an irreducible genuine cuspidal automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Let
$\mathcal{O}$ be the unipotent orbit of $\operatorname{SO}_{2k+1}$ such that

$\mathcal{O}=\left((2n_{1}+1)^{r_{1}}(2n_{2}+1)^{r_{2}}\cdots(2n_{p}+1)^{r_{p}}\right),\quad
n_{1}>n_{2}>\cdots>n_{p}\geqslant 0,\quad r_{i}>0\quad\forall i=1,2,\cdots,p.$

Let $l=r_{1}+r_{2}+\cdots+r_{p}$ be the length of the partition corresponding
to $\mathcal{O}$. Suppose $\Theta_{4k+2l+3}(\pi)$ is the automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+2l+2}(\mathbb{A})$
obtained by the theta lifting from $\pi$ via integrating functions in
$\mathcal{V}$ against the theta integral kernel $\Theta_{4k+2l+3}$ in the form
of (4.20). As we are only concerned about a fixed theta lifting in this
section, we suppress the subscript and simply let
$\Theta(\pi)=\Theta_{4k+2l+3}(\pi)$. We denote by $\mathcal{O}^{\prime}$ the
unipotent orbit of the group $\operatorname{SO}_{2k+2l+2}$ associated to the
partition

$\mathcal{O}^{\prime}=\left((2n_{1}+3)^{r_{1}}(2n_{2}+3)^{r_{2}}\cdots(2n_{p}+3)^{r_{p}}(1)\right)$

Recall that
$V_{\mathcal{O}^{\prime}}=V_{1,\mathcal{O}^{\prime}}=V_{2,\mathcal{O}^{\prime}}$
is the unipotent radical of the parabolic subgroup of
$\operatorname{SO}_{2k+2l+2}$ whose Levi subgroup is

$M(\mathcal{O}^{\prime})=\operatorname{GL}^{n_{1}-n_{2}}_{s_{1}}\times\operatorname{GL}^{n_{2}-n_{3}}_{s_{2}}\times\cdots\times\operatorname{GL}^{n_{p-1}-n_{p}}_{s_{p-1}}\times\operatorname{GL}^{n_{p}+1}_{l}\times\operatorname{SO}_{l+1}.$

Notice that

(5.1)
$V_{\mathcal{O}^{\prime}}/V_{\mathcal{O}^{\prime}}^{(1)}\cong\Bigg{(}\bigoplus_{j=1}^{p-1}\operatorname{Mat}_{s_{j}\times
s_{j+1}}\Bigg{)}\oplus\Bigg{(}\bigoplus_{j=1}^{p-1}\operatorname{Mat}_{s_{j}\times
s_{j}}^{n_{j}-n_{j+1}-1}\Bigg{)}\oplus\operatorname{Mat}_{l\times
l}^{n_{p}}\oplus\operatorname{Mat}_{l\times(l+1)}.$

We first define a character on
$[V_{\mathcal{O}^{\prime}}/V_{\mathcal{O}^{\prime}}^{(1)}]$ by specifying it
on each of the components, and then extend it trivially to a character on
$[V_{\mathcal{O}^{\prime}}]$. For any abelian group
$\operatorname{Mat}_{i\times j}$, we may identify the character group of
$[\operatorname{Mat}_{i\times j}]$ with $\operatorname{Mat}_{i\times j}(F)$
via

(5.2) $X\in\operatorname{Mat}_{i\times
j}(F)\mapsto\psi_{X}:Y\mapsto\psi(XY^{T}).$

Recall

(5.3) $s_{i}:=\sum_{j=1}^{i}r_{j},\quad i=1,2,\cdots,p.$

Let

$I_{s_{j},s_{j+1}}=\begin{pmatrix}I_{s_{j}}&0\end{pmatrix}\in\operatorname{Mat}_{s_{j}\times
s_{j+1}}(F),\quad j=1,2,\cdots,p-1,$

and

$I_{l,a}=\begin{pmatrix}I_{\frac{l-1}{2}}&&&\\\ &\frac{1}{2}&-a&\\\
&&&-I_{\frac{l-1}{2}}\end{pmatrix}\in\operatorname{Mat}_{l\times(l+1)}(F),\quad
a\in F^{\times}.$

Consider the following characters each defined on the respective component in
(5.1):

(5.4) $\begin{cases}\psi_{s_{j}\times
s_{j}}(v)=\psi(\operatorname{Tr}v)&\text{if}\;v\in\operatorname{Mat}_{s_{j}\times
s_{j}}(\mathbb{A}),j=1,2,\cdots,p,\\\ \psi_{s_{j}\times
s_{j+1}}(v)=\psi(\operatorname{Tr}(I_{s_{j},s_{j+1}}v^{T}))&\text{if}\;v\in\operatorname{Mat}_{s_{j}\times
s_{j+1}}(\mathbb{A}),j=1,2,\cdots,p-1,\\\
\psi_{l\times(l+1)}(v)=\psi(\operatorname{Tr}(I_{l,a}v^{T}))&\text{if}\;v\in\operatorname{Mat}_{l\times(l+1)}(\mathbb{A}).\end{cases}$

Pulling back each of these characters via the projection map onto the
respective component and taking the product afterwards, we obtain a character
on $[V_{\mathcal{O}^{\prime}}/V_{\mathcal{O}^{\prime}}^{(1)}]$. Extend it to a
character on $[V_{\mathcal{O}^{\prime}}]$ and denote the resulting character
by $\psi_{a,V_{\mathcal{O}^{\prime}}}$. This is a generic character attached
to the unipotent orbit $\mathcal{O}^{\prime}$.

For any automorphic function in $\Theta(\pi)$ of the form

$f(h)=\int_{\operatorname{SO}_{2k+1}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\varphi(g)\overline{\theta_{4k+2l+3}(h,g)}\,dg,$

we let

(5.5)
$F_{\psi_{a,V_{\mathcal{O}^{\prime}}}}(f)(h)=\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}]}\varphi(g)\overline{\theta_{4k+2l+3}(vh,g)}\psi_{a,V_{\mathcal{O}^{\prime}}}(v)\,dvdg.$

The maximal split torus of $\operatorname{SO}_{4k+2l+3}(F)$ normalizes
$V_{\mathcal{O}^{\prime}}$. Conjugating the variable $v$ in the inner
integration of (5.5) by
$\tau=\operatorname{diag}(t,\cdots,t,1,t^{-1},\cdots,t^{-1})),t\in F^{\times}$
leaves the integral unchanged. The automorphicity of $\theta_{4k+2l+3}$
implies that it is left invariant by $\tau$. After a change of variables by
$v\mapsto v\tau^{-1}$, we see that the Fourier coefficient depends only on the
square class of $a$ in $F^{\times}$. When $a$ is a square, the connected
component of the stabilizer of $\psi_{a,V_{\mathcal{O}^{\prime}}}$ in
$M(\mathcal{O}^{\prime})(F)$ is split. In this case, we call
$\psi_{a,V_{\mathcal{O}^{\prime}}}$ a split generic character, and denote it
by $\psi_{\mathcal{O}^{\prime}}$.

On the other hand, recall
$U_{\mathcal{O}}=V_{1,\mathcal{O}}=V_{2,\mathcal{O}}$ is the unipotent radical
of the maximal parabolic subgroup $P_{\mathcal{O}}$ of
$\operatorname{SO}_{2k+1}$ with the corresponding Levi part

$M(\mathcal{O})=\operatorname{GL}^{n_{1}-n_{2}}_{s_{1}}\times\operatorname{GL}^{n_{2}-n_{3}}_{s_{2}}\times\cdots\times\operatorname{GL}^{n_{p-1}-n_{p}}_{s_{p-1}}\times\operatorname{GL}_{l}^{n_{p}}\times\operatorname{SO}_{l}.$

In order to define a generic character on $[U_{\mathcal{O}}]$, it suffices to
specify the respective character on each of the components of the maximal
abelian quotient

(5.6)
$U_{\mathcal{O}}/U_{\mathcal{O}}^{(1)}\cong\Bigg{(}\bigoplus_{j=1}^{p-1}\operatorname{Mat}_{s_{j}\times
s_{j+1}}\Bigg{)}\oplus\Bigg{(}\bigoplus_{j=1}^{p-1}\operatorname{Mat}_{s_{j}\times
s_{j}}^{n_{j}-n_{j+1}-1}\Bigg{)}\oplus\operatorname{Mat}_{l\times
l}^{n_{p}-1}.$

We define these characters by

(5.7) $\begin{cases}\psi_{s_{j}\times
s_{j}}(v)=\psi(\operatorname{Tr}v)&\text{if}\;v\in\operatorname{Mat}_{s_{j}\times
s_{j}}(\mathbb{A}),j=1,2,\cdots,p,\\\ \psi_{s_{j}\times
s_{j+1}}(v)=\psi(\operatorname{Tr}(I_{s_{j},s_{j+1}}v^{T}))&\text{if}\;v\in\operatorname{Mat}_{s_{j}\times
s_{j+1}}(\mathbb{A}),j=1,2,\cdots,p-1.\end{cases}$

This gives a generic character
$\psi_{\mathcal{O}}:[U_{\mathcal{O}}]\rightarrow\mathbb{C}^{\times}$ .

######  Theorem 5.1.

Let $(\pi,\mathcal{V})$ be an irreducible cuspidal genuine automorphic
representation of $\widetilde{\operatorname{SO}}_{2k+1}(\mathbb{A})$. Suppose
the theta lifting $\Theta(\pi)$, as a representation of
$\widetilde{\operatorname{SO}}_{2k+2l+2}(\mathbb{A})$, has a non-zero Fourier
coefficient with respect to a split generic character associated with the
unipotent orbit

$\mathcal{O}^{\prime}=\left((2n_{1}+3)^{r_{1}}(2n_{2}+3)^{r_{2}}\cdots(2n_{p}+3)^{r_{p}}(1)\right).$

Then the representation $\pi$ has a non-zero Fourier coefficient with respect
to some generic character associated with the unipotent orbit

$\mathcal{O}=\left((2n_{1}+1)^{r_{1}}(2n_{2}+1)^{r_{2}}\cdots(2n_{p}+1)^{r_{p}}\right).$

###### Proof.

Throughout the proof, we identify any subgroup of $\operatorname{SO}_{2k+1}$
or $\operatorname{SO}_{2k+2l+2}$ with its embedded image in
$\operatorname{SO}_{4k+2l+3}$ via 2.1. The Fourier coefficient of
$\Theta(\pi)$ with respect to a generic character depends only on the square
class of $a$. Therefore, we may assume there exists data such that the
following integral is non-vanishing:

(5.8)
$F_{\psi_{\mathcal{O}^{\prime}}}(f)(1)=\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}]}\varphi(g)\overline{\theta}_{4k+2l+3}(v,g)\psi_{\mathcal{O}^{\prime}}(v)\,dvdg.$

