# Generalized Preservation Principle in Finite Theta Correspondence

Shu-Yen Pan  Department of Mathematics, National Tsing Hua University, Hsinchu
300, Taiwan<EMAIL_ADDRESS>
###### Abstract.

It is known that irreducible cuspidal characters satisfy the preservation
principle in the Howe correspondences of finite reductive dual pairs. In this
article, we generalize the preservation principle to any irreducible
characters of classical groups.

###### Key words and phrases:

Howe correspondence, finite reductive dual pair, preservation principle, first
occurrence

###### 1991 Mathematics Subject Classification:

Primary: 20C33; Secondary: 22E50

This work is partially supported by Taiwan MOST-grant 106-2115-M-007-015.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Generalized Preservation Principle of Unipotent Characters
  4. 4 General Generalized Preservation Principle
  5. 5 Pseudo-unipotent Cuspidal Characters

##  1\. Introduction

### 1.1.

Let $(\text{\bf G},\text{\bf G}^{\prime})$ be a reductive dual pair over a
finite field ${\mathbf{F}}_{q}$ of odd characteristic. By restricting the Weil
character (cf. [Gér77]) to the dual pair $(\text{\bf G},\text{\bf
G}^{\prime})$ with respect to a nontrivial additive character $\psi$ of
${\mathbf{F}}_{q}$, we have the _Weil character_ $\omega^{\psi}_{\text{\bf
G},\text{\bf G}^{\prime}}$ of $G\times G^{\prime}$, which has non-negative
integral decomposition

$\omega^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}=\sum_{\rho\in{\mathcal{E}}(G),\
\rho^{\prime}\in{\mathcal{E}}(G^{\prime})}m_{\rho,\rho^{\prime}}\rho\otimes\rho^{\prime}$

where ${\mathcal{E}}(G)$ denotes the set of irreducible characters of group of
rational points $G$ of G. We define the _Howe correspondence_

$\Theta^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}=\\{\,(\rho,\rho^{\prime})\in{\mathcal{E}}(G)\times{\mathcal{E}}(G^{\prime})\mid
m_{\rho,\rho^{\prime}}\neq 0\,\\}$

which is a relation between ${\mathcal{E}}(G)$ and
${\mathcal{E}}(G^{\prime})$. We say that an irreducible character
$\rho\in{\mathcal{E}}(G)$ _occurs_ in the correspondence for the dual pair
$(\text{\bf G},\text{\bf G}^{\prime})$ if there exists
$\rho^{\prime}\in{\mathcal{E}}(G^{\prime})$ such that
$(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$.

Now we consider the following types of _related Witt series_ :

  1. (I)

${\mathbf{v}}$ is a hermitian space (over a quadratic extension of
${\mathbf{F}}_{q}$), and $\\{{\mathbf{v}}^{\prime+}\\}$ (resp.
$\\{{\mathbf{v}}^{\prime-}\\}$) is the Witt series of even-dimensional (resp.
odd-dimensional) hermitian spaces;

  2. (II)

${\mathbf{v}}$ is a quadratic space, and $\\{{\mathbf{v}}^{\prime}\\}$ is the
Witt series of symplectic spaces;

  3. (III)

${\mathbf{v}}$ is a symplectic space, and $\\{{\mathbf{v}}^{\prime+}\\}$
(resp. $\\{{\mathbf{v}}^{\prime-}\\}$) is the Witt series of even-dimensional
quadratic spaces with trivial (resp. two-dimensional) anisotropic kernel;

  4. (IV)

${\mathbf{v}}$ is a symplectic space, and $\\{{\mathbf{v}}^{\prime}\\}$ is the
Witt series of odd-dimensional quadratic spaces.

For each case the pair $(G({\mathbf{v}}),G({\mathbf{v}}^{\prime}))$ (or
$(G({\mathbf{v}}),G({\mathbf{v}}^{\prime\epsilon}))$ where $\epsilon=\pm$)
forms a reductive dual pair over ${\mathbf{F}}_{q}$ where $G({\mathbf{v}})$
denotes the group of isometries of ${\mathbf{v}}$ with the given form.

Let $\rho\in{\mathcal{E}}(G({\mathbf{v}}))$. For (I) and (III), let
$n_{0}^{\prime\epsilon}(\rho)$ denote the minimal dimension of
${\mathbf{v}}^{\prime\epsilon}$ such that $\rho$ occurs in the Howe
correspondence for the dual pair
$(G({\mathbf{v}}),G({\mathbf{v}}^{\prime\epsilon}))$. For (II) and (IV), let
$n_{0}^{\prime}(\rho)$ denote the minimal dimension of ${\mathbf{v}}^{\prime}$
such that $\rho$ occurs in the Howe correspondence for the dual pair
$(G({\mathbf{v}}),G({\mathbf{v}}^{\prime}))$. The following _preservation
principle_ for cuspidal characters is from [Pan02] theorem 12.3:

######  Proposition 1.1.

Let $\rho$ be an irreducible cuspidal character of $G({\mathbf{v}})$.

  1. (i)

If $G({\mathbf{v}})={\rm U}({\mathbf{v}})$, then
$n^{\prime+}_{0}(\rho)+n^{\prime-}_{0}(\rho)=2\,\dim({\mathbf{v}})+1$.

  2. (ii)

If $G({\mathbf{v}})={\rm O}({\mathbf{v}})$, then
$n_{0}^{\prime}(\rho)+n_{0}^{\prime}(\rho\cdot{\rm
sgn})=2\,\dim({\mathbf{v}})$.

  3. (iii)

If $G({\mathbf{v}})={\rm Sp}({\mathbf{v}})$ and ${\mathbf{v}}^{\prime\pm}$
even-dimensional, then
$n^{\prime+}_{0}(\rho)+n^{\prime-}_{0}(\rho)=2\,\dim({\mathbf{v}})+2$.

  4. (iv)

If $G({\mathbf{v}})={\rm Sp}({\mathbf{v}})$ and ${\mathbf{v}}^{\prime}$ odd-
dimensional, then
$n^{\prime}_{0}(\rho)+n^{\prime}_{0}(\rho^{c})=2\,\dim({\mathbf{v}})+2$.

Note that here “${\rm sgn}$” denotes the sign character of ${\rm
O}({\mathbf{v}})$, and “$\rho^{c}$” denotes the character obtained from $\rho$
by conjugating certain element in the symplectic similitude group (cf. [Wal04]
§4.11).

### 1.2.

The purpose of this article is to generalize the above preservation principle
to all irreducible characters. To achieves this we need to recall Lusztig’s
parametrzation of irreducible characters of finite classical groups.

Suppose that $\text{\bf G}={\rm U}_{n}$, ${\rm Sp}_{2n}$, ${\rm
O}^{\epsilon}_{2n}$, or ${\rm SO}_{2n+1}$. Let ${\mathcal{E}}(G)_{s}$ denote
the _Lusztig series_ associated to the conjugacy class $(s)$ of a semisimple
element $s$ in the connected component $(G^{*})^{0}$ of the dual group $G^{*}$
of $G$. In particular, ${\mathcal{E}}(G)_{1}$ is the set of irreducible
unipotent characters of $G$. It is known that these Lusztig series partition
the set of irreducible characters:

${\mathcal{E}}(G)=\bigcup_{(s)\subset(G^{*})^{0}}{\mathcal{E}}(G)_{s}.$

In [Lus77] Lusztig shows that there exists a bijection
${\mathfrak{L}}_{s}\colon{\mathcal{E}}(G)_{s}\rightarrow{\mathcal{E}}(C_{G^{*}}(s))_{1}$
where $C_{G^{*}}(s)$ denotes the centralizer of $s$. Recall that we can define
groups $G^{(j)}$ (where $j=0,1$ of $G$ is a unitary group, and $j=0,1,2$ if
$G$ is an orthogonal group or a symplectic group), and then we have a
bijection

$\Xi_{s}\colon{\mathcal{E}}(G)_{s}\rightarrow\begin{cases}{\mathcal{E}}(G^{(0)}\times
G^{(1)})_{1},&\text{if $G$ is a unitary group};\\\ {\mathcal{E}}(G^{(0)}\times
G^{(1)}\times G^{(2)})_{1},&\text{otherwise}\end{cases}$

modified from ${\mathfrak{L}}_{s}$ above.

A _$\beta$ -set_ $A=\\{a_{1},a_{2},\ldots,a_{m}\\}$ is a finite subset of non-
negative integers with elements written in decreasing order, i.e.,
$a_{1}>a_{2}>\cdots>a_{m}$. A _symbol_ (by Lusztig) $\Lambda=\binom{A}{B}$ is
an ordered pair of two $\beta$-sets. Lusztig gives a parametrization of the
set ${\mathcal{E}}(G)_{1}$ of irreducible unipotent characters of a classical
group $G$ in terms of certain set of symbols denoted by
${\mathcal{S}}_{\text{\bf G}}$ (cf. Subsection 2.2, Subsection 2.4). The
unipotent character associated to $\Lambda$ is denoted by $\rho_{\Lambda}$.

For a $\beta$-set $A$ and a symbol $\Lambda=\binom{A}{B}$, we define

(1.2) $\displaystyle\begin{split}\delta(A)&=\begin{cases}0,&\text{if
$A=\emptyset$};\\\ \max(A)-|A|+1,&\text{if $A\neq\emptyset$}\end{cases}\\\
\delta(\Lambda)&=\delta(A)+\delta(B)\end{split}$

where $\max(A)$ denotes the maximal element of $A$, and $|A|$ denotes the
number of elements in $A$. For $\Lambda\in{\mathcal{S}}_{\text{\bf G}}$, we
define $\delta(\rho_{\Lambda})=\rho(\Lambda)$.

Now we consider types (I), (II), (III), (IV) of related Witt series given in
Subsection 1.1. Let $\rho\in{\mathcal{E}}(G({\mathbf{v}}))$. If ${\mathbf{v}}$
is not an odd-dimensional quadratic space, we suppose that
$\rho\in{\mathcal{E}}(G({\mathbf{v}}))_{s}$ for some $s$. If ${\mathbf{v}}$ is
an odd-dimensional quadratic space, we let $G^{0}({\mathbf{v}})$ denote the
connected component of $G({\mathbf{v}})$ and suppose that
$\rho|_{G^{0}({\mathbf{v}})}\in{\mathcal{E}}(G^{0}({\mathbf{v}}))_{s}$ for
some $s$. Now we write

$\begin{cases}\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)},&\text{if
$G({\mathbf{v}})$ is a unitary group};\\\
\Xi_{s}(\rho|_{G^{0}({\mathbf{v}})})=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)},&\text{if
$G({\mathbf{v}})$ is an odd-orthogonal group};\\\
\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)},&\text{otherwise}\end{cases}$

(cf. (2.15), (2.24)). Then we define

(1.3) $\delta(\rho)=\begin{cases}\delta(\rho^{(1)}),&\text{for types (I),
(IV)};\\\ \delta(\rho^{(2)}),&\text{for types (II), (III)}.\end{cases}$

It is known that $\delta(\rho)=\delta(\rho^{c})$ if $G$ is a symplectic group;
and $\delta(\rho)=\delta(\rho\cdot{\rm sgn})$ if $G$ is an orthogonal group.

Now we state the generalization of the preservation principle (cf. Theorems
4.1, 4.2, 4.6, and 4.9) for any irreducible characters:

######  Theorem 1.4.

Let $\rho$ be an irreducible character of $G({\mathbf{v}})$.

  1. (i)

If $G({\mathbf{v}})={\rm U}({\mathbf{v}})$, then
$n^{\prime+}_{0}(\rho)+n^{\prime-}_{0}(\rho)=2\,\dim({\mathbf{v}})-2\delta(\rho)+1$.

  2. (ii)

If $G({\mathbf{v}})={\rm O}({\mathbf{v}})$, then
$n_{0}^{\prime}(\rho)+n_{0}^{\prime}(\rho\cdot{\rm
sgn})=2\,\dim({\mathbf{v}})-2\delta(\rho)$.

  3. (iii)

If $G({\mathbf{v}})={\rm Sp}({\mathbf{v}})$ and
$\\{{\mathbf{v}}^{\prime\pm}\\}$ even-dimensional, then
$n^{\prime+}_{0}(\rho)+n^{\prime-}_{0}(\rho)=2\,\dim({\mathbf{v}})-2\delta(\rho)+2$.

  4. (iv)

If $G({\mathbf{v}})={\rm Sp}({\mathbf{v}})$ and $\\{{\mathbf{v}}^{\prime}\\}$
odd-dimensional, then
$n^{\prime}_{0}(\rho)+n^{\prime}_{0}(\rho^{c})=2\,\dim({\mathbf{v}})-2\delta(\rho)+2$.