Denote by $R_{s_{1}}=R_{s_{1},4k+2l+3}$ the unipotent radical of the maximal
parabolic subgroup of $\operatorname{SO}_{4k+2l+3}$ with Levi subgroup
$\operatorname{GL}_{s_{1}}\times\operatorname{SO}_{4k+2l-2s_{1}+3}$. Notice
that $V_{s_{1}}=V_{\mathcal{O}^{\prime}}\cap R_{s_{1}}$ is non-trivial. The
quotient $V_{s_{1}}\setminus R_{s_{1}}$ may be identified with the subgroup of
matrices of the form

$H_{s_{1}}=H_{s_{1},4k+2l+3}:=\left\\{\begin{pmatrix}I_{s_{1}}&&x&&\ast\\\
&I_{k+l-s_{1}+1}&&&\\\ &&I_{2k+1}&&x^{\ast}\\\ &&&I_{k+l-s_{1}+1}&\\\
&&&&I_{s_{1}}\end{pmatrix}\in\operatorname{SO}_{4k+2l+3}:x\in\operatorname{Mat}_{s_{1}\times(2k+1)}\right\\}.$

Although $H_{s_{1}}$ is not abelian, it is a Heisenberg group with
$Z(H_{s_{1}})$ corresponding to matrices of the form

$\left\\{\begin{pmatrix}I_{s_{1}}&0&z\\\ &I_{4k+2l-2s_{1}+3}&0\\\
&&I_{s_{1}}\end{pmatrix}:z^{T}J_{s_{1}}+J_{s_{1}}z=0\right\\}.$

Notice that the center $Z(H_{s_{1}})(\mathbb{A})\subset
V_{\mathcal{O}^{\prime}}(\mathbb{A})$ is included in the domain of integration
in (5.8). As a result, we expand the integral (5.8) against the abelian
quotient
$\left[\left(H_{s_{1}}/Z(H_{s_{1}})\right)\right]\cong[\operatorname{Mat}_{s_{1}\times(2k+1)}]$.

We may identify the character group of
$\left[\left(H_{s_{1}}/Z(H_{s_{1}})\right)\right]$ with
$\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$. We claim that the only non-zero
contributions from the expansion are those characters corresponding to
matrices in $\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$ with maximal rank and
totally-isotropic row space.

First, we look at the contribution from the trivial character, which
corresponds to the zero matrix in $\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$.
This is the constant term of the Fourier expansion given by

(5.9)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}]}\int_{[H_{s_{1}}/Z(H_{s_{1}})]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u(v,g)\right)\psi_{\mathcal{O}^{\prime}}(v)\,dudvdg.$

Write $v=v_{s_{1}}v^{1}$ where $v_{s_{1}}\in V_{s_{1}}(\mathbb{A})$ and
$v^{1}\in
V_{\mathcal{O}^{\prime}}^{1}(\mathbb{A}):=V_{\mathcal{O}^{\prime}}(\mathbb{A})\cap\operatorname{SO}_{2k+2l-2s_{1}+2}(\mathbb{A})$.
Combining the two variables $u$ and $v_{s_{1}}$, we obtain

(5.10)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s_{1}}]}\varphi(g)\overline{\theta_{4k+2l+3}\left(u(v^{1},g)\right)}\psi_{1}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg,$

where $\psi_{1}$ is the character on $[R_{s_{1}}]$ given by

(5.11) $\psi_{1}(u)=\psi(u_{1,s+1}+u_{2,s+2}+\cdots+u_{s+2s}).$

Applying Proposition 4.5, we see that (5.10) is equal to the integral

(5.12)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s^{2}_{1}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u(v^{1},g)\right)\psi_{1}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg,$

where we extend $\psi_{1}$ trivially to $[R_{s^{2}_{1}}]$ and
$R_{s^{2}_{1}}=R_{s_{1},4k+2l+3}R_{s_{1},4k+2l-2s_{1}+3}$ is the unipotent
radical of the maximal parabolic subgroup of $\operatorname{SO}_{4k+2l+3}$
with Levi subgroup

$\operatorname{GL}_{s_{1}}\times\operatorname{GL}_{s_{1}}\times\operatorname{SO}_{4k+2l-4s_{1}+3}.$

Notice that $R_{s^{2}_{1}}\cap V_{\mathcal{O}^{\prime}}^{1}$ is non-trivial.
Let $\beta$ be the root inside $\operatorname{SO}_{2k+2l+2}$ such that

(5.13)
$\beta=\begin{cases}\sum_{j=1}^{s_{2}}\alpha_{s_{1}+j}&\text{if}\;n_{1}-n_{2}=1\\\
\sum_{j=1}^{s_{1}}\alpha_{s_{1}+j}&\text{if}\;n_{1}-n_{2}>1,\end{cases}$

where we recall that $\alpha_{i}$’s are the positive simple roots of
$\operatorname{SO}_{2k+2l+2}$. By construction, the one parameter subgroup
$\\{x_{\beta}(r):r\in\mathbb{A}\\}$ associated to $\beta$ is in the
intersection $R_{s^{2}_{1}}(\mathbb{A})\cap
V_{\mathcal{O}^{\prime}}^{1}(\mathbb{A})$, and $\psi_{\mathcal{O}^{\prime}}$
is non-trivial on $x_{\beta}(r)$. We may write (5.12) as

$\int_{[\operatorname{SO}_{2k+1}]}\varphi(g)\int_{V_{\mathcal{O}^{\prime}}^{1}(F)x_{\beta}(\mathbb{A})\setminus
V_{\mathcal{O}^{\prime}}^{1}(\mathbb{A})}\int_{[R_{s^{2}_{1}}]}\int_{\mathbb{A}/F}\overline{\theta}_{4k+2l+3}\left(ux_{\beta}(r)(v^{1},g)\right)\psi_{1}(u)\psi(r)\psi_{\mathcal{O}^{\prime}}(v^{1})\,drdudv^{1}dg\\\
=\left(\int_{\mathbb{A}/F}\psi(r)\,dr\right)\int_{[\operatorname{SO}_{2k+1}]}\varphi(g)\int_{V_{\mathcal{O}^{\prime}}^{1}(F)x_{\beta}(\mathbb{A})\setminus
V_{\mathcal{O}^{\prime}}^{1}(\mathbb{A})}\int_{[R_{s^{2}_{1}}]}\overline{\theta}_{4k+2l+3}\left(u(v^{1},g)\right)\psi_{1}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg.$

This is zero since

$\int_{\mathbb{A}/F}\psi(r)\,dr=0$

for the non-trivial character $\psi$.

Second, we look at the contributions from the non-trivial characters. We
identify every non-trivial character $\psi_{\xi}$ of
$[\left(H_{s_{1}}/Z(H_{s_{1}})\right)]$ with some non-zero
$\xi\in\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$. The conjugation action by
the diagonally embedded group
$\operatorname{GL}_{s_{1}}(F)\times\operatorname{SO}_{2k+1}(F)$ on
$[\left(H_{s_{1}}/Z(H_{s_{1}})\right)]$ induces an action on
$\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$. If $\xi$ contains any row vector
in $F^{2k+1}$ that is non-isotropic, then the corresponding contribution is
the integral

(5.14)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s_{1}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u(v^{1},g)\right)\psi_{1,\xi}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg.$

Here, for any $u=u_{1}u_{2}\in R_{s_{1}}(\mathbb{A})$ with $u_{1}\in
V_{s_{1}}(\mathbb{A}),u_{2}\in H_{s_{1}}(\mathbb{A})$,
$\psi_{1,\xi}(u)=\psi_{1}(u_{1})\psi_{\xi}(u_{2})$ is a generic character
associated with the unipotent orbit corresponding to the partition
$(3^{s_{1}}1^{4k+2l+3-3s_{1}})$. Hence, (5.14) contains a Fourier coefficient
of $\theta_{4k+2l+3}$ associated to the unipotent orbit
$\mathcal{O}_{\xi}=(3^{s_{1}}1^{4k+2l+3-3s_{1}})$, which is zero by
Proposition 4.4.

Under the $\operatorname{GL}_{s_{1}}(F)\times\operatorname{SO}_{2k+1}(F)$
action, any $\xi\in\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$ whose row space
is not totally isotropic lies in the same orbit as the non-isotropic ones with
the same rank. The same argument shows that the contribution of a character
corresponding to any such $\xi$ is zero.

Thus, the only possible non-zero contributions are from $\psi_{\xi}$ with the
row space of the matrix $\xi$ being totally isotropic . All such matrices of a
given rank lie in the same orbit under the
$\operatorname{GL}_{s_{1}}(F)\times\operatorname{SO}_{2k+1}(F)$-action. Thus,
we may pick the representatives as

(5.15) $\xi_{q}=\begin{pmatrix}I_{q}&0\\\
0&0\end{pmatrix}\in\operatorname{Mat}_{s_{1}\times(2k+1)}(F),\quad
q=1,2,\cdots,s_{1}.$

For a given $\xi_{q}$, the contribution of the character
$\psi_{\xi_{q}}:[\left(H_{s_{1}}/Z(H_{s_{1}})\right)]\rightarrow\mathbb{C}^{\times}$
is

(5.16)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s_{1}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u(v^{1},g)\right)\psi_{1,\xi_{q}}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg,$

where $\psi_{1,\xi_{q}}=\psi_{1}\psi_{\xi_{q}}$ is the character on
$[R_{s_{1}}/Z(R_{s_{1}})]$ that corresponds to the matrix

$\begin{pmatrix}I_{s_{1}}&0_{s_{1}\times(k+l+1-2s_{1})}&\xi&0_{s_{1}\times(k+l-s_{1}+1)}\end{pmatrix}\in\operatorname{Mat}_{s_{1}\times(4k+2l-2s_{1}+3)}(F).$

Let $z_{q}\in\operatorname{SO}_{4k+2l+3}(F)$ be the unipotent element

$z_{q}=\begin{pmatrix}I_{s_{1}}&&&&\\\ &\mu_{q}&&&\\\ &&I_{2k+1-2q}&&\\\
&&&\mu_{q}^{\ast}&\\\
&&&&I_{s_{1}}\end{pmatrix},\mu_{q}=\begin{pmatrix}I_{q}&&-I_{q}\\\
&I_{k+l+1-s_{1}-q}&\\\ &&I_{q}\end{pmatrix}.$

Performing a change of variables by $u\mapsto uz_{q}$, (5.16) is equal to

(5.17)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s_{1}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(uz_{q}(v^{1},g)\right)\psi^{\prime}_{1,\xi_{q}}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg,$

where $\psi^{\prime}_{1,\xi_{q}}$ is the character on
$[\left(H_{s_{1}}/Z(H_{s_{1}})\right)]$ that corresponds to the matrix

$\begin{pmatrix}I^{\ast}_{s_{1}-q}&0_{s_{1}\times(k+l+1-2s_{1})}&\xi&0_{s_{1}\times(k+l-s+1)}\end{pmatrix}\in\operatorname{Mat}_{s_{1}\times(4k+2l-2s_{1}+3)}(F),$
$I^{\ast}_{s_{1}-q}=\begin{pmatrix}0&0\\\
0&I_{s_{1}-q}\end{pmatrix}\in\operatorname{Mat}_{s_{1}\times s_{1}}(F).$

Let $\omega_{q}$ be the Weyl group element given by

$\omega_{q}=\begin{pmatrix}I_{s_{1}}&&&&\\\ &\nu_{q}&&&\\\ &&I_{2k+1-2q}&&\\\
&&&\nu_{q}^{-1}&\\\
&&&&I_{s_{1}}\end{pmatrix}\in\operatorname{SO}_{4k+2l+3}(F),\nu_{q}=\begin{pmatrix}&&&I_{q}&\\\
&I_{s_{1}-q}&&&\\\ I_{q}&&&&\\\ &&I_{k+l+q+1-3s_{1}}&&\\\
&&&&I_{s_{1}-q}\end{pmatrix}.$

The conjugation action of $\omega_{q}$ stabilizes $[R_{s_{1}}]$. Therefore, we
change the variable $u\mapsto\omega^{-1}_{q}u\omega_{q}$ and use the
automorphicity of $\theta_{4k+2l+3}$ to obtain

(5.18)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s_{1}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\omega_{q}z_{q}(v^{1},g)\right)\psi_{1}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg.$

Apply Proposition 4.5 to the integral (5.18) to replace the integration on
$[R_{s_{1}}]$ by $[R_{s^{2}_{1}}]$. We obtain

(5.19)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s^{2}_{1}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\omega_{q}z_{q}(v^{1},g)\right)\psi_{1}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg.$

This integral is similar to (5.12) except for the presence of the Weyl group
element $\omega_{q}$. However, the same argument implies that the contribution
is zero as long as
$R_{s^{2}_{1}}\cap\omega_{q}V_{\mathcal{O}^{\prime}}^{1}\omega_{q}^{-1}$ is
non-trivial. This happens only when $\operatorname{Rank}(\xi_{q})<s_{1}$.
Hence, we conclude that the only non-zero contributions of the Fourier
expansion are from characters $\psi_{\xi}$ on
$[\left(H_{s_{1}}/Z(H_{s_{1}})\right)]$ corresponding to some
$\xi\in\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$ with rank $s_{1}$ and
totally isotropic row space. Notice that $\operatorname{SO}_{2k+1}(F)$ acts
transitively from the right on such matrices. If we take

$\xi_{s_{1}}=\begin{pmatrix}I_{s_{1}}&0\end{pmatrix}\in\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$

as a representative, then we conclude that the Fourier coefficient (5.8) is
equal to

(5.20)
$\int_{[\operatorname{SO}_{2k+1}]}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s^{2}_{1}}]}\sum_{\gamma\in
P_{s_{1}}^{0}(F)\setminus\operatorname{SO}_{2k+1}(F)}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\omega_{s_{1}}z_{s_{1}}\gamma(v^{1},g)\right)\psi_{1}(u)\psi_{\mathcal{O}}(v^{1})\,dudv^{1}dg.$

Here, $P_{s_{1}}=P_{s_{1},2k+1}$ is the standard maximal parabolic subgroup of
$\operatorname{SO}_{2k+1}$ with Levi part
$\operatorname{GL}_{s_{1}}\times\operatorname{SO}_{2k-2s_{1}+1}$. The upper
zero indicates that we omit the $\operatorname{GL}_{s_{1}}$ factor. In fact,
$P^{0}_{s_{1}}(F)$ is the stabilizer of $\psi_{\xi_{s_{1}}}$ under the
$\operatorname{SO}_{2k+1}(F)$ action. As any $\gamma\in
P_{s_{1}}^{0}(F)\setminus\operatorname{SO}_{2k+1}(F)$ commutes with any
$v^{1}\in V_{\mathcal{O}^{\prime}}^{1}(\mathbb{A})$, we can combine the
summation with the integration in (5.20) to rewrite it as

(5.21)
$\int_{P^{0}_{s_{1}}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\int_{[V_{\mathcal{O}^{\prime}}^{1}]}\int_{[R_{s^{2}_{1}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\omega_{s_{1}}z_{s_{1}}(v^{1},g)\right)\psi_{1}(u)\psi_{\mathcal{O}^{\prime}}(v^{1})\,dudv^{1}dg.$

To proceed, we start with the assumption that $n_{1}-n_{2}=1$ for notational
simplicity. The same argument applies to the other case. Consider the
unipotent subgroup
$R_{s_{2}}=R_{s_{2},4k+2l-4s_{1}+3}\subset\operatorname{SO}_{4k+2l+3}$, which
is the unipotent radical of the standard maximal parabolic subgroup whose Levi
part is
$\operatorname{GL}_{s_{1}}^{2}\times\operatorname{GL}_{s_{2}}\times\operatorname{SO}_{4k+2l-4s_{1}-2s_{2}+3}$.
The group $R_{s_{2}}$ consists of matrices of the form

(5.22) $\left\\{\begin{pmatrix}I_{2s_{1}}&&&&\\\ &I_{s_{2}}&x&\ast&\\\
&&I_{4k+2l-4s_{1}-2s_{2}+3}&x^{\ast}&\\\ &&&I_{s_{2}}&\\\
&&&&I_{2s_{1}}\end{pmatrix}\in\operatorname{SO}_{4k+2l+3}:x\in\operatorname{Mat}_{s_{2}\times(4k+2l-4s_{1}-2s_{2}+3)}\right\\}.$

We set
$V_{s_{2}}=R_{s_{2}}\cap\left(\omega_{s_{1}}z_{s_{1}}V_{\mathcal{O}^{\prime}}^{1}(\omega_{s_{1}}z_{s_{1}})^{-1}\right)$,
and the quotient $H_{s_{2}}=V_{s_{2}}\setminus R_{s_{2}}$. As the center
$[Z(H_{s_{2}})]\subset[V_{s_{2}}]$ is included in the domain of integration in
(5.21), we continue to expand (5.21) against $[H_{s_{2}}/Z(H_{s_{2}})]$. We
check on the contributions from each type of the characters on
$[H_{s_{2}}/Z(H_{s_{2}})]$ under the action of
$\operatorname{GL}_{s_{2}}(F)\times\operatorname{SO}_{4k+2l-4s_{1}-2s_{2}+3}(F)$.
By the same argument, it follows that the only contribution is from the orbit
of characters represented by $\psi_{\xi_{s_{2}}}$ corresponding to the matrix

$\xi_{s_{2}}=\begin{pmatrix}I_{s_{2}}&0\end{pmatrix}\in\operatorname{Mat}_{s_{2}\times(4k+2l-4s_{1}-2s_{2}+3)}(F).$

We continue the same argument repeatedly with the assumption that
$n_{i}-n_{i+1}=1$ for all $i=1,2,\cdots,p-1$ and $n_{p}=1$ for notational
simplicity. We deduce that (5.21) equals to

(5.23)
$\int_{P_{\mathcal{O}}^{0}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\int_{[V_{\mathcal{O}^{\prime}}^{p}]}\int_{[R_{s^{2}_{p}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\prod_{i=0}^{p-1}\omega_{s_{p-i}}z_{s_{p-i}}(v^{p},g)\right)\psi_{p}(u)\psi_{\mathcal{O}^{\prime}}(v^{p})\,dudv^{p}dg.$

We explain the notations here. First,
$P_{\mathcal{O}}=M(\mathcal{O})U_{\mathcal{O}}$ is the maximal parabolic
subgroup of $\operatorname{SO}_{2k+1}$ with the corresponding Levi
decomposition. The upper zero indicates that we omit all the
$\operatorname{GL}$-factors in the Levi factor $M(\mathcal{O})$. That is
$P_{\mathcal{O}}^{0}\cong\operatorname{SO}_{l}U_{\mathcal{O}}$. Next,
$V_{\mathcal{O}^{\prime}}^{p}=V_{\mathcal{O}^{\prime}}\cap\operatorname{SO}_{3l+1}$
with
$V_{\mathcal{O}^{\prime}}^{p}/(V_{\mathcal{O}^{\prime}}^{p})^{(1)}\cong\operatorname{Mat}_{l\times(l+1)}$.
Also, $R_{s^{2}_{p}}$ is the unipotent radical of the standard maximal
parabolic subgroup of $\operatorname{SO}_{4k+2l+3}$ with Levi part given by

$\operatorname{GL}^{2(n_{1}-n_{2})}_{s_{1}}\times\operatorname{GL}^{2(n_{2}-n_{3})}_{s_{2}}\times\cdots\times\operatorname{GL}^{2(n_{p-1}-n_{p})}_{s_{p-1}}\times\operatorname{GL}^{2n_{p}}_{l}\times\operatorname{SO}_{4l+1}.$

The character $\psi_{p}:[R_{s^{2}_{p}}]\rightarrow\mathbb{C}^{\times}$ is the
product of $\psi_{1}$ and the characters corresponding to the non-zero
contribution in each of the repeated steps. It is given by (similar to
$\psi_{n}$ in Corollary 4.6)

$\psi_{p}(u)=\psi\left(\sum_{i=1}^{s_{1}}u_{j,s_{1}+j}+\sum_{j=1}^{s_{2}}u_{2s_{1}+j,2s_{1}+s_{2}+j}+\cdots+\sum_{j=1}^{s_{p}}u_{2(s_{1}+\cdots+s_{p-1})+j,2(s_{1}+\cdots+s_{p-1})+s_{p}+j}\right).$

The term $\prod_{i=0}^{p-1}\omega_{s_{p-i}}z_{s_{p-i}}$ is the product of
$\omega_{s_{1}}z_{s_{1}}$ and the corresponding Weyl group elements and
unipotent elements produced during each of the repeated steps. To be more
precise, for each $1\leqslant j\leqslant p$, we have

$\omega_{s_{j}}=\begin{pmatrix}I_{2(s_{1}+\cdots+s_{j-1})}&&&&\\\
&\nu_{s_{j}}&&&\\\ &&I_{2k-2(s_{1}+\cdots+s_{j})+1}&&\\\
&&&\nu_{s_{j}}^{-1}&\\\ &&&&I_{2(s_{1}+\cdots+s_{j-1})}\end{pmatrix},$

$\nu_{s_{j}}=\begin{pmatrix}I_{s_{j}}&&\\\ &&I_{s_{j}}\\\
&I_{k+l+1-2(s_{1}+\cdots+s_{j-1})-s_{j}}&\end{pmatrix},$

and

$z_{s_{j}}=\begin{pmatrix}I_{2(s_{1}+\cdots+s_{j-1})}&&&&\\\
&\mu_{s_{j}}&&&\\\ &&I_{2k-2(s_{1}+\cdots+s_{j})+1}&&\\\
&&&\mu_{s_{j}}^{\ast}&\\\ &&&&I_{2(s_{1}+\cdots+s_{j-1})}\end{pmatrix},$
$\mu_{s_{j}}=\begin{pmatrix}I_{s_{j}}&&&&\\\ &I_{s_{j}}&&-I_{s_{j}}\\\
&&I_{k+l+1-2(s_{1}+\cdots+s_{j})}&\\\ &&&I_{s_{j}}\end{pmatrix}.$

Notice that we can conjugate all the $z_{j}$’s to the right of all the
$\omega_{j}$’s and rewrite (5.23) as