It is known that $\delta(\rho)=0$ if $\rho$ is cuspidal. Hence, the above
theorem is the generalization of the preservation principles for cuspidal
characters in Proposition 1.1. Now we list two applications of our main
theorem:

  1. (1)

For an irreducible character $\rho$ of a classical group $G$. A non-negative
integer called _tensor rank_ is attached to $\rho$ by Gurevich-Howe in [GH20].
The tensor rank is called _$\Theta$ -rank_ and denoted by $\Theta\text{\rm-
rk}(\rho)$ in [Pan21a]. Then Theorem 1.4 provides an upper bound for
$\Theta\text{\rm-rk}(\rho)$. More precisely, for $\rho\in{\mathcal{E}}(G)$, we
have

$\Theta\text{\rm-rk}(\rho)\leq\begin{cases}2n,&\text{if $\text{\bf G}={\rm
Sp}_{2n},{\rm O}^{\epsilon}_{2n},{\rm O}_{2n+1}$};\\\ n,&\text{if $\text{\bf
G}={\rm U}_{n}$}.\end{cases}$

More details can be found in [Pan21a].

  2. (2)

In [How] corollary 5.2.1, it is asserted that for each $0\leq k\leq n$, there
exists a unique $\rho_{k}\in{\mathcal{E}}({\rm Sp}_{2n}(q))$ such that
$\rho_{k}$ occurs in the Howe correspondences for both pairs $({\rm
Sp}_{2n},{\rm O}^{+}_{2k})$ and $({\rm Sp}_{2n},{\rm O}^{-}_{2(n-k)+2})$. From
Theorem 1.4 we can see that $\rho_{k}$ is in fact unipotent and parametrized
by the symbol $\binom{n-k+1,0}{k}$ (cf. Remark 4.8).

### 1.3.

Let $\text{\bf G}_{n}$ denote ${\rm Sp}_{2n}$, ${\rm O}_{2n+1}$, ${\rm
O}^{+}_{2n}$, ${\rm O}^{-}_{2n+2}$, ${\rm U}_{2n}$ or ${\rm U}_{2n+1}$. For
$(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf G}_{n},\text{\bf
G}^{\prime}_{n^{\prime}}}$, we denote

$\displaystyle\Theta_{\text{\bf G}^{\prime}_{n^{\prime}}}(\rho)$
$\displaystyle=\\{\,\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}^{\prime}_{n^{\prime}})\mid(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf
G}_{n},\text{\bf G}^{\prime}_{n^{\prime}}}\,\\},$
$\displaystyle\Theta_{\text{\bf G}_{n}}(\rho^{\prime})$
$\displaystyle=\\{\,\rho\in{\mathcal{E}}(\text{\bf
G}_{n})\mid(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf G}_{n},\text{\bf
G}^{\prime}_{n^{\prime}}}\,\\}.$

It is known that $\Theta_{\text{\bf
G}^{\prime}_{n^{\prime}}}(\rho)\neq\emptyset$ implies that $\Theta_{\text{\bf
G}^{\prime}_{n^{\prime\prime}}}(\rho)\neq\emptyset$ for any
$n^{\prime\prime}\geq n^{\prime}$. Therefore, we say that
$(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf G}_{n},\text{\bf
G}^{\prime}_{n^{\prime}}}$ is a _first occurrence_ if $\Theta_{\text{\bf
G}^{\prime}_{n^{\prime}-1}}(\rho)=\emptyset$ and $\Theta_{\text{\bf
G}_{n-1}}(\rho^{\prime})=\emptyset$. It is well known that if
$(\rho,\rho^{\prime})$ occurs in the correspondence and both
$\rho,\rho^{\prime}$ are cuspidal, then $(\rho,\rho^{\prime})$ is a first
occurrence. Of course, the converse implication is in general not true.

Let $\rho\in{\mathcal{E}}(G({\mathbf{v}}))$. It is shown in [Pan] that
$\delta(\rho)=0$ if and only if there exists
$\rho^{\prime}\in{\mathcal{E}}(G({\mathbf{v}}^{\prime}))$ for some
${\mathbf{v}}^{\prime}$ in its Witt series such that $(\rho,\rho^{\prime})$ is
a first occurrence.

######  Corollary 1.5.

Let $\rho$ be an irreducible character of a classical $G({\mathbf{v}})$. Then
$\rho$ satisfies the preservation principle in Proposition 1.1 if and only if
there exists $\rho^{\prime}\in{\mathcal{E}}(G({\mathbf{v}}^{\prime}))$ for
some ${\mathbf{v}}^{\prime}$ in its Witt series such that
$(\rho,\rho^{\prime})$ is a first occurrence in the Howe correspondence.

### 1.4.

It is known that the unipotent characters are not preserved by the
correspondence for synmplectic/odd-orthogonal dual pairs. To understand the
correspondence for this kind of dual pairs, we need to introduce some new
notations. An irreducible character $\rho$ of $G$ is called _pseudo-unipotent_
if $\langle\rho,R_{\text{\bf T}_{w},\theta_{w}}\rangle\neq 0$ for some
$\text{\bf T}_{w}$ where $\theta_{w}$ is an order $2$ character of $\text{\bf
T}_{w}$ (cf. [Pan16] §3.3).

When $G$ is a unitary group or an orthogonal group, an irreducible character
$\rho$ of $G$ is pseudo-unipotent if and only if $\rho\chi_{\text{\bf G}}$ is
unipotent where $\chi_{\text{\bf G}}$ is a fixed character of $G$ of order $2$
(cf. Lemma 5.1). However, the situation for symplectic groups is more
interesting.

######  Proposition 1.6.

A symplectic group ${\rm Sp}_{2n}(q)$ has a pseudo-unipotent cuspidal
character if and only if $n=m^{2}$ for some integer $m$. Moreover ${\rm
Sp}_{2m^{2}}(q)$, $m>0$, has exactly two irreducible pseudo-unipotent cuspidal
characters.

The description of the first occurrences of unipotent or pseudo-unipotent
cuspidal charatcers for the dual pairs $({\rm Sp}_{2n},{\rm
O}^{\epsilon}_{2n^{\prime}})$ or $({\rm Sp}_{2n},{\rm O}_{2n^{\prime}+1})$ are
given in Proposition 5.4, Proposition 5.5, and Proposition 5.6.

### 1.5.

The contents of this article are as follows. In Section 2 we first give the
definitions and notations which are used in this paper. Then we recall some
basic properties of Deligne-Lusztig virtual character and the characterization
of unipotent characters by Lusztig. In Section 3 we discuss the “ _generalized
preservation principle_ ” for the Howe correspondence of irreducible unipotent
characters. In Section 4 we deduce the generalized preservation principle for
general irreducible characters from the preservation principle of irreducible
unipotent characters discussed in Section 3. In the final section, we define
“pseudo-unipotent characters” for symplectic groups and give a complete
description of the Howe correspondence on unipotent and pseudo-unipotent
cuspidal characters.

When this article is nearly complete, the author has a chance to read the
preprint [LW20] by Liu and Wang in which some results are overlapped with the
results in Section 5 of this article. In particular, the “pseudo-unipotent
characters” here are called “$\theta$-representations” in [LW20]. However, the
approaches of two papers are somewhat different.

##  2\. Preliminaries

###  2.1. Lusztig’s symbols

Recall that a symbol $\Lambda=\binom{A}{B}$ is an ordered pair of two
$\beta$-sets (cf. Subsection 1.2). The set $A$ (resp. $B$) is called the
_first row_ (resp. the _second row_) of $\Lambda$ and is also denoted by
$\Lambda^{*}$ (resp. $\Lambda_{*}$). One the set of symbols, we define an
equivalence relation generated by

$\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}\sim\binom{a_{1}+1,a_{2}+1,\ldots,a_{m_{1}}+1,0}{b_{1}+1,b_{2}+1,\ldots,b_{m_{2}}+1,0}.$

The set of equivalence classes of symbols is denoted by ${\mathcal{S}}$.

For a symbol $\Lambda=\binom{a_{1},\ldots,a_{m_{1}}}{b_{1},\ldots,b_{m_{2}}}$,
we define its _rank_ and _defect_ by

(2.1) $\displaystyle\begin{split}{\rm
rk}(\Lambda)&=\sum_{i=1}^{m_{1}}a_{i}+\sum_{j=1}^{m_{2}}b_{j}-\biggl{\lfloor}\biggr{(}\frac{m_{1}+m_{2}-1}{2}\biggr{)}^{2}\biggr{\rfloor},\\\
{\rm def}(\Lambda)&=m_{1}-m_{2}.\end{split}$

It is easy to check that

${\rm rk}(\Lambda)\geq\biggl{\lfloor}\biggl{(}\frac{{\rm
def}(\Lambda)}{2}\biggr{)}^{2}\biggr{\rfloor}$

for any $\Lambda$. A symbol $\Lambda$ is called _cuspidal_ if the above
inequality is in fact an equality. It is known that $\Lambda$ is cuspidal if
and only if the unipotent character $\rho_{\Lambda}$ is cuspidal.

Recall that a _bi-partition_ $\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}$ of $n$ is an
ordered pair of two partitions $\mu,\nu$ such that
$\bigl{|}\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}\bigr{|}:=|\mu|+|\nu|=n$. We define
a mapping from symbols to bi-partitions by

(2.2)
$\Upsilon\colon\binom{a_{1},a_{2},\cdots,a_{m_{2}}}{b_{1},b_{2},\cdots,b_{m_{2}}}\mapsto\genfrac{[}{]}{0.0pt}{}{a_{1}-(m_{1}-1),a_{2}-(m_{2}-1),\ldots,a_{m_{1}}}{b_{1}-(m_{2}-1),b_{2}-(m_{2}-2),\ldots,b_{m_{2}}}.$

It is easy to see that $\Lambda_{1}\sim\Lambda_{2}$ if and only if
$\Upsilon(\Lambda_{1})=\Upsilon(\Lambda_{2})$. Moreover, it is not difficult
to check that

(2.3) ${\rm
rk}(\Lambda)=|\Upsilon(\Lambda)|+\biggl{\lfloor}\biggl{(}\frac{{\rm
def}(\Lambda)}{2}\biggr{)}^{2}\biggr{\rfloor}=|\Upsilon(\Lambda)|+\begin{cases}\bigl{(}\frac{{\rm
def}(\Lambda)}{2}\bigr{)}^{2},&\text{if ${\rm def}(\Lambda)$ even};\\\
\bigl{(}\frac{{\rm def}(\Lambda)-1}{2}\bigr{)}\bigl{(}\frac{{\rm
def}(\Lambda)+1}{2}\bigr{)},&\text{if ${\rm def}(\Lambda)$ odd}.\end{cases}$

Therefore $\Lambda$ is cuspidal if and only if $\Upsilon(\Lambda)$ is the
empty partition $\genfrac{[}{]}{0.0pt}{}{-}{-}$.

Recall that for a $\beta$-set $A$, a number $\delta(A)$ is given in Subsection
1.2. It is clear that $\delta(A)\geq 0$, and $\delta(A)=0$ if and only if
$A=\emptyset$ or $A=\\{m-1,m-2,\ldots,0\\}$ for some $m\geq 1$. For a symbol
$\Lambda=\binom{A}{B}$, it is clear that $\delta(\Lambda)=\delta(\Lambda^{\rm
t})$ where $\binom{A}{B}^{\rm t}=\binom{B}{A}$, and
$\delta(\Lambda)=\delta(\Lambda^{\prime})$ if $\Lambda\sim\Lambda^{\prime}$.

######  Lemma 2.4.

A symbol $\Lambda$ is cuspidal if and only if $\delta(\Lambda)=0$.

###### Proof.

Suppose that $\delta(\Lambda)=0$ and write $\Lambda=\binom{A}{B}$. Now
$\delta(\Lambda)=0$ implies that $\delta(A)=\delta(B)=0$. Hence $A$ is either
empty or $A=\\{m_{1},m_{1}-1,\ldots,1,0\\}$ for some $m_{1}\geq 0$. Similarly,
$B$ is either empty or $B=\\{m_{2},m_{2}-1,\ldots,1,0\\}$ for some $m_{2}\geq
0$. If $A$ or $B$ is empty, it is clear that $\Lambda$ is cuspidal. If $A$ and
$B$ are both non-empty, without loss of generality, may assume that $m_{1}\geq
m_{2}$, then

$\Lambda=\binom{m_{1},m_{1}-1,\ldots,1,0}{m_{2},m_{2}-1,\ldots,1,0}\sim\binom{m_{1}-m_{2}-1,m_{1}-m_{2}-2,\ldots,1,0}{-}.$

So $\Lambda$ is again cuspidal. On the other hand, it is also clear by
definition that $\delta(\Lambda)=0$ if $\Lambda$ is cuspidal. ∎

######  Lemma 2.5.

Suppose that ${\rm rk}(\Lambda)=n$. Then we have $\delta(\Lambda)\leq n$.
Moreover, $\delta(\Lambda)=n$ if and only if

$\Lambda\sim\begin{cases}\binom{k}{n-k+1,0},&\text{if\/ ${\rm
def}(\Lambda)=-1$};\\\ \binom{n-k}{k},&\text{if\/ ${\rm def}(\Lambda)=0$};\\\
\binom{n-k+1,0}{k},&\text{if\/ ${\rm def}(\Lambda)=1$}\end{cases}$

for some $0\leq k\leq n$.