(5.24)
$\int_{P_{\mathcal{O}}^{0}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\int_{[V_{\mathcal{O}^{\prime}}^{p}]}\int_{[R_{s^{2}_{p}}]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\omega_{l}z_{l}(v^{p},g)\right)\psi_{p}(u)\psi_{\mathcal{O}^{\prime}}(v^{p})\,dudv^{p}dg,$

with $\omega_{l}=\omega_{s_{p-1}}\omega_{s_{p-2}}\cdots\omega_{s_{1}}$ and

$z_{l}=\begin{pmatrix}I_{s_{1}}&&&&\\\ &\mu_{l}&&&\\\ &&I_{l}&&\\\
&&&\mu_{l}^{-1}&\\\
&&&&I_{s_{1}}\end{pmatrix},\mu_{l}=\begin{pmatrix}I_{s_{1}+\cdots+s_{p-1}}&&-I_{s_{1}+\cdots+s_{p-1}}\\\
&I_{k+l-2(s_{1}+\cdots+s_{p-1})+1}&\\\
&&I_{s_{1}+\cdots+s_{p-1}}\end{pmatrix}.$

Now we proceed to the last step. Let $R_{p}\subset\operatorname{SO}_{4l+1}$ be
the unipotent radical of the standard maximal parabolic subgroup of
$\operatorname{SO}_{4l+1}$ with Levi part
$\operatorname{GL}_{l}\times\operatorname{SO}_{2l+1}$. In terms of matrices,

$R_{p}=\left\\{\begin{pmatrix}I_{2(s_{1}+\cdots+s_{p-1})}&&&&\\\
&I_{l}&x&\ast&\\\ &&I_{2l+1}&x^{\ast}&\\\ &&&I_{l}&\\\
&&&&I_{2(s_{1}+\cdots+s_{p-1})}\end{pmatrix}\in\operatorname{SO}_{4k+2l+3}:x\in\operatorname{Mat}_{l\times(2l+1)}\right\\}.$

The subgroup $R_{p}$ is again a Heisenberg group with the maximal abelian
quotient $R_{p}/R^{(1)}_{p}\cong\operatorname{Mat}_{l\times(2l+1)}$. On the
other hand, for any $y=\begin{pmatrix}y_{1}&y_{2}\end{pmatrix}\in
V_{\mathcal{O}^{\prime}}^{p}/(V_{\mathcal{O}^{\prime}}^{p})^{(1)}\cong\operatorname{Mat}_{l\times(l+1)}$
where $y_{1},y_{2}\in\operatorname{Mat}_{l\times(\frac{l+1}{2})}$, we have

$\omega_{l}z_{l}y(\omega_{l}z_{l})^{-1}=\begin{pmatrix}y_{1}&0_{l\times
l}&y_{2}\end{pmatrix}\in R_{p}/R_{p}^{(1)}.$

Let $H_{p}=\left(\omega_{l}z_{l}V^{p}(\omega_{l}z_{l})^{-1}\right)\setminus
R_{p}$, which is also a Heisenberg group. As the center
$[Z(H_{p})]\subset[\left(\omega_{l}z_{l}V^{p}(\omega_{l}z_{l})^{-1}\right)]$
is included in the domain of integration in the integral (5.24), we further
expand (5.24) against $[\left(H_{p}/Z(H_{p})\right)]$.

We may identify the character group of $[\left(H_{p}/Z(H_{p})\right)]$ with

$\left(H_{p}/Z(H_{p})\right)(F)\cong\operatorname{Mat}_{l\times l}(F).$

In the expansion, we conjugate elements in $[V^{p}_{\mathcal{O}^{\prime}}]$ to
the left inside the function $\theta_{4k+2l+3}$ and combine this domain of
integration with $[\left(H_{p}/Z(H_{p})\right)]$. Denote the resulting domain
of integration by $[R_{p}]$. Let $R=R_{s^{2}_{p}}R_{p}$, which is the
unipotent subgroup that coincides with $V_{2,\mathcal{O}^{2}}$ where
$\mathcal{O}^{2}$ is the unipotent orbit in $\operatorname{SO}_{4k+2l+3}$
corresponding to the partition

$\left((4n_{1}+3)^{r_{1}}(4n_{2}+3)^{r_{2}}\cdots(4n_{p}+3)^{r_{p}}(1)^{l+1}\right).$

As a result, (5.24) is equal to

(5.25)
$\int_{P_{\mathcal{O}}^{0}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\int_{[R]}\sum_{\xi\in\operatorname{Mat}_{l\times
l}(F)}\varphi(g)\overline{\theta_{4k+2l+3}\left(u\omega_{l}z_{l}(1,g)\right)}\psi_{\mathcal{O}^{\prime},p,\xi}(u)\,dudg.$

Here, we combine all the characters involved to get the character
$\psi_{\mathcal{O}^{\prime},p,\xi}$ on $[R]$. For any $x=x_{1}x_{2}x_{3}\in
R(\mathbb{A})$ with $x_{1}\in R_{s^{2}_{p}}(\mathbb{A}),x_{2}\in
V_{\mathcal{O}^{\prime}}^{p}(\mathbb{A})$ and $x_{3}\in
H_{p}(\mathbb{A})/(Z(H_{p}))(\mathbb{A})$,

$\psi_{\mathcal{O}^{\prime},p,\xi}(x)=\psi_{p}(x_{1})\psi_{\mathcal{O}^{\prime}}(x_{2})\psi_{\xi}(x_{3}).$

By Proposition 4.3, the inner integral of (5.25) factors as the product of an
integral of a theta function in the theta representation
$\Theta_{\operatorname{GL}}$ of the double cover
$\widetilde{\operatorname{GL}}_{2(s_{1}+\cdots+s_{p-1})}(\mathbb{A})$ with
respect to the character $\psi_{p}$ and an integral of a theta function in
$\Theta_{4l+1}$ of $\widetilde{\operatorname{SO}}_{4l+1}(\mathbb{A})$ with
respect to the character $\psi_{\mathcal{O}^{\prime}}\psi_{\xi}$. When the
character $\psi_{\mathcal{O}^{\prime}}\psi_{\xi}$ is generic with respect to
the unipotent orbit associated with the partition $(3^{l}1^{l+1})$, the second
integral is a Fourier coefficient of the theta representation $\Theta_{4l+1}$
with respect to the unipotent orbit associated to $(3^{l}1^{l+1})$. By
Proposition 4.4, such an integral must be zero. Thus, the non-zero
contributions in the expansion (5.25) come from those $\psi_{\xi}$
corresponding to $\xi\in\operatorname{Mat}_{l\times l}(F)$ such that
$\psi_{\mathcal{O}^{\prime}}\psi_{\xi}$ is not generic.

To proceed, let us fix a

$\xi=\begin{pmatrix}\lambda_{1}\\\ \vdots\\\
\lambda_{l}\end{pmatrix}\in\operatorname{Mat}_{l\times l}(F),$

with row vectors $\lambda_{1},\cdots,\lambda_{l}\in F^{l}$. By construction,
the character $\psi_{\eta}:=\psi_{\mathcal{O}^{\prime}}\psi_{\xi}$ corresponds
to the matrix

$\eta=\begin{pmatrix}1&&&&\lambda_{1}&&&&\\\ &\ddots&&&\vdots&&&&\\\
&&1&&\lambda_{\frac{l-1}{2}}&&&&\\\
&&&\frac{1}{2}&\lambda_{\frac{l+1}{2}}&-1&&&\\\
&&&&\lambda_{\frac{l+3}{2}}&&-1&&\\\ &&&&\vdots&&&\ddots&\\\
&&&&\lambda_{l}&&&&-1\end{pmatrix}\in\operatorname{Mat}_{l\times(2l+1)}(F).$

Notice that $\psi_{\eta}$ is generic as long as the row space of $\eta$ is not
totally isotropic with respect to the non-degenerate bilinear form on
$F^{2l+1}$ defined by $J_{2l+1}$. As a result, the contribution of
$\psi_{\eta}$ is non-zero only if the row space of $\eta$ is totally isotropic
and of rank $l$. That is, the row space of $\eta$ is a maximal totally
isotropic subspace of $F^{2l+1}$. Consequently, we only need to consider $\xi$
with row vectors $\lambda_{1},\lambda_{2},\cdots\lambda_{l}$ satisfying the
following conditions:

  1. (1)

Each $\lambda_{i}$ except $\lambda_{\frac{l+1}{2}}$ is non-zero isotropic.

  2. (2)

Each pair $\lambda_{i}$ and $\lambda_{l+1-i}$ for $i=1,2,\cdots,\frac{l-1}{2}$
is a hyperbolic pair, i.e.

$(\lambda_{i},\lambda_{l+1-i})=1,\;i=1,2,\cdots,\frac{l-1}{2}.$

All these hyperbolic pairs are mutually orthogonal.

  3. (3)

The vector $\lambda_{\frac{l+1}{2}}$ is non-isotropic with
$(\lambda_{\frac{l+1}{2}},\lambda_{\frac{l+1}{2}})=1$.

The group $\operatorname{SO}_{l}(F)$ acts transitively from the right on the
set of $\xi$’s and we pick the identity matrix
$I_{l}\in\operatorname{Mat}_{l\times l}(F)$ as a representative. As a result,
the integral (5.25) is equal to

(5.26)
$\int_{P_{\mathcal{O}}^{0}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\int_{[R]}\sum_{\gamma\in\operatorname{SO}_{2l+1}(F)}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\gamma\omega_{l}z_{l}(1,g)\right)\psi_{\mathcal{O}^{\prime},p,I_{l}}(u)\,dudg.$

Since $\gamma$ commutes with the product $\omega_{l}z_{l}$, we can conjugate
it to the right and collapse the summation with the outer integration to
obtain

(5.27)
$\int_{U_{\mathcal{O}}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\int_{[R]}\varphi(g)\overline{\theta}_{4k+2l+3}\left(u\omega_{l}z_{l}(1,g)\right)\psi_{\mathcal{O}^{\prime},p,I_{l}}(u)\,dudg.$

We further decompose the domain of integration of the outer integral as

$U_{\mathcal{O}}(F)\setminus\operatorname{SO}_{2k+1}(\mathbb{A})=\left(U_{\mathcal{O}}(F)\setminus
U_{\mathcal{O}}(\mathbb{A})\right)\left(U_{\mathcal{O}}(\mathbb{A})\setminus\operatorname{SO}_{2k+1}(\mathbb{A})\right).$

We obtain

(5.28) $\int_{U_{\mathcal{O}}(F)\setminus
U_{\mathcal{O}}(\mathbb{A})}\int_{U_{\mathcal{O}}(\mathbb{A})\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\int_{[R]}\varphi(ng)\overline{\theta}_{4k+2l+3}\left(u\omega_{l}z_{l}(1,ng)\right)\psi_{\mathcal{O}^{\prime},p,I_{l}}(u)\,dudgdn.$

Observe that $n_{0}:=\omega_{l}z_{l}n(\omega_{l}z_{l})^{-1}\in R(\mathbb{A})$
for any $n\in U_{\mathcal{O}}(\mathbb{A})$, and
$\psi_{\mathcal{O}^{\prime},p,I_{l}}(n_{0})=\psi_{\mathcal{O}}(n)$. Hence, we
conjugate the variable $n$ inside the function $\theta_{4k+2l+3}$ to the left
and perform a change of variable by $u\mapsto un_{0}^{-1}$ to finally obtain
that (5.28) is equal to

(5.29)
$\int_{U_{\mathcal{O}}(\mathbb{A})\setminus\operatorname{SO}_{2k+1}(\mathbb{A})}\left(\int_{U_{\mathcal{O}}(F)\setminus
U_{\mathcal{O}}(\mathbb{A})}\varphi(ng)\psi_{\mathcal{O}}(n)dn\right)\int_{[R]}\overline{\theta}_{4k+2l+3}\left(u\omega_{l}z_{l}(1,g)\right)\psi_{\mathcal{O}^{\prime},p,I_{l}}(u)\,dudg.$

Here
$F_{\psi_{\mathcal{O}},U_{\mathcal{O}}}(\varphi)(g)=\int_{U_{\mathcal{O}}(F)\setminus
U_{\mathcal{O}}(\mathbb{A})}\varphi(ng)\psi_{\mathcal{O}}(n)dn$ is a Fourier
coefficient of $\varphi$ associated to the unipotent orbit $\mathcal{O}$ and
the generic character $\phi_{\mathcal{O}}$ on $[U_{\mathcal{O}}]$. The fact
that (5.8) is non-vanishing implies that
$F_{\psi_{\mathcal{O}},U_{\mathcal{O}}}(\varphi)$ is non-vanishing, which
completes the proof. ∎

##  6\. Local theory

In this section, we establish the local counterpart to Theorem 5.1. Let $F$ be
a non-archimedean local field. We are still concerned with the two unipotent
orbits

$\mathcal{O}=\left((2n_{1}+1)^{r_{1}}(2n_{2}+1)^{r_{2}}\cdots(2n_{p}+1)^{r_{p}}\right)$

and

$\mathcal{O}^{\prime}=\left((2n_{1}+3)^{r_{1}}(2n_{2}+3)^{r_{2}}\cdots(2n_{p}+3)^{r_{p}}(1)\right)$

of the two groups $\operatorname{SO}_{2k+1}$ and $\operatorname{SO}_{2k+2l+2}$
respectively, where $l=r_{1}+r_{2}+\cdots r_{p}$.

Recall that we can associate the two orbits $\mathcal{O}$ and
$\mathcal{O}^{\prime}$ with the unipotent subgroups
$U_{\mathcal{O}}=V_{1,\mathcal{O}}=V_{2,\mathcal{O}}$ and
$V_{\mathcal{O}^{\prime}}=V_{1,\mathcal{O}^{\prime}}=V_{2,\mathcal{O}^{\prime}}$
respectively. Define the generic characters
$\psi_{\mathcal{O}}:U_{\mathcal{O}}(F)\rightarrow\mathbb{C}^{\times}$ and
$\psi_{\mathcal{O}^{\prime}}:V_{\mathcal{O}^{\prime}}(F)\rightarrow\mathbb{C}^{\times}$
similarly to the global ones given by (5.4) and (5.7) respectively.

Suppose $\pi$ is an irreducible genuine admissible representation of
$\widetilde{\operatorname{SO}}_{2k+1}(F)$. Recall from Section 4.1 that
$\Theta_{4k+2l+3}$ is the local theta representation of
$\widetilde{\operatorname{SO}}_{4k+2l+3}(F)$. Suppose there exists an
irreducible genuine admissible representation
$\Theta(\pi)=\Theta_{4k+2l+3}(\pi)$ of
$\widetilde{\operatorname{SO}}_{2k+2l+2}(F)$ such that the
$\operatorname{Hom}$-space

(6.1)
$\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times\widetilde{\operatorname{SO}}_{2k+2l+2}}\left(\Theta_{4k+2l+3},\pi\otimes\Theta(\pi)\right)$

is non-zero. Here, we restrict $\Theta_{4k+2l+3}$ to a representation of the
product subgroup
$\widetilde{\operatorname{SO}}_{2k+1}(F)\times\widetilde{\operatorname{SO}}_{2k+2l+2}(F)$
which is the preimage of the product
$\operatorname{SO}_{2k+1}(F)\times\operatorname{SO}_{2k+2l+2}(F)$ in
$\widetilde{\operatorname{SO}}_{4k+2l+3}(F)$ via the embedding (2.1).

We extend the definition of twisted Jacquet modules in Section 3.4. If
$\mathcal{O}_{0}$ is an odd unipotent orbit, then $U:=V_{2,\mathcal{O}_{0}}$
is equal to the unipotent radical of some maximal parabolic subgroup of
$\operatorname{SO}_{2k+1}$. If the unipotent radical $U(F)$ is a Heisenberg
group with its center $Z(U)(F)$ acting trivially on $\mathcal{V}$ and
$\psi_{0}$ is trivial on $Z(U)(F)$, then the vector subspace
$\mathcal{V}(U,\psi_{0})=\mathcal{V}(U/Z(U),\psi_{0})$. Denote the quotient
space $\mathcal{V}/\mathcal{V}(U/Z(U),\psi_{0})$ by $J_{U/Z(U),\psi}(\pi)$.

######  Theorem 6.1.

Let $\pi$ be an irreducible admissible representation of
$\widetilde{\operatorname{SO}}_{2k+1}(F)$. Suppose there exists an irreducible
admissible representation $\Theta(\pi)$ of
$\widetilde{\operatorname{SO}}_{2k+2l+2}(F)$ such that, as representations of
the group
$\widetilde{\operatorname{SO}}_{2k+1}(F)\times\widetilde{\operatorname{SO}}_{2k+2l+2}(F)$,

(6.2)
$\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times\widetilde{\operatorname{SO}}_{2k+2l+2}}\left(\Theta_{4k+2l+3},\pi\otimes\Theta(\pi)\right)\neq
0.$

Furthermore, suppose the twisted Jacquet module of $\Theta(\pi)$ with respect
to the unipotent orbit $\mathcal{O}^{\prime}$ and the generic character
$\psi_{\mathcal{O}^{\prime}}$ is non-zero, i.e.

(6.3)
$J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\neq 0.$

Then the twisted Jacquet module of $\pi$ with respect to the unipotent orbit
$\mathcal{O}$ and the generic character $\psi_{\mathcal{O}}$ is also non-zero,
i.e.

(6.4) $J_{U_{\mathcal{O}},\psi_{\mathcal{O}}}(\pi)\neq 0.$

In order to prove Theorem 6.1, we need the following Proposition which is the
local version of Proposition 4.5. For a positive integer $s<k/4$, let
$R_{s,2k+1}$ be the unipotent radical of the standard maximal parabolic
subgroup of $\operatorname{SO}_{2k+1}$ with Levi factor
$\operatorname{GL}_{s}\times\operatorname{SO}_{2k-2s+1}$. Similarly, let
$R_{2s,2k+1}$ be the unipotent radical of the maximal parabolic subgroup of
$\operatorname{SO}_{2k+1}$ with Levi factor
$\operatorname{GL}_{2s}\times\operatorname{SO}_{2k-4s+1}$. Define the
character $\psi_{1}:R_{s,2k+1}(F)\rightarrow\mathbb{C}^{\times}$ by

$\psi_{1}(u)=\psi(\sum_{j=1}^{s}u_{j,j+s}),\quad u=(u_{i,j})\in
R_{s,2k+1}(F).$

######  Proposition 6.2.

Consider the local theta representation $\Theta_{2k+1}$ of
$\widetilde{\operatorname{SO}}_{2k+1}(F)$. There is a surjection of
$\widetilde{\operatorname{GL}}_{s}^{\Delta}\times\widetilde{\operatorname{SO}}_{2k-4s+1}$
-modules

$J_{R_{2s,2k+1}}(\Theta_{2k+1})\twoheadrightarrow
J_{R_{s,2k+1},\psi_{1}}(\Theta_{2k+1}),$

where $\operatorname{GL}_{s}^{\Delta}\times\operatorname{SO}_{2k-4s+1}$ is the
subgroup of $\operatorname{GL}_{2s}\times\operatorname{SO}_{2k-4s+1}$ with the
$\operatorname{GL}_{s}$-factor embedded in $\operatorname{GL}_{2s}$
diagonally.

###### Proof.

By Proposition 4.2, it suffices to show that the unipotent subgroup

$R_{s,2k-2s+1}(F)\subset\operatorname{SO}_{2k-2s+1}(F)\xhookrightarrow{\iota}\operatorname{SO}_{2k+1}(F)$

acts trivially on the Jacquet module $J_{R_{s,2k+1},\psi_{1}}(\Theta_{2k+1})$.
The proof then proceeds in a similar fashion to the global case.

Let $R_{s,2k-2s+1}$ be the unipotent radical of the maximal parabolic subgroup
of $\operatorname{SO}_{2k-2s+1}$ with the Levi factor
$\operatorname{GL}_{s}\times\operatorname{SO}_{2k-4s+1}$. We identify
$R_{s,2k-2s+1}$ with its embedded image in $\operatorname{SO}_{2k+1}$ via
(2.1). Note that $R_{s,2k-2s+1}(F)$ is a Heisenberg group. We claim that its
center $Z(R_{s,2k-2s+1})(F)$ acts trivially on
$J_{R_{s,2k+1},\psi_{1}}(\Theta_{2k+1})$.

Under the action of the Levi subgroup, any non-trivial character on
$Z(R_{s,2k-2s+1})(F)$ may be represented by $\psi_{\xi_{t}}$ ($0<2t\leqslant
s$) associated to a matrix $\xi_{t}$ of the form

$\xi_{t}=\begin{pmatrix}0&z_{t}\\\
0&0\end{pmatrix}\in\operatorname{Mat}_{s\times s}(F),$

where
$z_{t}=\operatorname{diag}(\lambda_{1},\cdots,\lambda_{t},\lambda_{t}^{-1},\cdots,\lambda_{1}^{-1})\in\operatorname{Mat}_{2t\times
2t}(F)$ with $\lambda_{i}\in F^{\times},\,\forall i=1,\cdots,t$. If the claim
is not true, then there must be a non-trivial character $\psi_{\xi_{t}}$ on
$Z(R_{s,2k-2s+1})(F)$ such that

(6.5)
$J_{Z(R_{s,2k-2s+1}),\psi_{\xi_{t}}}\left(J_{R_{s,2k+1,},\psi_{1}}(\Theta_{2k+1})\right)\neq
0.$

However, the product of $\psi_{1}$ and $\psi_{\xi_{t},Z}$ is a generic
character attached to the unipotent orbit
$\mathcal{O}_{t}=(4^{2t}3^{s-2t}1^{2k-2t-3s+1})$. Hence, the resulting twisted
Jacquet module (6.5) is zero by Proposition 4.1. We get a contradiction.