###### Proof.

Suppose that ${\rm rk}(\Lambda)=n$ and write
$\Lambda=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}$.
Because $\delta(\Lambda)$ is invariant on the similar class of $\Lambda$, we
may assume that $m_{1},m_{2}$ are both nonzero. Now by definition we have

$\delta(\Lambda)=(a_{1}-m_{1}+1)+(b_{1}-m_{2}+1)\leq|\Upsilon(\Lambda)|={\rm
rk}(\Lambda)-\biggl{\lfloor}\biggl{(}\frac{{\rm
def}(\Lambda)}{2}\biggr{)}^{2}\biggr{\rfloor}\leq n.$

It is easy to check that if $\Lambda$ is similar to one of the symbols listed
in the lemma, then $\delta(\Lambda)=n$. On the other hand, suppose that
$\delta(\Lambda)=n$. Then we must have $\bigl{\lfloor}\bigl{(}\frac{{\rm
def}(\Lambda)}{2}\bigr{)}^{2}\bigr{\rfloor}=0$ and
$\Upsilon(\Lambda)=\genfrac{[}{]}{0.0pt}{}{a_{1}-m_{1}+1}{b_{1}-m_{2}+1}=\genfrac{[}{]}{0.0pt}{}{n-k}{k}$
for some $0\leq k\leq n$. This implies that ${\rm def}(\Lambda)=0,\pm 1$ and
$\Lambda$ is similar to one the symbols listed in the lemma. ∎

######  Lemma 2.6.

Suppose that ${\rm rk}(\Lambda)=n$ and ${\rm def}(\Lambda)=\pm 2$. Then we
have $\delta(\Lambda)\leq n-1$. Moreover, $\delta(\Lambda)=n-1$ if and only if

$\Lambda\sim\textstyle\binom{k}{n-k+1,1,0},\binom{n-k+1,1,0}{k}$

for some $0\leq k\leq n-1$.

###### Proof.

Write
$\Lambda=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}$.
From the proof of the previous lemma, we see that

$\delta(\Lambda)\leq|\Upsilon(\Lambda)|={\rm
rk}(\Lambda)-\biggl{\lfloor}\biggl{(}\frac{{\rm
def}(\Lambda)}{2}\biggr{)}^{2}\biggr{\rfloor}=n-1.$

Moreover, if $\delta(\Lambda)=n-1$, we need
$\Upsilon(\Lambda)=\genfrac{[}{]}{0.0pt}{}{a_{1}-m_{1}+1}{b_{1}-m_{2}+1}=\genfrac{[}{]}{0.0pt}{}{n-k-1}{k}$
for some $0\leq k\leq n-1$, i.e.,
$\Lambda\sim\binom{n-k+1,1,0}{k},\binom{k}{n-k+1,1,0}$ for some $0\leq k\leq
n-1$. ∎

###  2.2. Lusztig parametrizations for classical groups

In this subsection, we recall the basic results of Lusztig’s parametrization
of irreducible characters of finite classical groups. First we define

$\displaystyle{\mathcal{S}}_{{\rm O}^{+}_{\rm even}}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}\mid{\rm def}(\Lambda)\equiv
0\pmod{4}\,\\},$ $\displaystyle{\mathcal{S}}_{{\rm O}_{2n}^{+}}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}_{{\rm O}^{+}}\mid{\rm
rk}(\Lambda)=n\,\\};$ $\displaystyle{\mathcal{S}}_{\rm Sp}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}\mid{\rm def}(\Lambda)\equiv
1\pmod{4}\,\\},$ $\displaystyle{\mathcal{S}}_{{\rm Sp}_{2n}}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}_{\rm Sp}\mid{\rm
rk}(\Lambda)=n\,\\};$ $\displaystyle{\mathcal{S}}_{{\rm O}^{-}_{\rm even}}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}\mid{\rm def}(\Lambda)\equiv
2\pmod{4}\,\\},$ $\displaystyle{\mathcal{S}}_{{\rm O}_{2n}^{-}}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}_{{\rm O}^{-}}\mid{\rm
rk}(\Lambda)=n\,\\}.$

Then we have the following parametrization of irreducible unipotent characters
from [Lus77] theorem 8.2:

######  Proposition 2.7 (Lusztig).

Let $\text{\bf G}={\rm Sp}_{2n}$ or ${\rm O}^{\epsilon}_{2n}$ for
$\epsilon=\pm$. Then there exists a bijective correspondence between
${\mathcal{S}}_{\text{\bf G}}$ and ${\mathcal{E}}(G)_{1}$.

The following result (cf. [DM91] theorem 13.23, remark 13.24) is fundamental
for the classification of ${\mathcal{E}}(G)$:

######  Proposition 2.8 (Lusztig).

Let $\text{\bf G}={\rm Sp}_{2n}$, ${\rm SO}_{2n+1}$, or ${\rm
O}^{\epsilon}_{2n}$ for $\epsilon=\pm$, and let $s\in(G^{*})^{0}$. There is a
bijection
${\mathfrak{L}}_{s}\colon{\mathcal{E}}(G)_{s}\rightarrow{\mathcal{E}}(C_{G^{*}}(s))_{1}$
satisfying the condition

$\langle\rho,\epsilon_{\text{\bf G}}R^{\text{\bf G}}_{\text{\bf
T}^{*},s}\rangle_{G}=\langle{\mathfrak{L}}_{s}(\rho),\epsilon_{C_{\text{\bf
G}^{*}}(s)}R^{C_{\text{\bf G}^{*}}(s)}_{\text{\bf T}^{*},{\bf
1}}\rangle_{C_{G^{*}}(s)}$

for any rational maximal torus $\text{\bf T}^{*}$ containing $s$ where
$C_{G^{*}}(s)$ denotes the centralizer of $s$ in $G^{*}$, $\epsilon_{\text{\bf
G}}=(-1)^{r}$ and $r$ is the ${\mathbf{F}}_{q}$-rank of G, and
$\langle,\rangle_{G}$ denotes the inner product on the space of class
functions on $G$.

Such a bijection ${\mathfrak{L}}_{s}$ is called a _Lusztig correspondence_.
The following proposition follows from [Lus84] (9.9.1):

######  Proposition 2.9 (Lusztig).

The Lusztig series ${\mathcal{E}}(G)_{s}$ has a cuspidal character if and only
if ${\mathcal{E}}(C_{G^{*}}(s))_{1}$ has a (unipotent) cuspidal character and
the largest ${\mathbf{F}}_{q}$-split torus in the center of $C_{G^{*}}(s)$
coincides with the largest ${\mathbf{F}}_{q}$-split torus in the center of
$G^{*}$. In this case, $\rho$ is cuspidal if and only if
${\mathfrak{L}}_{s}(\rho)$ is cuspidal.

###  2.3. Theta correspondence for $({\rm Sp}_{2n},{\rm
O}^{\epsilon}_{2n^{\prime}})$ and $({\rm Sp}_{2n},{\rm O}_{2n^{\prime}+1})$

Let $\lambda=[\lambda_{1},\ldots,\lambda_{k}]$ and
$\mu=[\mu_{1},\ldots,\mu_{l}]$ be two partitions with
$\lambda_{1}\geq\cdots\geq\lambda_{k}\geq 0$ and
$\mu_{1}\geq\cdots\geq\mu_{l}\geq 0$. We denote

$\lambda\preccurlyeq\mu\quad\text{ if \
}\mu_{1}\geq\lambda_{1}\geq\mu_{2}\geq\lambda_{2}\geq\cdots.$

Then we define two relations on the set ${\mathcal{S}}$:

(2.10)
$\displaystyle\begin{split}{\mathcal{B}}^{+}&=\\{\,(\Lambda,\Lambda^{\prime})\in{\mathcal{S}}\times{\mathcal{S}}\mid\Upsilon(\Lambda_{*})\preccurlyeq\Upsilon(\Lambda^{\prime*}),\
\Upsilon(\Lambda^{\prime}_{*})\preccurlyeq\Upsilon(\Lambda^{*})\,\\};\\\
{\mathcal{B}}^{-}&=\\{\,(\Lambda,\Lambda^{\prime})\in{\mathcal{S}}\times{\mathcal{S}}\mid\Upsilon(\Lambda^{*})\preccurlyeq\Upsilon(\Lambda^{\prime}_{*}),\
\Upsilon(\Lambda^{\prime*})\preccurlyeq\Upsilon(\Lambda_{*})\,\\}.\end{split}$

Then we define

(2.11) $\displaystyle\begin{split}{\mathcal{B}}_{{\rm Sp},{\rm
O}^{\epsilon}_{\rm
even}}&=\\{\,(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}^{+}\cap({\mathcal{S}}_{\rm
Sp}\times{\mathcal{S}}_{{\rm O}^{\epsilon}_{\rm even}})\mid{\rm
def}(\Lambda^{\prime})=-{\rm def}(\Lambda)+\epsilon\cdot 1\,\\},\\\
{\mathcal{B}}_{{\rm Sp}_{2n},{\rm
O}_{2n^{\prime}}^{\epsilon}}&={\mathcal{B}}_{{\rm Sp},{\rm O}^{\epsilon}_{\rm
even}}\cap({\mathcal{S}}_{{\rm Sp}_{2n}}\times{\mathcal{S}}_{{\rm
O}_{2n^{\prime}}^{\epsilon}})\end{split}$

where $\epsilon=\pm$.

Recall that $\omega_{\text{\bf G},\text{\bf G}^{\prime},1}$ denotes the
unipotent part of Weil character $\omega^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}$ for the dual pair $(\text{\bf G},\text{\bf G}^{\prime})=({\rm
Sp}_{2n},{\rm O}^{\epsilon}_{2n^{\prime}})$. The following proposition is from
[Pan19a] theorem 3.4:

######  Proposition 2.12.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm Sp}_{2n},{\rm
O}^{\epsilon}_{2n^{\prime}})$. Then we have the decomposition

$\omega_{\text{\bf G},\text{\bf
G}^{\prime},1}=\sum_{(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}_{\text{\bf
G},\text{\bf G}^{\prime}}}\rho_{\Lambda}\otimes\rho_{\Lambda^{\prime}}.$

Let G be ${\rm Sp}_{2n}$, ${\rm O}^{\epsilon}_{2n}$ or ${\rm SO}_{2n+1}$. For
$s\in(G^{*})^{0}$, we define

(2.13) $\displaystyle\begin{split}\text{\bf G}^{(0)}=\text{\bf
G}^{(0)}(s)&=\prod_{\langle\xi\rangle\subset\\{\xi_{1},\ldots,\xi_{n}\\},\
\xi\neq\pm 1}\text{\bf G}_{[\xi]}(s);\\\ \text{\bf G}^{(1)}=\text{\bf
G}^{(1)}(s)&=\text{\bf G}_{[-1]}(s);\\\ \text{\bf G}^{(2)}=\text{\bf
G}^{(2)}(s)&=\begin{cases}\text{\bf G}_{[1]}(s),&\text{if $\text{\bf G}$ is an
orthogonal group};\\\ (\text{\bf G}_{[1]}(s))^{*},&\text{if $\text{\bf G}$ is
a symplectic group}.\end{cases}\end{split}$

where $\text{\bf G}_{[\xi]}(s)$ is given in [AMR96] subsection 1.B (see also
[Pan19b] subsection 2.2). We know that $\text{\bf G}^{(0)}$ is a product of
general linear groups or unitary group, and

(2.14) $(\text{\bf G}^{(1)},\text{\bf G}^{(2)})=\begin{cases}({\rm
Sp}_{2n^{(1)}},{\rm Sp}_{2n^{(2)}}),&\text{if $\text{\bf G}={\rm
SO}_{2n+1}$};\\\ ({\rm O}^{\epsilon^{(1)}}_{2n^{(1)}},{\rm
Sp}_{2n^{(2)}}),&\text{if $\text{\bf G}={\rm Sp}_{2n}$};\\\ ({\rm
O}^{\epsilon^{(1)}}_{2n^{(1)}},{\rm O}^{\epsilon^{(2)}}_{2n^{(2)}}),&\text{if
$\text{\bf G}={\rm O}^{\epsilon}_{2n}$}\end{cases}$

for some non-negative integers $n^{(1)},n^{(2)}$ depending on $s$, and some
$\epsilon^{(1)},\epsilon^{(2)}$. Note that if $\text{\bf G}={\rm
O}^{\epsilon}_{2n}$, then $\epsilon^{(1)},\epsilon^{(2)}$ also depend on $s$
(and $\epsilon$), if $\text{\bf G}={\rm Sp}_{2n}$, then $\epsilon^{(1)}$ can
be $+$ or $-$ for each $s$ such that $n^{(1)}\geq 1$. Now the element $s$ can
be written as $s=s^{(0)}\times s^{(1)}\times s^{(2)}$ where $s^{(1)}$ (resp.
$s^{(2)}$) is the part whose eigenvalues are all equal to $-1$ (resp. $1$),
and $s^{(0)}$ is the part whose eigenvalues do not contain $1$ or $-1$. There
is a natural bijection
${\mathcal{E}}(C_{G^{*}}(s))_{1}\simeq{\mathcal{E}}(G^{(0)}\times
G^{(1)}\times G^{(2)})_{1}$, and so from Proposition 2.7, we have a bijection

(2.15)
$\displaystyle\begin{split}\Xi_{s}\colon{\mathcal{E}}(G)_{s}&\rightarrow{\mathcal{E}}(G^{(0)}\times
G^{(1)}\times G^{(2)})_{1}\\\
\rho&\mapsto\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}\end{split}$

where $\rho^{(j)}\in{\mathcal{E}}(G^{(j)})_{1}$, and $\Xi_{s}$ is called a
_modified Lusztig correspondence_.