Therefore, it remains to establish that the abelian quotient
$R_{s,2k-2s+1}/Z(R_{s,2k-2s+1})(F)$ also acts trivially on
$J_{R_{s,2k+1},\psi_{1}}(\Theta_{2k+1})$. We may identify the character group
of $R_{s,2k-2s+1}/Z(R_{s,2k-2s+1})(F)$ with
$\operatorname{Mat}_{s\times(2k-4s+1)}(F)$. Under the action of Levi subgroup
$\operatorname{GL}_{s}(F)\times\operatorname{SO}_{2k-4s+1}(F)$, any character
on $R_{s,2k-2s+1}/Z(R_{s,2k-2s+1})(F)$ may be represented by $\psi_{2,z_{t}}$,
corresponding to a matrix $z_{t}$ of the form

$z_{t}=\begin{pmatrix}I_{t}&0\\\
0&0\end{pmatrix}\in\operatorname{Mat}_{s\times(2k-4s+1)}(F),t=1,\cdots,s.$

If the action of $R_{s,2k-2s+1}/Z(R_{s,2k-2s+1})(F)$ on
$J_{R_{s,2k+1},\psi_{1}}(\Theta_{2k+1})$ is not trivial, then there exists a
character $\psi_{2,z_{t}}$ such that

(6.6)
$J_{R_{s,2k-2s+1}/Z(R_{s,2k-2s+1}),\psi_{2,z_{t}}}\left(J_{Z(R_{s,2k-2s+1})}\left(J_{R_{s,2k+1,},\psi_{1}}(\Theta_{2k+1})\right)\right)\cong
J_{R_{s^{2},2k+1},\psi_{1}\psi_{2,z_{t}}}(\Theta)\neq 0,$

where $R_{s^{2},2k+1}=R_{s,2k+1}R_{s,2k-2s+1}$.

Proceed with the twisted Jacquet module
$J_{R_{s^{2},2k+1},\psi_{1}\psi_{2,z_{t}}}(\Theta_{2k+1})$. Apply the same
argument to see that $Z(R_{s,2k-4s+1})(F)$ acts on it trivially. Hence, it
suffices to check the action of the abelian quotient
$R_{s,2k-4s+1}/Z(R_{s,2k-4s+1})(F)$ on
$J_{R_{s^{2},2k+1},\psi_{1}\psi_{2,z_{t}}}(\Theta_{2k+1})$. If the action is
trivial, Proposition 4.2 implies that there is a
$\widetilde{\operatorname{GL}}_{3s}(F)\times\widetilde{\operatorname{SO}}_{2k-6s+1}(F)$-module
isomorphism

(6.7)
$J_{R_{s,2k-4s+1}}\left(J_{R_{s^{2},2k+1},\psi_{1}\psi_{2,z_{t}}}(\Theta_{2k+1})\right)\cong
J_{N_{s^{3}},\psi_{1}\psi_{2,z_{t}}}\left(\Theta_{\operatorname{GL}_{3s}(F)}\right)\otimes\Theta_{2k-6s+1}.$

Here $\Theta_{\operatorname{GL}_{3s}(F)}$ is the local theta representation of
the double cover $\widetilde{\operatorname{GL}}_{3s}(F)$, and $N_{s^{3}}$ is
the unipotent radical of the parabolic subgroup of $\operatorname{GL}_{3s}$
with Levi subgroup $\operatorname{GL}_{s}^{3}$. However, by Corollary 3.34 of
[6], the twisted Jacquet module
$J_{N_{s^{3}},\psi_{1}\psi_{2,t}}\left(\Theta_{\operatorname{GL}_{3s}(F)}\right)$
is zero. This is a contradiction to (6.6). Therefore, there is a non-trivial
character on $R_{s,2k-4s+1}(F)$ such that the twisted Jacquet module of
$J_{R_{s^{2},2k+1},\psi_{1}\psi_{2,t}}(\Theta_{2k+1})$ with respect to
$R_{s,2k-4s+1}(F)$ and this character is non-zero.

We continue by the same argument repeatedly. For each step, the corresponding
unipotent radical acts trivially due to Corollary 3.34 of [6]. Eventually, we
obtain a non-zero twisted Jacquet module of $\Theta_{2k+1}$ with respect to
some unipotent orbit that is not comparable to $\mathcal{O}(\Theta_{2k+1})$.
By Proposition 4.1, such a twisted Jacquet module must be zero, which is a
contradiction. Thus, the action of $R_{s,2k-2s+1}/Z(R_{s,2k-2s+1})(F)$ on
$J_{R_{s,2k+1},\psi_{1}}(\Theta_{2k+1})$ must be trivial, which completes the
proof. ∎

With enough tools at our disposal, we are now ready to prove Theorem 6.1.
Throughout the proof, we identify any subgroup of $\operatorname{SO}_{2k+1}$
or $\operatorname{SO}_{2k+2l+2}$ with its embedded image in
$\operatorname{SO}_{4k+2l+3}$ via 2.1. Similar to the proof of Theorem 5.1, we
only discuss the case of $n_{i}-n_{i+1}=1$ for all $i=1,2,\cdots,p-1$ and
$n_{p}=1$ for notational simplicity.

###### Proof.

Fix a non-zero
$\Psi_{1}\in\operatorname{Hom}\left(\Theta_{4k+2l+3},\pi\otimes\Theta(\pi)\right)$.
It is clear that $\Psi_{1}$ must be surjective because both $\pi$ and
$\Theta(\pi)$ are irreducible so that $\pi\otimes\Theta(\pi)$ is irreducible
as a representation of the group
$\widetilde{\operatorname{SO}}_{2k+1}(F)\times\widetilde{\operatorname{SO}}_{2k+2l+2}(F)$.
As a result, $\Psi_{1}$ factors through the non-zero twisted Jacquet module
$J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))$. In
other words, $\Psi_{1}$ induces a non-zero
$\widetilde{\operatorname{SO}}_{2k+1}(F)\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})(F)V_{\mathcal{O}^{\prime}}(F)$-equivariant
morphism which we still denote by $\Psi_{1}$:

(6.8) $\Psi_{1}:\Theta_{4k+2l+3}\rightarrow\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi)).$

Here, $M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})(F)$ denotes the
stabilizer of $\psi_{\mathcal{O}^{\prime}}$ in $M(\mathcal{O}^{\prime})(F)$.

Recall we have $V_{s_{1}}=V_{\mathcal{O}^{\prime}}\cap R_{s_{1}}$ where
$R_{s_{1}}=R_{s_{1},4k+2l+3}$ is the unipotent radical of the maximal
parabolic subgroup of $\operatorname{SO}_{4k+2l+3}$ with Levi subgroup
$\operatorname{GL}_{s_{1}}\times\operatorname{SO}_{4k+2l-2s_{1}+3}$. Notice
that $V_{s_{1}}(F)$ acts on
$J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))$ by the
character $\psi_{1}$, and $\Psi_{1}$ is $V_{s_{1}}(F)$-equivariant. As a
result, $\Psi_{1}$ must factor through the non-zero twisted Jacquet module of
$\Theta_{4k+2l+3}$ with respect to the unipotent subgroup $V_{s_{1}}(F)$ and
the character $\psi_{\mathcal{O}^{\prime}}$ restricted on $V_{s_{1}}(F)$,
which we denote $\psi_{1}$. If we denote the resulting map by
$\Psi_{1}^{\prime}$, then

(6.9)
$\Psi_{1}^{\prime}:J_{V_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3})\rightarrow\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))$

is non-zero.

Consider the Heisenberg group $H_{s_{1}}=V_{s_{1}}\setminus R_{s_{1}}$, which
may be identified with the subgroup of matrices of the form

$H_{s_{1}}=H_{s_{1},4k+2l+3}:=\left\\{\begin{pmatrix}I_{s_{1}}&&x&&\ast\\\
&I_{k+l-s_{1}+1}&&&\\\ &&I_{2k+1}&&x^{\ast}\\\ &&&I_{k+l-s_{1}+1}&\\\
&&&&I_{s_{1}}\end{pmatrix}\in\operatorname{SO}_{4k+2l+3}:x\in\operatorname{Mat}_{s_{1}\times(2k+1)}\right\\}.$

As $Z(H_{s_{1}})(F)\subset V_{s_{1}}(F)$ acts trivially on
$J_{V_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3})$, we consider the action of the
abelian quotient $H_{s_{1}}/Z(H_{s_{1}})(F)$ on the twisted Jacquet module
$J_{V_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3})$. If there exists a non-zero vector
in $J_{V_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3})$ on which
$H_{s_{1}}/Z(H_{s_{1}})(F)$ acts by a character $\psi^{\prime}$, then
$\Psi_{1}^{\prime}$ must factor through the non-trivial twisted Jacquet module

(6.10)
$J_{H_{s_{1}}/Z(H_{s_{1}})(F),\psi^{\prime}}\left(J_{Z(H_{s_{1}})(F)}(J_{V_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3}))\right)\cong
J_{R_{s_{1}},\psi_{1}\psi^{\prime}}(\Theta_{4k+2l+3}).$

We may identify the character group of $H_{s_{1}}/Z(H_{s_{1}})(F)$ with
$\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$. Under the action of
$\operatorname{GL}_{s_{1}}(F)\times\operatorname{SO}_{2k+1}(F)$, suppose
$\psi^{\prime}$ corresponds to a matrix in
$\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$ that lies in the same conjugacy
class with a matrix that contains a non-isotropic row vector. Then the product
of $\psi_{1}$ and $\psi^{\prime}$ is a generic character attached to the
unipotent orbit associated with the partition $(3^{s_{1}}1^{4k+2l+3-3s_{1}})$.
By Proposition 4.1, the twisted Jacquet module corresponding to this character
(6.10) is zero.

It remains to examine over those characters on $H_{s_{1}}/Z(H_{s_{1}})(F)$
corresponding to a matrix in $\operatorname{Mat}_{s_{1}\times(2k+1)}(F)$ with
totally isotropic row space. Any such $\psi^{\prime}$ lies in the same
conjugacy class with a character $\psi_{\xi_{q}}$ corresponding to a matrix
$\xi_{q}$ of the form

(6.11) $\xi_{q}=\begin{pmatrix}I_{q}&0\\\
0&0\end{pmatrix}\in\operatorname{Mat}_{s_{1}\times(2k+1)}(F),\quad
q=0,1,\cdots,s_{1}.$

For a fixed $q$, we check on $\Psi_{1}^{\prime}$ restricted to the twisted
Jacquet module of the form

$J_{H_{s_{1}}/Z(H_{s_{1}})(F),\psi_{\xi_{q}}}\left(J_{Z(H_{s_{1}})(F)}(J_{V_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3}))\right)\cong
J_{R_{s_{1}},\psi_{1,\xi_{q}}}(\Theta_{4k+2l+3}),$

where $\psi_{1,\xi_{q}}=\psi_{1}\psi_{\xi_{q}}$. Consider the elements
$w_{q},z_{q}\in\operatorname{SO}_{4k+2l+3}(F)$ of the form

$z_{q}=\begin{pmatrix}I_{s_{1}}&&&&\\\ &\mu_{q}&&&\\\ &&I_{2k+1-2q}&&\\\
&&&\mu_{q}^{\ast}&\\\
&&&&I_{s_{1}}\end{pmatrix},\mu_{q}=\begin{pmatrix}I_{q}&&-I_{q}\\\
&I_{k+l+1-s_{1}-q}&\\\ &&I_{q}\end{pmatrix},$

and

$\omega_{q}=\begin{pmatrix}I_{s_{1}}&&&&\\\ &\nu_{q}&&&\\\ &&I_{2k+1-2q}&&\\\
&&&\nu_{q}^{-1}&\\\
&&&&I_{s_{1}}\end{pmatrix},\nu_{q}=\begin{pmatrix}&&&I_{q}&\\\
&I_{s_{1}-q}&&&\\\ I_{q}&&&&\\\ &&I_{k+l+q+1-3s_{1}}&&\\\
&&&&I_{s_{1}-q}\end{pmatrix}.$

Note that $z_{0}$ and $\omega_{0}$ are both the identity matrix.