Now suppose that $\text{\bf G}={\rm O}_{2n+1}$. Then $\text{\bf G}^{0}={\rm
SO}_{2n+1}$, $G\simeq G^{0}\times\\{\pm 1\\}$, and we have

${\mathcal{E}}(G)=\\{\,\rho\otimes{\bf 1},\rho\otimes{\rm
sgn}\mid\rho\in{\mathcal{E}}(G^{0})\,\\}.$

For $s\in(G^{0})^{*}$ and $\epsilon=\pm$, we define

$\displaystyle{\mathcal{E}}(G)_{s,\epsilon}=\begin{cases}\\{\,\rho\otimes{\bf
1}\mid\rho\in{\mathcal{E}}(G^{0})_{s}\,\\},&\text{if $\epsilon=+$};\\\
\\{\,\rho\otimes{\rm sgn}\mid\rho\in{\mathcal{E}}(G^{0})_{s}\,\\},&\text{if
$\epsilon=-$}.\end{cases}$

Each ${\mathcal{E}}(G)_{s,\epsilon}$ is regarded as a Lusztig series for
$\text{\bf G}={\rm O}_{2n+1}$. It is clear that

${\mathcal{E}}(G)=\bigcup_{(s)\subset(G^{0})^{*},\
\epsilon=\pm}{\mathcal{E}}(G)_{s,\epsilon}.$

Suppose that $\text{\bf G}={\rm Sp}_{2n}$ or ${\rm O}_{2n+1}$. For
$\rho\in{\mathcal{E}}(G)$, we define $\epsilon_{\rho}=\pm 1$ by
$\rho(-1)=\epsilon_{\rho}\rho(1)$. If $\text{\bf G}={\rm O}_{2n+1}$ and
$\rho\in{\mathcal{E}}(G)_{s,\epsilon}$, then it is obvious that
$\epsilon_{\rho}=\epsilon$.

The following two propositions concerning the commutativity between the
modified Lusztig correspondence and the theta correspondence are from [Pan19b]
proposition 8.3 and proposition 8.8:

######  Proposition 2.16.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm Sp}_{2n},{\rm
O}_{2n^{\prime}}^{\epsilon})$ where $\epsilon=+$ or $-$, and let
$\rho\in{\mathcal{E}}(G)_{s}$ and
$\rho^{\prime}\in{\mathcal{E}}(G^{\prime})_{s^{\prime}}$ for some semisimple
$s\in G^{*}$ and $s^{\prime}\in(G^{\prime*})^{0}$. Write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$ and
$\Xi_{s^{\prime}}(\rho^{\prime})=\rho^{\prime(0)}\otimes\rho^{\prime(1)}\otimes\rho^{\prime(2)}$
where $\Xi_{s},\Xi_{s^{\prime}}$ are the modified Lusztig correspondences for
G and $\text{\bf G}^{\prime}$ respectively given in [Pan22]. Then
$(\rho,\rho^{\prime})$ occurs in $\Theta_{\text{\bf G},\text{\bf
G}^{\prime}}^{\psi}$ if and only if

  * •

$s^{(0)}=s^{\prime(0)}$ (up to conjugation), and
$\rho^{(0)}=\rho^{\prime(0)}$;

  * •

$\text{\bf G}^{(1)}\simeq\text{\bf G}^{\prime(1)}$, and
$\rho^{(1)}=\rho^{\prime(1)}$;

  * •

$(\rho^{(2)},\rho^{\prime(2)})$ occurs $\Theta_{\text{\bf G}^{(2)},\text{\bf
G}^{\prime(2)},1}$

where $\Theta_{\text{\bf G}^{(2)},\text{\bf G}^{\prime(2)},1}$ denotes the
unipotent part of $\Theta^{\psi}_{\text{\bf G}^{(2)},\text{\bf
G}^{\prime(2)}}$.

######  Proposition 2.17.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm Sp}_{2n},{\rm
O}_{2n^{\prime}+1})$, and let $\rho\in{\mathcal{E}}(G)_{s}$ and
$\rho^{\prime}\in{\mathcal{E}}(G^{\prime})_{s^{\prime},\epsilon^{\prime}}$ for
some semisimple $s\in G^{*}$ and $s^{\prime}\in G^{\prime*}$, and some
$\epsilon^{\prime}=\pm$. Write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$,
$\Xi_{s^{\prime}}(\rho^{\prime}|_{G^{\prime
0}})=\rho^{\prime(0)}\otimes\rho^{\prime(1)}\otimes\rho^{\prime(2)}$ where
$\Xi_{s},\Xi_{s^{\prime}}$ are the modified Lusztig correspondences for G and
$\text{\bf G}^{\prime}$ respectively given in [Pan22]. Then
$(\rho,\rho^{\prime})$ occurs in $\Theta_{\text{\bf G},\text{\bf
G}^{\prime}}^{\psi}$ if and only if

  * •

$s^{(0)}=-s^{\prime(0)}$ (up to conjugation), and
$\rho^{(0)}=\rho^{\prime(0)}$;

  * •

$\text{\bf G}^{(2)}\simeq\text{\bf G}^{\prime(1)}$ and
$\rho^{(2)}=\rho^{\prime(1)}$;

  * •

$(\rho^{(1)},\rho^{\prime(2)})$ occurs in $\Theta_{\text{\bf
G}^{(1)},\text{\bf G}^{\prime(2)},1}$;

  * •

$\epsilon_{\rho}=\epsilon^{\prime}$.

###  2.4. Theta correspondence for unitary dual pairs

For a partition $\lambda$ of $n$ (we write $|\lambda|=n$), let
$\lambda_{\infty}$ denote its _$2$ -core_ and let $\Lambda_{\lambda}$ denote
the symbol given in [Pan19b] §5. It is known that

(2.18) $|\lambda|=|\lambda_{\infty}|+2|\Upsilon(\Lambda_{\lambda})|$

(cf. [FS90] p.233). We define

${\mathcal{S}}_{{\rm U}_{n}}=\\{\,\Lambda_{\lambda}\mid\lambda\text{ a
partition of }n\,\\}.$

Define ${\mathcal{B}}^{+}_{{\rm U},{\rm U}}$ to be the set consisting of pairs
of symbols $(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}^{+}$ (cf. (2.10)) such
that

(2.19) ${\rm def}(\Lambda^{\prime})=\begin{cases}-{\rm def}(\Lambda),-{\rm
def}(\Lambda)+1,&\text{if ${\rm def}(\Lambda)$ is even};\\\ -{\rm
def}(\Lambda)+1,-{\rm def}(\Lambda)+2,&\text{if ${\rm def}(\Lambda)$ is
odd},\end{cases}$

Similarly, define ${\mathcal{B}}^{-}_{{\rm U},{\rm U}}$ to be the set
consisting of pairs of symbols
$(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}^{-}$ such that

(2.20) ${\rm def}(\Lambda^{\prime})=\begin{cases}-{\rm def}(\Lambda)-2,-{\rm
def}(\Lambda)-1,&\text{if ${\rm def}(\Lambda)$ is even};\\\ -{\rm
def}(\Lambda)-1,-{\rm def}(\Lambda),&\text{if ${\rm def}(\Lambda)$ is
odd},\end{cases}$

Then we define

(2.21) ${\mathcal{B}}_{{\rm U}_{n},{\rm
U}_{n^{\prime}}}=\\{(\Lambda_{\lambda},\Lambda_{\lambda^{\prime}})\in{\mathcal{B}}^{+}_{{\rm
U},{\rm U}}\cup{\mathcal{B}}^{-}_{{\rm U},{\rm U}}\mid|\lambda|=n,\
|\lambda^{\prime}|=n^{\prime}\,\\}.$

From [Pan19b] lemma 5.1, it is known that

(2.22) ${\mathcal{B}}_{{\rm U}_{n},{\rm
U}_{n^{\prime}}}\subset\begin{cases}{\mathcal{B}}^{+}_{{\rm U},{\rm
U}},&\text{if $n+n^{\prime}$ is even};\\\ {\mathcal{B}}^{-}_{{\rm U},{\rm
U}},&\text{if $n+n^{\prime}$ is odd}.\end{cases}$

The following proposition is rephrased from [AMR96] théorème 5.15 (cf.
[Pan19b] proposition 5.13).

######  Proposition 2.23.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm U}_{n},{\rm U}_{n^{\prime}})$.
Then we have the decomposition

$\omega_{\text{\bf G},\text{\bf
G}^{\prime},1}=\sum_{(\Lambda_{\lambda},\Lambda_{\lambda^{\prime}})\in{\mathcal{B}}_{\text{\bf
G},\text{\bf G}^{\prime}}}\rho_{\lambda}\otimes\rho_{\lambda^{\prime}}$

where $\omega_{\text{\bf G},\text{\bf G}^{\prime},1}$ denotes the unipotent
part of $\omega^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$.

For $s\in G^{*}$, we know that $C_{\text{\bf
G}^{*}}(s)=\prod_{\langle\xi\rangle}\text{\bf G}_{[\xi]}(s)$ where each
$\text{\bf G}_{[\xi]}(s)$ is a unitary group or a general linear group. Define

$\displaystyle\text{\bf G}^{(0)}$ $\displaystyle=\prod_{\langle\xi\rangle,\
\xi\neq 1}\text{\bf G}_{[\xi]}(s)$ $\displaystyle\text{\bf G}^{(1)}$
$\displaystyle=\text{\bf G}_{[1]}(s)$

where $\text{\bf G}_{[\xi]}(s)$ is given as in Subsection 2.3. Then
$C_{\text{\bf G}^{*}}(s)=\text{\bf G}^{(0)}\times\text{\bf G}^{(1)}$ and hence
there exists a (unique) one-to-one correspondence

(2.24)
$\displaystyle\begin{split}\Xi_{s}\colon{\mathcal{E}}(G)_{s}&\rightarrow{\mathcal{E}}(G^{(0)}\times
G^{(1)})_{1}\\\ \rho&\mapsto\rho^{(0)}\otimes\rho^{(1)}\end{split}$

by Proposition 2.7. Then $s$ can be written as $s=s^{(0)}\times s^{(1)}$ where
$s^{(1)}$ is the part whose eigenvalues are all equal to $1$, and $s^{(0)}$ is
the part whose eigenvalues do not contain $1$.

The following proposition can be extracted from [AMR96] théorème 2.6 (cf.
[Pan19c] theorem 3.10).

######  Proposition 2.25.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm U}_{n},{\rm U}_{n^{\prime}})$,
and let $\rho\in{\mathcal{E}}(G)_{s}$ and
$\rho^{\prime}\in{\mathcal{E}}(G^{\prime})_{s^{\prime}}$ for some semisimple
$s,s^{\prime}$. Write $\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}$ and
$\Xi_{s^{\prime}}(\rho^{\prime})=\rho^{\prime(0)}\otimes\rho^{\prime(1)}$.
Then $(\rho,\rho^{\prime})$ occurs in $\Theta^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}$ if and only if

  * •

$s^{(0)}=s^{\prime(0)}$, and $\rho^{(0)}=\rho^{\prime(0)}$,

  * •

$(\rho^{(1)},\rho^{\prime(1)})$ occurs in $\Theta_{\text{\bf
G}^{(1)},\text{\bf G}^{\prime(1)},1}$.

##  3\. Generalized Preservation Principle of Unipotent Characters

###  3.1. Preservation principle for unipotent cuspidal characters

Irreducible unipotent cuspidal characters of classical groups (other than the
general linear groups) are characterized by Lusztig as follows:

  1. (1)

${\rm U}_{n}(q)$ has an irreducible unipotent character if and only if
$n=\frac{1}{2}m(m+1)$ for some $m\geq 0$. In this case, there is a unique
irreducible unipotent cuspidal character.

  2. (2)

${\rm O}_{2n+1}(q)$ has an irreducible unipotent character if and only if
$n=m(m+1)$ for some $m\geq 0$. In this case, there are two irreducible
unipotent cuspidal characters different by the sign character.

  3. (3)

${\rm Sp}_{2n}(q)$ has an irreducible unipotent character if and only if
$n=m(m+1)$ for some $m\geq 0$. In this case, there is a unique irreducible
unipotent cuspidal character.

  4. (4)

${\rm O}^{\epsilon}_{2n}(q)$ has an irreducible unipotent character if and
only if $n=m^{2}$ and $\epsilon={\rm sgn}((-1)^{m})$ for some $m\geq 0$. In
this case, there are two irreducible unipotent cuspidal characters different
by the sign character.

When $(\text{\bf G},\text{\bf G}^{\prime})=({\rm U}_{n},{\rm U}_{n^{\prime}})$
or $({\rm Sp}_{2n},{\rm O}^{\epsilon}_{2n^{\prime}})$, the unipotent
characters are preserved by the Howe correspondence, and the first occurrence
of irreducible unipotent cuspidal characters are completely determined in
[AM93] theorems 4.1 and 5.2. And we have the following preservation principle
of unipotent cuspidal characters:

  1. (I)

For the unipotent cuspidal character $\zeta^{\rm U}_{m}$ of ${\rm
U}_{\frac{m(m+1)}{2}}(q)$, we have

$n_{0}^{\prime+}(\zeta_{m}^{\rm U})+n_{0}^{\prime-}(\zeta_{m}^{\rm
U})=\tfrac{1}{2}m(m-1)+\tfrac{1}{2}(m+1)(m+2)=m(m+1)+1.$

  2. (II)

For the unipotent cuspidal characters $\zeta^{\rm I}_{m},\zeta^{\rm II}_{m}$
of ${\rm O}^{\epsilon}_{2m^{2}}(q)$, we have

$n_{0}^{\prime}(\zeta_{m}^{\rm I})+n_{0}^{\prime}(\zeta_{m}^{\rm
II})=2m(m-1)+2m(m+1)=4m^{2}.$

  3. (III)

For the unipotent cuspidal character $\zeta_{m}^{\rm Sp}$ of ${\rm
Sp}_{2m(m+1)}(q)$ and all the orthogonal spaces ${\mathbf{v}}^{\prime\pm}$ are
even-dimensional, we have

$n_{0}^{\prime+}(\zeta_{m}^{\rm Sp})+n_{0}^{\prime-}(\zeta_{m}^{\rm
Sp})=2m^{2}+2(m+1)^{2}=4m(m+1)+2.$

Note that unipotent characters are not preserved by the Howe correspondence
for the dual pair $({\rm Sp}_{2n},{\rm O}_{2n^{\prime}+1})$.

###  3.2. Generalized preservation principle for symplectic groups

For a symbol
$\Lambda=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}\in{\mathcal{S}}_{\rm
Sp}$ and $\epsilon=\pm$, we define $\theta^{\epsilon}_{0}(\Lambda)$ by

$\displaystyle\theta_{0}^{+}(\Lambda)$
$\displaystyle=\begin{cases}\binom{b_{1},b_{2},b_{3},\ldots,b_{m_{2}}}{a_{2},a_{3},\ldots,a_{m_{1}}}=\Lambda^{\rm
t}\smallsetminus\binom{-}{a_{1}},&\text{if $m_{1}>0$};\\\
\binom{b_{1}+1,b_{2}+1,\ldots,b_{m_{2}}+1,0}{-},&\text{if
$m_{1}=0$}.\end{cases}$ $\displaystyle\theta_{0}^{-}(\Lambda)$
$\displaystyle=\begin{cases}\binom{b_{2},b_{3},\ldots,b_{m_{2}}}{a_{1},a_{2},\ldots,a_{m_{1}}}=\Lambda^{\rm
t}\smallsetminus\binom{b_{1}}{-},&\text{if $m_{2}>0$};\\\
\binom{-}{a_{1}+1,a_{2}+1,\ldots,a_{m_{1}}+1,0},&\text{if
$m_{2}=0$}.\end{cases}$

It is known that ${\rm def}(\Lambda)\equiv 1\pmod{4}$, so it is easy to see
that ${\rm def}(\theta_{0}^{+}(\Lambda))\equiv 0\pmod{4}$ and ${\rm
def}(\theta_{0}^{-}(\Lambda))\equiv 2\pmod{4}$. Then we have
$\theta_{0}^{\epsilon}(\Lambda)\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{\rm
even}}$.

######  Lemma 3.1.

For $\Lambda\in{\mathcal{S}}_{\rm Sp}$ and $\epsilon=\pm$,
$\theta^{\epsilon}_{0}(\Lambda)$ is a symbol of minimal rank in

$\Theta^{\epsilon}(\Lambda)=\\{\,\Lambda^{\prime}\in{\mathcal{S}}_{{\rm
O}^{\epsilon}_{\rm
even}}\mid(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}^{\epsilon}\,\\},$

i.e., $\theta^{\epsilon}_{0}(\Lambda)\in\Theta^{\epsilon}(\Lambda)$ and ${\rm
rk}(\theta^{\epsilon}_{0}(\Lambda))\leq{\rm rk}(\Lambda^{\prime})$ for any
$\Lambda^{\prime}\in\Theta^{\epsilon}(\Lambda)$.

###### Proof.

From [Pan21b] lemma 2.6, it is not difficult to see that
$(\Lambda,\theta^{\epsilon}_{0}(\Lambda))\in{\mathcal{B}}^{\epsilon}$ and
$\theta^{\epsilon}_{0}(\Lambda)$ is of minimal rank in
$\Theta^{\epsilon}(\Lambda)$. ∎

######  Lemma 3.2.

For $\Lambda\in{\mathcal{S}}_{\rm Sp}$, we have

${\rm rk}(\theta_{0}^{+}(\Lambda))+{\rm rk}(\theta_{0}^{-}(\Lambda))=2\,{\rm
rk}(\Lambda)-\delta(\Lambda)+1.$

###### Proof.

Write
$\Lambda=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}\in{\mathcal{S}}_{\rm
Sp}$. Note that $m_{1}-m_{2}\equiv 1\pmod{4}$. Now $m_{1}+m_{2}$ is odd and
hence

$\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}}{2}\bigr{)}^{2}\bigr{\rfloor}-\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}-1}{2}\bigr{)}^{2}\bigr{\rfloor}=\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}-1}{2}\bigr{)}^{2}\bigr{\rfloor}-\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}-2}{2}\bigr{)}^{2}\bigr{\rfloor}=\tfrac{1}{2}(m_{1}+m_{2}-1).$

If $m_{1}=0$, then $m_{2}\neq 0$ and

$\displaystyle{\rm rk}(\theta_{0}^{+}(\Lambda))$ $\displaystyle={\rm
rk}(\Lambda)+m_{2}-\tfrac{1}{2}(m_{1}+m_{2}-1);$ $\displaystyle{\rm
rk}(\theta_{0}^{-}(\Lambda))$ $\displaystyle={\rm
rk}(\Lambda)-b_{1}+\tfrac{1}{2}(m_{1}+m_{2}-1)$

by (2.1). If $m_{2}=0$, then $m_{1}\neq 0$ and

$\displaystyle{\rm rk}(\theta_{0}^{+}(\Lambda))$ $\displaystyle={\rm
rk}(\Lambda)+m_{1}-\tfrac{1}{2}(m_{1}+m_{2}-1);$ $\displaystyle{\rm
rk}(\theta_{0}^{-}(\Lambda))$ $\displaystyle={\rm
rk}(\Lambda)-a_{1}+\tfrac{1}{2}(m_{1}+m_{2}-1).$

If both $m_{1},m_{2}$ are nonzero, then

$\displaystyle{\rm rk}(\theta_{0}^{+}(\Lambda))$ $\displaystyle={\rm
rk}(\Lambda)-a_{1}+\tfrac{1}{2}(m_{1}+m_{2}-1);$ $\displaystyle{\rm
rk}(\theta_{0}^{-}(\Lambda))$ $\displaystyle={\rm
rk}(\Lambda)-b_{1}+\tfrac{1}{2}(m_{1}+m_{2}-1).$

For all three cases, we have

${\rm rk}(\theta_{0}^{+}(\Lambda))+{\rm rk}(\theta_{0}^{-}(\Lambda))=2\,{\rm
rk}(\Lambda)-\delta(\Lambda)+1.$

∎

######  Corollary 3.3.

For an irreducible unipotent character $\rho_{\Lambda}$ of ${\rm Sp}_{2n}(q)$,
we have

$n^{+}_{0}(\rho_{\Lambda})+n_{0}^{-}(\rho_{\Lambda})=4n-2\delta(\Lambda)+2.$

###### Proof.

We know that ${\rm rk}(\Lambda)=n$ and
$n_{0}^{\epsilon}(\rho_{\Lambda})=2\,{\rm rk}(\theta^{\epsilon}_{0}(\Lambda))$
by Lemma 3.1. So the corollary follows from Lemma 3.2 immediately. ∎

###  3.3. Generalized preservation principle for even orthogonal groups

For a symbol
$\Lambda=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}\in{\mathcal{S}}_{{\rm
O}^{\epsilon}_{\rm even}}$, we define $\theta_{0}(\Lambda)$ by

$\theta_{0}(\Lambda)=\begin{cases}\binom{b_{1},b_{2},b_{3},\ldots,b_{m_{2}}}{a_{2},a_{3},\ldots,a_{m_{1}}}=\Lambda^{\rm
t}\smallsetminus\binom{-}{a_{1}},&\text{if $\epsilon=+$ and $m_{1}>0$};\\\
\binom{b_{1}+1,b_{2}+1,\ldots,b_{m_{2}}+1,0}{-},&\text{if $\epsilon=+$ and
$m_{1}=0$};\\\
\binom{b_{2},b_{3},\ldots,b_{m_{2}}}{a_{1},a_{2},\ldots,a_{m_{1}}}=\Lambda^{\rm
t}\smallsetminus\binom{b_{1}}{-},&\text{if $\epsilon=-$ and $m_{2}>0$};\\\
\binom{-}{a_{1}+1,a_{2}+1,\ldots,a_{m_{1}}+1,0},&\text{if $\epsilon=-$ and
$m_{2}=0$}.\end{cases}$

If $\epsilon=+$, then ${\rm def}(\Lambda)\equiv 0\pmod{4}$; if $\epsilon=-$,
then ${\rm def}(\Lambda)\equiv 2\pmod{4}$. Then we have ${\rm
def}(\theta_{0}(\Lambda))\equiv 1\pmod{4}$ for both cases, and hence
$\theta_{0}(\Lambda)\in{\mathcal{S}}_{\rm Sp}$.

######  Lemma 3.4.

For $\Lambda\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{\rm even}}$,
$\theta_{0}(\Lambda)$ is a symbol of minimal rank in

$\Theta(\Lambda)=\\{\,\Lambda^{\prime}\in{\mathcal{S}}_{\rm
Sp}\mid(\Lambda^{\prime},\Lambda)\in{\mathcal{B}}^{\epsilon}\,\\}.$

###### Proof.

The proof is similar to that of Lemma 3.1. ∎

######  Lemma 3.5.

For $\Lambda\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{\rm even}}$, we have

${\rm rk}(\theta_{0}(\Lambda))+{\rm rk}(\theta_{0}(\Lambda^{\rm t}))=2\,{\rm
rk}(\Lambda)-\delta(\Lambda).$

###### Proof.

Write
$\Lambda=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}$.
Note that now $m_{1}+m_{2}$ is even. Suppose first that $m_{1}=m_{2}=0$. Then
$\Lambda=\Lambda^{\rm t}=\binom{-}{-}$ and $\epsilon=+$. It is clearly that
$\theta_{0}(\Lambda)=\theta_{0}(\Lambda^{\rm t})=\binom{0}{-}$, and the
assertion is obvious.

Now we suppose that $m_{1}+m_{2}\geq 2$ and we have

$\displaystyle\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}}{2}\bigr{)}^{2}\bigr{\rfloor}-\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}-1}{2}\bigr{)}^{2}\bigr{\rfloor}$
$\displaystyle=\tfrac{1}{2}(m_{1}+m_{2});$
$\displaystyle\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}-1}{2}\bigr{)}^{2}\bigr{\rfloor}-\bigl{\lfloor}\bigl{(}\tfrac{m_{1}+m_{2}-2}{2}\bigr{)}^{2}\bigr{\rfloor}$
$\displaystyle=\tfrac{1}{2}(m_{1}+m_{2}-2).$

If $m_{1}=0$, then $m_{2}\neq 0$ and

${\rm rk}(\theta_{0}(\Lambda))+{\rm rk}(\theta_{0}(\Lambda^{\rm t}))={\rm
rk}(\Lambda)+m_{2}-\tfrac{1}{2}(m_{1}+m_{2})+{\rm
rk}(\Lambda)-b_{1}+\tfrac{1}{2}(m_{1}+m_{2}-2).$

If $m_{2}=0$, then $m_{1}\neq 0$ and

${\rm rk}(\theta_{0}(\Lambda))+{\rm rk}(\theta_{0}(\Lambda^{\rm t}))={\rm
rk}(\Lambda)+m_{1}-\tfrac{1}{2}(m_{1}+m_{2})+{\rm
rk}(\Lambda)-a_{1}+\tfrac{1}{2}(m_{1}+m_{2}-2).$

If both $m_{1},m_{2}$ are nonzero, then

${\rm rk}(\theta_{0}(\Lambda))+{\rm rk}(\theta_{0}(\Lambda^{\rm t}))=2\,{\rm
rk}(\Lambda)-(a_{1}+b_{1})+(m_{1}+m_{2}-2).$

For all cases we have

${\rm rk}(\theta_{0}(\Lambda))+{\rm rk}(\theta_{0}(\Lambda^{\rm t}))=2\,{\rm
rk}(\Lambda)-\delta(\Lambda).$

∎

######  Corollary 3.6.

For an irreducible unipotent character $\rho_{\Lambda}$ of ${\rm
O}^{\epsilon}_{2n}(q)$, we have

$n^{\prime}_{0}(\rho_{\Lambda})+n^{\prime}_{0}(\rho_{\Lambda}\cdot{\rm
sgn})=4n-2\delta(\Lambda).$

###### Proof.

Now ${\rm rk}(\Lambda)={\rm rk}(\Lambda^{\rm t})=n$, $\rho_{\Lambda^{\rm
t}}=\rho_{\Lambda}\cdot{\rm sgn}$, and $n^{\prime}_{0}(\rho_{\Lambda})=2\,{\rm
rk}(\theta_{0}(\Lambda))$ by Lemma 3.4. So the corollary follows from Lemma
3.5 immediately. ∎

###  3.4. Generalized preservation principle for unitary groups

Suppose that $(\rho_{\lambda},\rho_{\lambda^{\prime\epsilon}})$ occurs in the
Howe correspondence for the dual pair $({\rm U}_{n},{\rm
U}_{n^{\prime\epsilon}})$ where $|\lambda|=n$,
$|\lambda^{\prime\epsilon}|=n^{\prime\epsilon}$, $\epsilon=\pm$ and
$n^{\prime+}$ is an even integer, $n^{\prime-}$ is an odd integer.