Notice that the conjugation action of $w_{q}z_{q}$ on
$\widetilde{\operatorname{SO}}_{4k+2l+3}(F)$ preserves $R_{s_{1}}(F)$. As a
result, we have

(6.12) $J_{R_{s_{1}},\psi_{1,\xi}}(\Theta_{4k+2l+3})\cong
J_{R_{s_{1}},\psi_{1}}(\Theta^{w_{q}z_{q}}_{4k+2l+3}),$

where $\Theta^{w_{q}z_{q}}_{4k+2l+3}$ is the representation of the group
$\widetilde{\operatorname{SO}}_{4k+2l+3}(F)$ obtained by pulling back the
representation $\Theta_{4r+2l+3}$ via the conjugation by $w_{q}z_{q}$ on
$\widetilde{\operatorname{SO}}_{4k+2l+3}(F)$. For any
$g\in\widetilde{\operatorname{SO}}_{4k+2l+3}(F)$ and any function $\theta$ in
the representation $\Theta_{4k+2l+3}$, the action of $g$ on $\theta$ is
replaced by the action of $(w_{q}z_{q})^{-1}gw_{q}z_{q}$ on $\theta$.
Moreover, by Proposition 6.2,

(6.13)
$\Psi_{1}^{\prime}:J_{R_{s_{1}},\psi_{1}}(\Theta^{w_{q}z_{q}}_{4k+2l+3})\cong
J_{R_{s_{1}},\psi_{1,\xi_{q}}}(\Theta_{4k+2l+3})\rightarrow\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))$

factors through the Jacquet module
$J_{R_{s^{2}_{1}},\psi_{1}}(\Theta^{w_{q}z_{q}}_{4k+2l+3})$ where we recall
that $R_{s^{2}_{1}}=R_{s_{1}}R_{s_{1},4k+2l-2s_{1}+3}$ and
$R_{s_{1},4k+2l-2s_{1}+3}$ is the unipotent radical of the maximal parabolic
subgroup of $\operatorname{SO}_{4k+2l-2s_{1}+3}$ with Levi subgroup
$\operatorname{GL}_{s_{1}}\times\operatorname{SO}_{4k+2l-4s_{1}+3}$. If we
denote the resulting map by $\Psi_{1}^{{}^{\prime\prime}}$, then

(6.14)
$\Psi_{1}^{{}^{\prime\prime}}:J_{R_{s^{2}_{1}},\psi_{1}}(\Theta^{w_{q}z_{q}}_{4k+2l+3})\rightarrow\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))$

is non-zero.

Let
$V_{\mathcal{O}^{\prime}}^{1}=V_{\mathcal{O}^{\prime}}\cap\operatorname{SO}_{2k+2l-2s_{1}+2}$.
The intersection $R_{s_{1},4k+2l-2s_{1}+3}\cap
w_{q}z_{q}V_{\mathcal{O}^{\prime}}^{1}(w_{q}z_{q})^{-1}$ is non-trivial as
long as $q<s_{1}$. It contains the one parameter subgroup
$\\{x_{\beta}(r):r\in F\\}$ associated to $\beta$ given by (5.13). The root
group $x_{\beta}(r)$ acts trivially on
$J_{R_{s^{2}_{1}},\psi_{1}}(\Theta^{w_{q}z_{q}}_{4k+2l+3})$, but acts by the
non-trivial character $\psi_{\mathcal{O}^{\prime}}$ on
$J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))$. This
implies that $\Psi_{1}^{{}^{\prime\prime}}$ must be zero when $q<s_{1}$.

It remains only the case of $q=s_{1}$. Note that
$\psi_{\xi}=\psi_{\xi_{s_{1}}}$ is generic. Recall that the center
$Z(H_{s_{1}}(F))$ acts trivially on $\Theta_{4k+2l+3}$. Hence,
$J_{Z(H_{s_{1}})}(\Theta_{4k+2l+3})\cong\Theta_{4k+2l+3}$. By Proposition
5.12(d) of [1] or Lemma A.1 of [16], there exists a short exact sequence of
$Q_{s_{1}}(F)\times\widetilde{\operatorname{SO}}_{2k+1}(F)$-modules

(6.15) $0\rightarrow\operatorname{ind}_{Q_{s_{1}}\times
P^{0}_{s_{1}}}^{Q_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{H_{s_{1}},\psi_{\xi}}(\Theta_{4k+2l+3})\right)\rightarrow
J_{Z(H_{s_{1}})}(\Theta_{4k+2l+3})\rightarrow
J_{H_{s_{1}}}(\Theta_{4k+2l+3})\rightarrow 0,$

where
$Q_{s_{1}}=(\widetilde{\operatorname{GL}}_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2})V_{s_{1}}$
is a maximal parabolic subgroup of $\widetilde{\operatorname{SO}}_{2k+2l+2}$.
The product $Q_{s_{1}}(F)\times\widetilde{\operatorname{SO}}_{2k+1}(F)$ is the
normalizer of $H_{s_{1}}(F)$ in $\widetilde{\operatorname{SO}}_{4k+2l+3}(F)$,
while the stablizer of $\psi_{\xi}$ in
$Q_{s_{1}}(F)\times\widetilde{\operatorname{SO}}_{2k+1}(F)$ is

$(\widetilde{\operatorname{GL}}_{s_{1}}^{\Delta}\times\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2})V_{s_{1}}\times
P^{0}_{s_{1}}\cong Q_{s_{1}}\times P^{0}_{s_{1}},$

where $\widetilde{\operatorname{GL}}_{s_{1}}$ is diagonally embeded into the
Levi subgroups of the two parabolic subgroups $Q_{s_{1}}$ and $P_{s_{1}}$.
Here, $\operatorname{ind}_{Q_{s_{1}}\times
P^{0}_{s_{1}}}^{Q_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+1}}$ is the
induction with compact support as in [2].

The two functors $J_{V_{s_{1}},\psi_{1}}$ and
$\operatorname{ind}_{Q_{s_{1}}\times
P^{0}_{s_{1}}}^{Q_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+1}}$ satisfy
the following relation

$J_{V_{s_{1}},\psi_{1}}\circ\operatorname{ind}_{Q_{s_{1}}\times
P^{0}_{s_{1}}}^{Q_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+1}}\cong\operatorname{ind}_{\widetilde{\operatorname{GL}}_{s_{1}}\times(Q^{\prime}_{s_{1}})^{0}\times
P^{0}_{s_{1}}}^{\widetilde{\operatorname{GL}}_{s_{1}}\times(Q^{\prime}_{s_{1}})^{0}\times\widetilde{\operatorname{SO}}_{2k+1}}\circ
J_{V_{s_{1}},\psi_{1}}.$

Here,
$\widetilde{\operatorname{GL}}^{\Delta}_{s_{1}}(F)\times(Q^{\prime}_{s_{1}})^{0}(F)$
is the stablizer of $\psi_{1}$ in the Levi factor
$\widetilde{\operatorname{GL}}_{s_{1}}(F)\times\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2}(F)$,
where $Q^{\prime}_{s_{1}}$ is the parabolic subgroup of
$\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2}$ with Levi subgroup
$\widetilde{\operatorname{GL}}_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+2l-4s_{1}+2}$.
Hence,

(6.16) $J_{V_{s_{1}},\psi_{1}}\left(\operatorname{ind}_{Q_{s_{1}}\times
P^{0}_{s_{1}}}^{Q_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{H_{s_{1}},\psi_{\xi}}(\Theta_{4k+2l+3})\right)\right)$

is isomorphic to

(6.17)
$\operatorname{ind}_{\widetilde{\operatorname{GL}}^{\Delta}_{s_{1}}\times(Q^{\prime}_{s_{1}})^{0}\times
P^{0}_{s_{1}}}^{\widetilde{\operatorname{GL}}_{s_{1}}\times(Q^{\prime}_{s_{1}})^{0}\times\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s_{1}},\psi_{1,\xi}}(\Theta_{4k+2l+3})\right).$

Moreover, $w_{s_{1}}z_{s_{1}}$ acts on $\Theta_{4k+2l+3}$ and preserves
$R_{s_{1}}$. By (6.12), we further deduce that (6.17) is isomorphic to

(6.18)
$\operatorname{ind}_{\widetilde{\operatorname{GL}}^{\Delta}_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2}\times
P^{0}_{s_{1}}}^{\widetilde{\operatorname{GL}}_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2}\times\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s_{1}},\psi_{1}}(\Theta^{w_{s_{1}}z_{s_{1}}}_{4k+2l+3})\right).$

Hereafter, we denote
$\operatorname{ind}_{\widetilde{\operatorname{GL}}^{\Delta}_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2}\times
P^{0}_{s_{1}}}^{\widetilde{\operatorname{GL}}_{s_{1}}\times\widetilde{\operatorname{SO}}_{2k+2l-2s_{1}+2}\times\widetilde{\operatorname{SO}}_{2k+1}}$
by $\operatorname{ind}_{P^{0}_{s_{1}}}^{\widetilde{\operatorname{SO}}_{2k+1}}$
for simplicity.