  1. (1)

Suppose that $n$ is even. Then we know that
$(\Lambda_{\lambda},\Lambda_{\lambda^{\prime+}})\in{\mathcal{B}}^{+}$ and
$(\Lambda_{\lambda},\Lambda_{\lambda^{\prime-}})\in{\mathcal{B}}^{-}$ by
(2.22). To achieve the minimal rank, we see that
$\Lambda_{\lambda^{\prime\epsilon}}$ must be equal to
$\theta^{\epsilon}_{0}(\Lambda_{\lambda})$ given in Lemma 3.1, or
equivalently, equal to $\theta_{0}(\Lambda_{\lambda})$ with given $\epsilon$
in Lemma 3.4.

  2. (2)

Suppose that $n$ is odd. Then we know that
$(\Lambda_{\lambda},\Lambda_{\lambda^{\prime-}})\in{\mathcal{B}}^{+}$ and
$(\Lambda_{\lambda},\Lambda_{\lambda^{\prime+}})\in{\mathcal{B}}^{-}$. To
achieve the minimal rank, we see that $\Lambda_{\lambda^{\prime\epsilon}}$
must be equal to $\theta^{-\epsilon}_{0}(\Lambda_{\lambda})$ given in Lemma
3.1, or equivalently, equal to $\theta_{0}(\Lambda_{\lambda})$ with given
$-\epsilon$ in Lemma 3.4.

Then from Proposition 3.2 and Proposition 3.5, we know that

(3.7) ${\rm rk}(\theta_{0}^{+}(\Lambda_{\lambda}))+{\rm
rk}(\theta_{0}^{-}(\Lambda_{\lambda}))=\begin{cases}2\,{\rm
rk}(\Lambda_{\lambda})-\delta(\Lambda_{\lambda})+1,&\text{if ${\rm
def}(\Lambda_{\lambda})$ is odd};\\\ 2\,{\rm
rk}(\Lambda_{\lambda})-\delta(\Lambda_{\lambda}),&\text{if ${\rm
def}(\Lambda_{\lambda})$ is even}.\end{cases}$

######  Proposition 3.8.

For an irreducible unipotent character $\rho_{\lambda}$ of ${\rm U}_{n}(q)$,
we have

$n_{0}^{\prime+}(\rho_{\lambda})+n_{0}^{\prime-}(\rho_{\lambda})=2n-2\delta(\rho_{\lambda})+1.$

###### Proof.

Let $\lambda$ be a partition of $n$, and suppose
$|\lambda_{\infty}|=\frac{1}{2}d(d+1)$ for some integer $d$. If ${\rm
def}(\Lambda_{\lambda})$ is odd, then by (2.3), (2.18) and (3.7), we conclude
that

$\displaystyle n_{0}^{+}(\rho_{\lambda})+n_{0}^{-}(\rho_{\lambda})$
$\displaystyle=\frac{1}{2}d(d-1)+\frac{1}{2}(d+1)(d+2)+2\,{\rm
rk}(\theta_{0}^{+}(\Lambda))+2\,{\rm rk}(\theta_{0}^{-}(\Lambda))$
$\displaystyle\qquad-2\biggl{[}\frac{-{\rm
def}(\Lambda_{\lambda})-1}{2}\biggr{]}^{2}-2\biggl{[}\frac{-{\rm
def}(\Lambda_{\lambda})+1}{2}\biggr{]}^{2}$ $\displaystyle=d(d+1)+1+4\,{\rm
rk}(\Lambda_{\lambda})-4\biggl{[}\frac{{\rm
def}(\Lambda_{\lambda})-1}{2}\biggr{]}\biggl{[}\frac{{\rm
def}(\Lambda_{\lambda})+1}{2}\biggr{]}-2\delta(\Lambda_{\lambda})$
$\displaystyle=2\,|\lambda|-2\delta(\Lambda_{\lambda})+1.$

If ${\rm def}(\Lambda_{\lambda})$ is even, then similarly we conclude that

$\displaystyle n_{0}^{+}(\rho_{\lambda})+n_{0}^{-}(\rho_{\lambda})$
$\displaystyle=\frac{1}{2}d(d-1)+\frac{1}{2}(d+1)(d+2)+2\,{\rm
rk}(\theta_{0}^{+}(\Lambda))+2\,{\rm rk}(\theta_{0}^{-}(\Lambda))$
$\displaystyle\quad-2\biggl{[}\frac{-{\rm
def}(\Lambda_{\lambda})-2}{2}\biggr{]}\biggl{[}\frac{-{\rm
def}(\Lambda_{\lambda})}{2}\biggr{]}-2\biggl{[}\frac{-{\rm
def}(\Lambda_{\lambda})}{2}\biggr{]}\biggl{[}\frac{-{\rm
def}(\Lambda_{\lambda})+2}{2}\biggr{]}$ $\displaystyle=d(d+1)+1+4\,{\rm
rk}(\Lambda_{\lambda})-4\biggl{[}\frac{{\rm
def}(\Lambda_{\lambda})}{2}\biggr{]}^{2}-2\delta(\Lambda_{\lambda})$
$\displaystyle=2\,|\lambda|-2\delta(\Lambda_{\lambda})+1.$

Note that $\delta(\rho_{\lambda})$ is defined to be
$\delta(\Lambda_{\lambda})$. ∎

##  4\. General Generalized Preservation Principle

Now we can deduce the generalized preservation principle for general
irreducible characters from Proposition 2.25, Proposition 2.16, Proposition
2.17 and the result for unipotent characters in Section 3.

###  4.1. For unitary groups

In this subsection, we consider type (I), i.e., $G=G({\mathbf{v}})$ is a
unitary group.

######  Theorem 4.1.

Let $\rho$ be an irreducible character of ${\rm U}({\mathbf{v}})$. Then

$n_{0}^{\prime+}(\rho)+n_{0}^{\prime-}(\rho)=2\dim({\mathbf{v}})-2\delta(\rho)+1.$

###### Proof.

Let $G={\rm U}({\mathbf{v}})$ and suppose that $\rho\in{\mathcal{E}}(G)_{s}$
for some $s$ and write $\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}$ by (2.24).
Suppose that $\rho$ first occurs in the correspondences for both dual pairs
$(G({\mathbf{v}}),G({\mathbf{v}}^{\prime+}))$ and
$(G({\mathbf{v}}),G({\mathbf{v}}^{\prime-}))$. Recall that $C_{\text{\bf
G}^{*}}=\prod_{\langle\xi\rangle}\text{\bf G}_{[\xi]}$ and we identify
$\text{\bf G}^{*}=\text{\bf G}$. Then we have decompositions

${\mathbf{v}}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq
1}{\mathbf{v}}_{[\xi]}\Biggr{]}\oplus{\mathbf{v}}_{[1]},\qquad{\mathbf{v}}^{\prime\pm}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\
\xi\neq
1}{\mathbf{v}}_{[\xi]}^{\prime\pm}\Biggr{]}\oplus{\mathbf{v}}_{[1]}^{\prime\pm}.$

By Proposition 2.25 we have
$\dim({\mathbf{v}}_{[\xi]})=\dim({\mathbf{v}}_{[\xi]}^{\prime\pm})$ for
$\xi\neq 1$. Moreover the unipotent character $\rho^{(1)}$ first occurs in the
correspondences for both dual pairs
$(G({\mathbf{v}}_{[1]}),G({\mathbf{v}}_{[1]}^{\prime+}))$ and
$(G({\mathbf{v}}_{[1]}),G({\mathbf{v}}_{[1]}^{\prime-}))$. Note that
${\mathbf{v}}_{[1]}^{\prime+}$ and ${\mathbf{v}}_{[1]}^{\prime-}$ are in
different Witt series. Hence we have

$\dim({\mathbf{v}}^{\prime+}_{[1]})+\dim({\mathbf{v}}^{\prime-}_{[1]})=2\dim({\mathbf{v}}_{[1]})-2\delta(\rho^{(1)})+1$

by Proposition 3.8. Therefore

$\displaystyle\dim({\mathbf{v}}^{\prime+})+\dim({\mathbf{v}}^{\prime-})$
$\displaystyle=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{\prime+})+\dim({\mathbf{v}}_{[1]}^{\prime+})+\sum_{\langle\xi\rangle,\
\xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{\prime-})+\dim({\mathbf{v}}_{[1]}^{\prime-})$
$\displaystyle=2\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{[\xi]})+2\dim({\mathbf{v}}_{[1]})-2\delta(\rho^{(1)})+1$
$\displaystyle=2\dim({\mathbf{v}})-2\delta(\rho)+1.$

Note that $\delta(\rho)=\delta(\rho^{(1)})$ by definition in (1.3). ∎

###  4.2. For orthogonal groups

In this subsection, we consider type (II), i.e., $G=G({\mathbf{v}})$ is an
orthogonal group.

######  Theorem 4.2.

Let $\rho$ be an irreducible character of ${\rm O}({\mathbf{v}})$. Then

$n_{0}^{\prime}(\rho)+n_{0}^{\prime}(\rho\cdot{\rm
sgn})=2\dim({\mathbf{v}})-2\delta(\rho).$

###### Proof.

Let $G={\rm O}({\mathbf{v}})$, and suppose that $(\rho,\rho^{\prime})$ (resp.
$(\rho\cdot{\rm sgn},\rho^{\prime\prime})$) first occurs in the correspondence
for the dual pair $({\rm O}({\mathbf{v}}),{\rm Sp}({\mathbf{v}}^{\prime}))$
(resp. $({\rm O}({\mathbf{v}}),{\rm Sp}({\mathbf{v}}^{\prime\prime}))$).

  1. (1)

Suppose that ${\mathbf{v}}$ is even-dimensional and identify $G({\mathbf{v}})$
with its dual group. Suppose that $\rho\in{\mathcal{E}}(G)_{s}$ for some $s$
and write $\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$. It is
known that $\rho\cdot{\rm sgn}\in{\mathcal{E}}(G)_{s}$ and

$\Xi_{s}(\rho\cdot{\rm sgn})=\rho^{(0)}\otimes(\rho^{(1)}\cdot{\rm
sgn})\otimes(\rho^{(2)}\cdot{\rm sgn}).$

Now we have a decomposition

${\mathbf{v}}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq\pm
1}{\mathbf{v}}_{[\xi]}\Biggr{]}\oplus{\mathbf{v}}_{[-1]}\oplus{\mathbf{v}}_{[1]}$

such that

$G^{(0)}=\prod_{\langle\xi\rangle,\ \xi\neq\pm 1}G({\mathbf{v}}_{[\xi]}),\quad
G^{(1)}=G({\mathbf{v}}_{[-1]}),\quad
G^{(2)}=G({\mathbf{v}}_{[1]})^{*}=G({\mathbf{v}}_{[1]}).$

Note that $G({\mathbf{v}}_{[1]})$ is an even orthogonal group.

Let ${\mathbf{v}}^{\prime*}$ be the odd-dimensioanl quadratic space such that
$G({\mathbf{v}}^{\prime*})$ is the dual group of $G({\mathbf{v}}^{\prime})$.
Then we have

${\mathbf{v}}^{\prime*}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq\pm
1}{\mathbf{v}}_{[\xi]}^{\prime*}\Biggr{]}\oplus{\mathbf{v}}_{[-1]}^{\prime*}\oplus{\mathbf{v}}_{[1]}^{\prime*}.$

Now ${\mathbf{v}}_{[1]}^{\prime*}$ is an odd-dimensional quadratic space. Let
${\mathbf{v}}_{[1]}^{\prime}$ be a symplectic space such that
$G({\mathbf{v}}_{[1]}^{\prime})$ are the dual groups of
$G({\mathbf{v}}_{[1]}^{\prime*})$. By Proposition 2.16, we have
$\dim({\mathbf{v}}_{[\xi]})=\dim({\mathbf{v}}_{[\xi]}^{\prime*})$ for $\xi\neq
1$. Moreover, $\rho^{(2)}$ first occurs in the correspondences for the dual
pairs $(G({\mathbf{v}}_{[1]}),G({\mathbf{v}}_{[1]}^{\prime}))$. Similarly, we
have a decomposition of ${\mathbf{v}}^{\prime\prime*}$ as above, and
$\rho^{(2)}\cdot{\rm sgn}$ first occurs in the correspondences for the dual
pairs $(G({\mathbf{v}}_{[1]}),G({\mathbf{v}}_{[1]}^{\prime\prime}))$. Hence by
Corollary 3.6, we have

$\dim({\mathbf{v}}_{[1]}^{\prime})+\dim({\mathbf{v}}_{[1]}^{\prime\prime})=2\dim({\mathbf{v}}_{[1]})-2\delta(\rho^{(2)}),$

and therefore

$\displaystyle\dim({\mathbf{v}}^{\prime})+\dim({\mathbf{v}}^{\prime\prime})$
$\displaystyle=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}^{\prime*}_{[\xi]})+\dim({\mathbf{v}}^{\prime}_{[1]})+\sum_{\langle\xi\rangle,\
\xi\neq
1}\dim({\mathbf{v}}^{\prime\prime*}_{[\xi]})+\dim({\mathbf{v}}^{\prime\prime}_{[1]})$
$\displaystyle=2\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{[\xi]})+2\dim({\mathbf{v}}_{[1]})-2\delta(\rho^{(2)})$
$\displaystyle=2\dim({\mathbf{v}})-2\delta(\rho).$

  2. (2)

Suppose that ${\mathbf{v}}$ is odd-dimensional, and $G={\rm O}({\mathbf{v}})$,
$G^{0}={\rm SO}({\mathbf{v}})$. Suppose that
$\rho\in{\mathcal{E}}(G)_{s,\epsilon}$ for some $s$ and some $\epsilon=\pm$.
Then $\rho\cdot{\rm sgn}\in{\mathcal{E}}(G)_{s,-\epsilon}$, and we can write
$\Xi_{s}(\rho|_{G^{0}})=\Xi_{s}(\rho\cdot{\rm
sgn}|_{G^{0}})=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$.

Now ${\mathbf{v}}^{*}$ is an even-dimensional symplectic space with
$\dim({\mathbf{v}}^{*})=\dim({\mathbf{v}})-1$ such that
$G({\mathbf{v}}^{*})=G^{0}({\mathbf{v}})^{*}$, and there is a decomposition

${\mathbf{v}}^{*}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq\pm
1}{\mathbf{v}}^{*}_{[\xi]}\Biggr{]}\oplus{\mathbf{v}}_{[1]}^{*}\oplus{\mathbf{v}}_{[-1]}^{*}$

where both ${\mathbf{v}}^{*}_{[1]}$ and ${\mathbf{v}}^{*}_{[-1]}$ are
symplectic spaces, and hence

(4.3) $\dim({\mathbf{v}})=\dim({\mathbf{v}}^{*})+1=\sum_{\langle\xi\rangle,\
\xi\neq-1}\dim({\mathbf{v}}^{*}_{[\xi]})+\dim({\mathbf{v}}_{[-1]}^{*})+1.$

Suppose that $(\rho,\rho^{\prime})$ first occurs in the correspondence for
$({\rm O}({\mathbf{v}}),{\rm Sp}({\mathbf{v}}^{\prime}))$. Then
${\mathbf{v}}^{\prime*}$ is odd-dimensional, and there are decompositions

${\mathbf{v}}^{\prime*}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq\pm
1}{\mathbf{v}}^{\prime*}_{[\xi]}\Biggr{]}\oplus{\mathbf{v}}_{[1]}^{\prime*}\oplus{\mathbf{v}}_{[-1]}^{\prime*}$

where ${\mathbf{v}}^{\prime*}_{[1]}$ is odd-dimensional and
${\mathbf{v}}^{\prime*}_{[-1]}$ is even-dimensional. Let
${\mathbf{v}}_{[1]}^{\prime}$ be the symplectic space such that
$G({\mathbf{v}}_{[1]}^{\prime})$ is the dual group of
$G({\mathbf{v}}_{[1]}^{\prime*})$. Then

$\dim({\mathbf{v}}^{\prime})=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}^{\prime*}_{[\xi]})+\dim({\mathbf{v}}^{\prime}_{[1]}).$

By Proposition 2.17, we have
$\dim({\mathbf{v}}^{*}_{[\xi]})=\dim({\mathbf{v}}^{\prime*}_{[\xi]})$ for
$\xi\neq\pm 1$,
$\dim({\mathbf{v}}_{[-1]}^{*})=\dim({\mathbf{v}}^{\prime}_{[1]})$, and
$\rho^{(2)}$ first occurs in the correspondence for
$(G({\mathbf{v}}^{*}_{[1]}),G({\mathbf{v}}^{\prime*}_{[-1]}))$.

Suppose that $(\rho\cdot{\rm sgn},\rho^{\prime\prime})$ first occurs in the
correspondence for $({\rm O}({\mathbf{v}}),{\rm
Sp}({\mathbf{v}}^{\prime\prime}))$. Similarly, we have
$\dim({\mathbf{v}}^{*}_{[\xi]})=\dim({\mathbf{v}}^{\prime\prime*}_{[\xi]})$
for $\xi\neq\pm 1$,
$\dim({\mathbf{v}}_{[-1]}^{*})=\dim({\mathbf{v}}^{\prime\prime}_{[1]})$, and
$\rho^{(2)}$ first occurs in the correspondence for
$(G({\mathbf{v}}^{*}_{[1]}),G({\mathbf{v}}^{\prime\prime*}_{[-1]}))$.

Now from [Pan19b] lemma 8.6 and proposition 8.7, we known that the two even-
dimensional quadratic spaces ${\mathbf{v}}^{\prime*}_{[-1]}$ and
${\mathbf{v}}^{\prime\prime*}_{[-1]}$ are in different Witt series. Hence

$\dim({\mathbf{v}}^{\prime*}_{[-1]})+\dim({\mathbf{v}}^{\prime\prime*}_{[-1]})=2\dim({\mathbf{v}}^{*}_{[1]})-2\delta(\rho^{(2)})+2$

by Corollary 3.3. Therefore by (4.3), we have

$\displaystyle\dim({\mathbf{v}}^{\prime})+\dim({\mathbf{v}}^{\prime\prime})$
$\displaystyle=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{\xi}^{\prime*})+\dim({\mathbf{v}}^{\prime}_{[1]})+\sum_{\langle\xi\rangle,\
\xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{\prime\prime*})+\dim({\mathbf{v}}^{\prime\prime}_{[1]})$
$\displaystyle=2\sum_{\langle\xi\rangle,\ \xi\neq\pm
1}\dim({\mathbf{v}}_{[\xi]}^{*})+2\dim({\mathbf{v}}^{*}_{[-1]})+2\dim({\mathbf{v}}^{*}_{[1]})-2\delta(\rho^{(2)})+2$
$\displaystyle=2\dim({\mathbf{v}})-2\delta(\rho).$

Note that $\delta(\rho)=\delta(\rho^{(2)})$ for this situation. ∎

######  Remark 4.4.

Suppose that $G({\mathbf{v}})={\rm O}^{+}_{2n}(q)$, and let
$\rho\in{\mathcal{E}}(G)_{s}$ for some $s$. Write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$. From (1.3) and
Lemma 2.5, we know that $\delta(\rho)=\delta(\rho^{(2)})\leq n$. This means
that if $n^{\prime}+n^{\prime\prime}<n$, no irreducible character $\rho$ of
$G$ such that $\rho$ occurs in the correspondences for $({\rm O}_{2n}^{+},{\rm
Sp}_{2n^{\prime}})$ and $\rho\cdot{\rm sgn}$ occurs in the correspondence for
$({\rm O}_{2n}^{+},{\rm Sp}_{2n^{\prime\prime}})$. Moreover, for $0\leq k\leq
n$, the unipotent character $\rho_{\binom{n-k}{k}}$ is the only irreducible
character $\rho$ of $G$ such that $\rho$ occurs in the correspondences for
$({\rm O}_{2n}^{+},{\rm Sp}_{2k})$ and $\rho\cdot{\rm sgn}$ occurs in the
correspondence for $({\rm O}_{2n}^{+},{\rm Sp}_{2(n-k)})$.

######  Remark 4.5.

Suppose that $G({\mathbf{v}})={\rm O}^{-}_{2n}(q)$, and let
$\rho\in{\mathcal{E}}(G)$ for some $s$. Write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$. Now we know that
$\delta(\rho)=\delta(\rho^{(2)})\leq n-1$. This means that if
$n^{\prime}+n^{\prime\prime}<n+1$, no irreducible character $\rho$ of $G$ such
that $\rho$ occurs in the correspondences for $({\rm O}_{2n}^{-},{\rm
Sp}_{2n^{\prime}})$ and $\rho\cdot{\rm sgn}$ occurs in the correspondence for
$({\rm O}_{2n}^{-},{\rm Sp}_{2n^{\prime\prime}})$. Moreover, by Lemma 2.6 the
unipotent character $\rho_{\binom{k}{n-k+1,1,0}}$ is the only irreducible
character $\rho$ of $G$ such that $\rho$ occurs in the correspondences for
$({\rm O}_{2n}^{-},{\rm Sp}_{2k})$ and $\rho\cdot{\rm sgn}$ occurs in the
correspondence for $({\rm O}_{2n}^{-},{\rm Sp}_{2(n-k)+2})$.

###  4.3. For symplectic groups I

In this subsection, we consider type (III), i.e., $G=G({\mathbf{v}})$ is a
symplectic group, and $G({\mathbf{v}}^{\prime\pm})$ are even orthogonal
groups.

######  Theorem 4.6.

Let $\rho$ be an irreducible character of ${\rm Sp}({\mathbf{v}})$. Suppose
that $\\{{\mathbf{v}}^{\prime\pm}\\}$ are Witt series of even-dimensional
orthogonal spaces. Then

(4.7)
$n_{0}^{\prime+}(\rho)+n_{0}^{\prime-}(\rho)=2\dim({\mathbf{v}})-2\delta(\rho)+2.$

###### Proof.

Suppose that $(\rho,\rho^{\prime+})$ (resp. $(\rho,\rho^{\prime-})$) first
occurs in the correspondence for the dual pair $({\rm Sp}({\mathbf{v}}),{\rm
O}({\mathbf{v}}^{\prime+}))$ (resp. $({\rm Sp}({\mathbf{v}}),{\rm
O}({\mathbf{v}}^{\prime-}))$). Suppose that $\rho$ is in
${\mathcal{E}}(G)_{s}$ for some $s$ and write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$. We know that $s$
determines a decomposition

${\mathbf{v}}^{*}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq\pm
1}{\mathbf{v}}^{*}_{[\xi]}\Biggr{]}\oplus{\mathbf{v}}^{*}_{[-1]}\oplus{\mathbf{v}}^{*}_{[1]}$

such that

$G^{(0)}=\prod_{\langle\xi\rangle,\ \xi\neq\pm
1}G({\mathbf{v}}^{*}_{[\xi]}),\quad G^{(1)}=G({\mathbf{v}}_{[-1]}^{*}),\quad
G^{(2)}=G({\mathbf{v}}_{[1]}^{*})^{*}.$

Let ${\mathbf{v}}_{[1]}$ be a $2\nu_{1}$-dimensional symplectic space such
that $G({\mathbf{v}}_{[1]})=G({\mathbf{v}}_{[1]}^{*})^{*}$. Note that

$\dim({\mathbf{v}})=\dim({\mathbf{v}}^{*})-1=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{*})+\dim({\mathbf{v}}_{[1]}).$

Let ${\mathbf{v}}_{[\xi]}^{\prime*\pm}$, ${\mathbf{v}}_{[-1]}^{\prime*\pm}$,
and ${\mathbf{v}}_{[1]}^{\prime\pm}$ be defined similarly, and hence

$\dim({\mathbf{v}}^{\prime\pm})=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{\prime*\pm})+\dim({\mathbf{v}}_{[1]}^{\prime\pm}).$

By Proposition 2.16, we know that
$\dim({\mathbf{v}}^{*}_{[\xi]})=\dim({\mathbf{v}}^{\prime*\pm}_{[\xi]})$ for
$\xi\neq 1$. Moreover, $\rho^{(2)}$ first occurs in the correspondences for
the dual pairs $(G({\mathbf{v}}_{[1]}),G({\mathbf{v}}_{[1]}^{\prime\pm}))$.
Note that ${\mathbf{v}}_{[1]}^{\prime+}$ and ${\mathbf{v}}_{[1]}^{\prime-}$
are both even-dimensional quadratic spaces and are in different Witt series.
Then by Corollary 3.3, we have

$\dim({\mathbf{v}}^{\prime+}_{[1]})+\dim({\mathbf{v}}^{\prime-}_{[1]})=2\dim({\mathbf{v}}_{[1]})-2\delta(\rho^{(2)})+2.$