Applying the functor $J_{V_{s_{1}},\psi_{1}}$ to (6.15), we obtain the short
exact sequence

$0\rightarrow\operatorname{ind}_{P^{0}_{s_{1}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s_{1}},\psi_{1}}(\Theta^{w_{s_{1}}z_{s_{1}}}_{4k+2l+3})\right)\rightarrow
J_{V_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3})\rightarrow
J_{R_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3})\rightarrow 0.$

We have just shown that, corresponding to $q=0$,

$\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})V_{\mathcal{O}^{\prime}}}\left(J_{R_{s_{1}},\psi_{1}}(\Theta_{4k+2l+3}),\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\right)=0.$

Then, by (6.9), (6.15) and (6.18), we have

(6.19) $\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})V_{\mathcal{O}^{\prime}}}\left(\operatorname{ind}_{P^{0}_{s_{1}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s_{1}},\psi_{1}}(\Theta^{w_{s_{1}}z_{s_{1}}}_{4k+2l+3})\right),\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\right)\neq
0.$

By Proposition 6.2, we further deduce that

(6.20) $\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})V}\left(\operatorname{ind}_{P^{0}_{s_{1}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s^{2}_{1}},\psi_{1}}(\Theta^{w_{s_{1}}z_{s_{1}}}_{4k+2l+3})\right),\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\right)\neq
0.$

Continue the same argument by replacing (6.8) by any non-zero $\Psi_{2}$ in
the $\operatorname{Hom}$-space (6.20). Recall the unipotent subgroup
$R_{s_{2}}$ defined by (5.22) and the Heisenberg quotient
$H_{s_{2}}=V_{s_{2}}\setminus R_{s_{2}}$. Consider the twisted Jacquet modules
with respect to $H_{s_{2}}(F)$ whose center acts trivially on
$J_{R_{s^{2}_{1}},\psi_{1}}(\Theta^{w_{s_{1}}z_{s_{1}}}_{4k+2l+3})$. We note
that the normalizer of $H_{s_{2}}$ in $\widetilde{\operatorname{SO}}_{2k+1}$
is $\widetilde{\operatorname{SO}}_{2k-2s_{1}+1}\subset P^{0}_{s_{1}}$. By a
similar argument, any non-zero homomorphism

$\Psi_{2}:\operatorname{ind}_{P^{0}_{s_{1}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s^{2}_{1}},\psi_{1}}(\Theta^{w_{s_{1}}z_{s_{1}}}_{4k+2l+3})\right)\rightarrow\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))$

must factor through the non-zero twisted Jacquet module with respect to a
generic character on $H_{s_{2}}(F)$, which is

$\operatorname{ind}_{P^{0}_{s_{1}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(\operatorname{ind}_{P^{0}_{s_{2}}}^{\widetilde{\operatorname{SO}}_{2k-2s_{1}+1}}J_{R_{s_{2}},\psi^{\circ}_{2}}\left(J_{R_{s^{2}_{1}},\psi_{1}}(\Theta^{w_{s_{2}}z_{s_{2}}}_{4k+2l+3})\right)\right).$

Here, $\psi_{2}^{\circ}$ is $\psi_{\mathcal{O}^{\prime}}$ restricted to
$V_{s_{2}}$, and $P_{s_{2}}$ is the standard maximal parabolic subgroup of
$\widetilde{\operatorname{SO}}_{2k-2s_{1}+1}$ with Levi subgroup
$\widetilde{\operatorname{GL}}_{s_{2}}\times\widetilde{\operatorname{SO}}_{2k-2(s_{1}+s_{2})+1}$.

By the transitivity of induction

$\operatorname{ind}_{P^{0}_{s_{1}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\operatorname{ind}_{P^{0}_{s_{2}}}^{\widetilde{\operatorname{SO}}_{2k-2s_{1}+1}}\cong\operatorname{ind}_{P^{0}_{s_{2}}}^{\widetilde{\operatorname{SO}}_{2k+1}}.$

By Proposition 6.2, we obtain that

(6.21) $\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})V_{\mathcal{O}^{\prime}}}\left(\operatorname{ind}_{P^{0}_{s_{2}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s^{2}_{2}},\psi_{2}}(\Theta^{w_{s_{2}}z_{s_{2}}}_{4k+2l+3})\right),\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\right)$

is non-zero. Continuing the same argument repeatedly, we obtain that

(6.22) $\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})V_{\mathcal{O}^{\prime}}}\left(\operatorname{ind}_{\widetilde{\operatorname{SO}}_{l}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s^{2}_{p}},\psi_{p}}(\Theta^{w_{l}z_{l}}_{4k+2l+3})\right),\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\right)\neq
0.$

Here, $R_{s^{2}_{p}}$ is the unipotent radical of the standard maximal
parabolic subgroup of $\operatorname{SO}_{4k+2l+3}$ with Levi part

$\operatorname{GL}^{2(n_{1}-n_{2})}_{s_{1}}\times\operatorname{GL}^{2(n_{2}-n_{3})}_{s_{2}}\times\cdots\times\operatorname{GL}^{2(n_{p-1}-n_{p})}_{s_{p-1}}\times\operatorname{GL}^{2n_{p}}_{l}\times\operatorname{SO}_{4l+1}.$

The character $\psi_{p}:R_{s^{2}_{p}}(F)\rightarrow\mathbb{C}^{\times}$ is the
product of $\psi_{1}$ and the characters corresponding to the non-zero twisted
Jacquet modules in each of the repeated steps.

We now proceed to the last step. Let $R_{p}$ be the unipotent radical of the
maximal parabolic subgroup of $\operatorname{SO}_{4k+1}$ with Levi part
$\operatorname{GL}_{l}\times\operatorname{SO}_{2l+1}$. Consider the action of
the abelian quotient
$H_{p}(F)=\omega_{l}z_{l}V^{p}_{\mathcal{O}^{\prime}}(\omega_{l}z_{l})^{-1}\setminus
R_{p}$ on

(6.23)
$\operatorname{ind}_{\widetilde{\operatorname{SO}}_{l}U}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R_{s^{2}_{p}},\psi_{p}}(\Theta^{w_{l}z_{l}}_{4k+2l+3})\right).$

If there is any non-zero vector in (6.23) on which $H_{p}(F)$ act by a
character, then any non-zero $\Psi_{p}$ in (6.22) must factor through the non-
zero twisted Jacquet module of (6.23) with respect to $H_{p}(F)$ and the
corresponding character. Following the same argument as in the proof of
Theorem 5.1, we only need to consider characters $\psi_{\xi}$ on
$H_{p}/Z(H_{p})(F)$ such that the product of $\psi_{\mathcal{O}^{\prime}}$ and
$\psi_{\xi}$ restricted to $R_{p}(F)$ is non-generic. That is, the matrix in
$\operatorname{Mat}_{l\times(2l+1)}(F)$ corresponding to
$\psi_{\mathcal{O}^{\prime}}\psi_{\xi}$ has totally isotropic row space. We
deduce that such a character $\psi_{\xi}$ must correspond to a matrix given by

$\xi=\begin{pmatrix}\lambda_{1}\\\ \vdots\\\
\lambda_{l}\end{pmatrix}\in\operatorname{Mat}_{l\times l}(F)$

satisfying the following conditions:

  1. (1)

Each $\lambda_{i}$ except $\lambda_{\frac{l+1}{2}}$ is non-zero isotropic.

  2. (2)

Each pair $\lambda_{i}$ and $\lambda_{l+1-i}$ for $i=1,2,\cdots,\frac{l-1}{2}$
is a hyperbolic pair, i.e.

$(\lambda_{i},\lambda_{l+1-i})=1,\;i=1,2,\cdots,\frac{l-1}{2}.$

These hyperbolic pairs are mutually orthogonal.

  3. (3)

$\lambda_{\frac{l+1}{2}}$ is non-isotropic with unit length.

The group $\operatorname{SO}_{l}(F)$ acts transitively on the set of these
matrices. This allows us to pick the identity matrix $I_{l}$ as a
representative. Denote the corresponding character on $H_{p}/Z(H_{p})(F)$ by
$\psi_{I_{l}}$. Any non-zero $\Psi_{p}$ in (6.22) factors through the non-zero
twisted Jacquet module of (6.23) given by

(6.24)
$\operatorname{ind}_{\widetilde{\operatorname{SO}}_{l}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(\operatorname{ind}_{\widetilde{\operatorname{SO}}^{\Delta}_{l}}^{\widetilde{\operatorname{SO}}_{l}\times\widetilde{\operatorname{SO}}_{l}}\left(J_{R_{p},\psi_{V_{\mathcal{O}^{\prime}},I_{l}}}J_{R_{s^{2}_{p}},\psi_{p}}(\Theta^{w_{l}z_{l}}_{4k+2l+3})\right)\right)\cong\operatorname{ind}_{U_{\mathcal{O}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R,\psi_{V_{\mathcal{O}^{\prime}},p,I_{l}}}(\Theta^{w_{l}z_{l}}_{4k+2l+3})\right).$

We note that the normalized induction
$\operatorname{ind}_{U_{\mathcal{O}}}^{\widetilde{\operatorname{SO}}_{2k+1}}$
is $\operatorname{ind}_{L^{\Delta}(\mathcal{O})\times
U_{\mathcal{O}}}^{L(\mathcal{O})\times\widetilde{\operatorname{SO}}_{2k+1}}$
where

$L(\mathcal{O})\cong\widetilde{\operatorname{GL}}_{s_{1}}^{n_{1}-n_{2}}\times\widetilde{\operatorname{GL}}_{s_{2}}^{n_{2}-n_{3}}\times\cdots\times\widetilde{\operatorname{GL}}_{l}^{n_{p}}\times\widetilde{\operatorname{SO}}_{l},$

and

$L(\mathcal{O})^{\Delta}\cong\left(\widetilde{\operatorname{GL}}^{\Delta}_{s_{1}}\right)^{n_{1}-n_{2}}\times\left(\widetilde{\operatorname{GL}}^{\Delta}_{s_{2}}\right)^{n_{2}-n_{3}}\times\cdots\times\left(\widetilde{\operatorname{GL}}^{\Delta}_{l}\right)^{n_{p}}\times\widetilde{\operatorname{SO}}^{\Delta}_{l}.$

Thus, we conclude that

(6.25) $\operatorname{Hom}_{\widetilde{\operatorname{SO}}_{2k+1}\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})V_{\mathcal{O}^{\prime}}}\left(\operatorname{ind}_{U_{\mathcal{O}}}^{\widetilde{\operatorname{SO}}_{2k+1}}\left(J_{R,\psi_{V_{\mathcal{O}^{\prime}},p,I_{l}}}(\Theta^{w_{l}z_{l}}_{4k+2l+3})\right),\pi\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\right)\neq
0.$

We now examine the action of $U_{\mathcal{O}}(F)$ on (6.25). The group
$U_{\mathcal{O}}(F)$ acts on the left by the character $\psi_{\mathcal{O}}$,
since $(w_{l}z_{l})^{-1}U_{\mathcal{O}}w_{l}z_{l}\subset R$ and
$\psi_{V_{\mathcal{O}^{\prime}},p,I_{l}}$ is defined on $R(F)$. As a result,
there exists a non-zero vector in $\pi$ on which $U_{\mathcal{O}}(F)$ acts by
the same character $\psi_{\mathcal{O}}$, and

(6.26) $\operatorname{Hom}_{M^{\psi_{\mathcal{O}}}(\mathcal{O})\times
M^{\psi_{\mathcal{O}^{\prime}}}(\mathcal{O}^{\prime})}\left(J_{R,\psi_{V_{\mathcal{O}^{\prime}},p,I_{l}}}(\Theta^{w_{l}z_{l}}_{4k+2l+3}),J_{U_{\mathcal{O}},\psi_{\mathcal{O}}}\left(\pi\right)\otimes
J_{V_{\mathcal{O}^{\prime}},\psi_{\mathcal{O}^{\prime}}}(\Theta(\pi))\right)\neq
0.$

Thus, $J_{U_{\mathcal{O}},\psi_{\mathcal{O}}}\left(\pi\right)$ is non-zero.
This completes the proof. ∎

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