Therefore,

$\displaystyle\dim({\mathbf{v}}^{\prime+})+\dim({\mathbf{v}}^{\prime-})$
$\displaystyle=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}^{\prime*+}_{[\xi]})+\dim({\mathbf{v}}^{\prime+}_{[1]})+\sum_{\langle\xi\rangle,\
\xi\neq
1}\dim({\mathbf{v}}^{\prime*-}_{[\xi]})+\dim({\mathbf{v}}^{\prime-}_{[1]})$
$\displaystyle=2\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{*})+2\dim({\mathbf{v}}_{[1]})-2\delta(\rho^{(2)})+2$
$\displaystyle=2\dim({\mathbf{v}})-2\delta(\rho)+2.$

Note that now $\delta(\rho)=\delta(\rho^{(2)})$. ∎

######  Remark 4.8.

Suppose that $G({\mathbf{v}})={\rm Sp}_{2n}(q)$ and both
${\mathbf{v}}^{\prime\pm}$ are even-dimensional. Let
$\rho\in{\mathcal{E}}(G)_{s}$ for some $s$, and write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$. From (1.3) and
Lemma 2.5, we know that $\delta(\rho)=\delta(\rho^{(2)})\leq n$. This means
that if $n^{\prime}+n^{\prime\prime}<n$, no irreducible character $\rho$ of
$G$ such that $\rho$ occurs in the correspondences for both $({\rm
Sp}_{2n},{\rm O}_{2n^{\prime}}^{+})$ and $({\rm Sp}_{2n},{\rm
O}^{-}_{2n^{\prime\prime}})$. Moreover, if $\delta(\Lambda)=n$, then
$\Lambda\sim\Lambda_{k}:=\binom{n-k+1,0}{k}$ for some $0\leq k\leq n$. Note
that $\theta^{+}_{0}(\Lambda_{k})=\binom{k}{0}\in{\mathcal{S}}_{{\rm
O}^{+}_{2k}}$ and
$\theta^{-}_{0}(\Lambda_{k})=\binom{n-k+1,0}{-}\in{\mathcal{S}}_{{\rm
O}^{-}_{2(n-k)+2}}$. Hence the unipotent character $\rho_{\Lambda_{k}}$ is the
only irreducible character of $G$ which occurs in the correspondences for both
$({\rm Sp}_{2n},{\rm O}^{+}_{2k})$ and $({\rm Sp}_{2n},{\rm
O}^{-}_{2(n-k)+2})$. This is an refinement of [How] corollary 5.2.1.

###  4.4. For symplectic groups II

In this subsection, we consider type (III), i.e., $G=G({\mathbf{v}})$ is a
symplectic group, and $G({\mathbf{v}}^{\prime})$ is an odd orthogonal group.

######  Theorem 4.9.

Let $\rho$ be an irreducible character of ${\rm Sp}({\mathbf{v}})$. Suppose
that $\\{{\mathbf{v}}^{\prime}\\}$ is a Witt series of odd-dimensional
orthogonal spaces. Then

(4.10)
$n_{0}^{\prime}(\rho)+n_{0}^{\prime}(\rho^{c})=2\dim({\mathbf{v}})-2\delta(\rho)+2.$

###### Proof.

Let ${\mathbf{v}}$ be a symplectic space, and let ${\mathbf{v}}^{*}$ be an
odd-dimensional orthogonal space such that
$G({\mathbf{v}}^{*})=G({\mathbf{v}})^{*}$. We have a decomposition

${\mathbf{v}}^{*}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq\pm
1}{\mathbf{v}}^{*}_{[\xi]}\Biggr{]}\oplus{\mathbf{v}}^{*}_{[-1]}\oplus{\mathbf{v}}^{*}_{[1]}.$

Now ${\mathbf{v}}_{[1]}^{*}$ is a $(2\nu_{1}+1)$-dimensional orthogonal space.
Let ${\mathbf{v}}_{[1]}$ the $2\nu_{1}$-dimensional symplectic space such that
$G({\mathbf{v}}_{[1]})$ is the dual group of $G({\mathbf{v}}_{[1]}^{*})$. Then
we have

$G^{(0)}=\prod_{\langle\xi\rangle,\ \xi\neq\pm
1}G({\mathbf{v}}^{*}_{[\xi]}),\qquad G^{(1)}=G({\mathbf{v}}_{[-1]}^{*}),\qquad
G^{(2)}=G({\mathbf{v}}_{[1]}).$

So now

$\dim({\mathbf{v}})=\dim({\mathbf{v}}^{*})-1=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}^{*}_{[\xi]})+\dim({\mathbf{v}}_{[1]}),$

and ${\mathbf{v}}^{*}_{[-1]}$ is an even-dimensional orthogonal space.

Also ${\mathbf{v}}^{\prime}$ is an odd-dimensional orthogonal space, then
${\mathbf{v}}^{\prime*}$ is a symplectic space, and we have a decomposition

${\mathbf{v}}^{\prime*}=\Biggl{[}\bigoplus_{\langle\xi\rangle,\ \xi\neq\pm
1}{\mathbf{v}}^{\prime*}_{[\xi]}\Biggr{]}\oplus{\mathbf{v}}^{\prime*}_{[-1]}\oplus{\mathbf{v}}^{\prime*}_{[1]}.$

Therefore, we have

$\dim({\mathbf{v}}^{\prime})=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}^{\prime*}_{[\xi]})+\dim({\mathbf{v}}^{\prime*}_{[1]})+1,$

and ${\mathbf{v}}^{\prime*}_{[-1]}$ is symplectic space.

Let $\rho$ be an irreducible character in ${\mathcal{E}}(G)_{s}$ for some $s$
and write $\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$. From
[Pan22] §8, we know $\rho^{c}\in{\mathcal{E}}(G)_{s}$ and

(4.11) $\Xi_{s}(\rho^{c})=\rho^{(0)}\otimes(\rho^{(1)}\cdot{\rm
sgn})\otimes\rho^{(2)}.$

Now suppose that $\rho$ (resp. $\rho^{c}$) first occurs in the correspondence
for the pair $(G({\mathbf{v}}),G({\mathbf{v}}^{\prime}))$ (resp.
$(G({\mathbf{v}}),G({\mathbf{v}}^{\prime\prime}))$) where
${\mathbf{v}}^{\prime},{\mathbf{v}}^{\prime\prime}$ are in the same Witt
series of odd-dimensional quadratic spaces. By Proposition 2.17, we have that
$\dim({\mathbf{v}}^{*}_{[\xi]})=\dim({\mathbf{v}}^{\prime*}_{[\xi]})=\dim({\mathbf{v}}^{\prime\prime*}_{[\xi]})$
for $\xi\neq 1$. Moreover, $\rho^{(1)}$ (resp. $\rho^{(1)}\cdot{\rm sgn}$)
first occurs in the correspondence for the dual pair
$(G({\mathbf{v}}^{*}_{[-1]}),G({\mathbf{v}}^{\prime*}_{[-1]}))$ (resp.
$(G({\mathbf{v}}^{*}_{[-1]}),G({\mathbf{v}}^{\prime\prime*}_{[-1]}))$). Hence

$\dim({\mathbf{v}}^{\prime*}_{[-1]})+\dim({\mathbf{v}}^{\prime\prime*}_{[-1]})=2\dim({\mathbf{v}}^{*}_{[-1]})-2\delta(\rho^{(1)})$

by Corollary 3.6. Therefore

$\displaystyle\dim({\mathbf{v}}^{\prime})+\dim({\mathbf{v}}^{\prime\prime})$
$\displaystyle=\sum_{\langle\xi\rangle,\ \xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{\prime*})+\dim({\mathbf{v}}_{[1]}^{\prime*})+1+\sum_{\langle\xi\rangle,\
\xi\neq
1}\dim({\mathbf{v}}_{[\xi]}^{\prime\prime*})+\dim({\mathbf{v}}_{[1]}^{\prime\prime*})+1$
$\displaystyle=2\sum_{\langle\xi\rangle,\ \xi\neq\pm
1}\dim({\mathbf{v}}_{[\xi]}^{*})+2\dim({\mathbf{v}}_{[1]})+2+2\dim({\mathbf{v}}^{*}_{[-1]})-2\delta(\rho^{(1)})$
$\displaystyle=2\dim({\mathbf{v}})-2\delta(\rho)+2.$

Note that now $\delta(\rho^{c})=\delta(\rho^{(1)}\cdot{\rm
sgn})=\delta(\rho^{(1)})=\delta(\rho)$. ∎

Let $\\{{\mathbf{v}}^{\prime\pm}\\}$ be the two Witt series of odd-dimensional
quadratic spaces. For $\rho\in{\mathcal{E}}({\rm Sp}({\mathbf{v}}))$, let
$n_{0}^{\prime\pm}(\rho)$ denote the minimal dimension of
${\mathbf{v}}^{\prime\pm}$ such that $\rho$ occurs in the correspondence for
the dual pair $(G({\mathbf{v}}),G({\mathbf{v}}^{\prime\pm}))$.

######  Lemma 4.12.

Let $\rho\in{\mathcal{E}}({\rm Sp}({\mathbf{v}}))$. Then
$n_{0}^{\prime-}(\rho^{c})=n_{0}^{\prime+}(\rho)$.

###### Proof.

Let $\psi$ be a non-trivial additive character of ${\mathbf{F}}_{q}$. It is
well known that $\omega^{\psi_{a}}_{{\rm Sp}({\mathbf{v}}),{\rm
O}({\mathbf{v}}^{\prime+})}=\omega^{\psi}_{{\rm Sp}({\mathbf{v}}),{\rm
O}({\mathbf{v}}^{\prime-})}$ where $\psi_{a}$ is another additive character of
${\mathbf{F}}_{q}$ given by $\psi_{a}(x)=\psi(ax)$ and $a$ is a non-square
element in ${\mathbf{F}}_{q}^{\times}$. Then by [Pan22] lemma 5.2, we know
that $\rho$ occurs in the correspondence for the dual pair $({\rm
Sp}({\mathbf{v}}),{\rm O}({\mathbf{v}}^{\prime+}))$ if and only if $\rho^{c}$
occurs in the correspondence for the dual pair $({\rm Sp}({\mathbf{v}}),{\rm
O}({\mathbf{v}}^{\prime-}))$. Hence the lemma is proved. ∎

Now we have another formulation of Theorem 4.9:

######  Corollary 4.13.

Let $\rho$ be an irreducible character of ${\rm Sp}({\mathbf{v}})$. Suppose
that $\\{{\mathbf{v}}^{\prime\pm}\\}$ are the two Witt series of odd-
dimensional quadratic spaces. Then

$n_{0}^{\prime+}(\rho)+n_{0}^{\prime-}(\rho)=2\dim({\mathbf{v}})-2\delta(\rho)+2.$

######  Remark 4.14.

Suppose that $G({\mathbf{v}})={\rm Sp}_{2n}(q)$ and ${\mathbf{v}}^{\prime}$ is
odd-dimensional. Let $\rho\in{\mathcal{E}}(G)_{s}$ for some $s$, and write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$. From (1.3) and
Lemma 2.5, we know that $\delta(\rho)=\delta(\rho^{(1)})\leq n$. This means
that if $n^{\prime}+n^{\prime\prime}<n$, no irreducible character $\rho$ of
$G$ such that $\rho$ occurs in the correspondences for $({\rm Sp}_{2n},{\rm
O}_{2n^{\prime}+1})$ and $\rho^{c}$ occurs in the correspondences for $({\rm
Sp}_{2n},{\rm O}_{2n^{\prime\prime}+1})$.

Recall that an irreducible character $\rho$ of $G$ is pseudo-unipotent if
$\rho\in{\mathcal{E}}(G)_{s}$ such that both $G^{(0)}(s),G^{(2)}(s)$ are
trivial. In this case, we have $G^{(1)}(s)={\rm O}^{\epsilon}_{2n}(q)$ where
$\epsilon=\pm$ (cf. Lemma 5.2). Then by (1.3) and Lemma 2.3, we see that
$\delta(\rho)=n$ if and only if $\rho\in{\mathcal{E}}(G)_{s}$ such that
$G^{(1)}(s)={\rm O}^{+}_{2n}(q)$ and $\rho^{(1)}=\rho_{\binom{n-k}{k}}$ for
some $0\leq k\leq n$. Denote this pseudo-unipotent character by $\rho_{k}$.
Now $\rho_{k}$ is the only irreducible character $\rho$ of $G$ such that
$\rho$ occurs in the correspondences for $({\rm Sp}_{2n},{\rm O}_{2k+1})$ and
$\rho^{c}$ occurs in the correspondences for $({\rm Sp}_{2n},{\rm
O}_{2(n-k)+1})$.

##  5\. Pseudo-unipotent Cuspidal Characters

###  5.1. Pseudo-unipotent cuspidal characters

Pseudo-unipotent characters are defined in Subsection 1.4. By convention, we
regard the trivial character of the trivial group ${\rm Sp}_{0}(q)={\rm
O}_{0}^{+}(q)={\rm SO}_{1}(q)$ to be both unipotent and pseudo-unipotent.

Recall that for a general linear group, a unitary group, a special orthogonal
group or an orthogonal group G, a character $\chi_{\text{\bf G}}$ of order $2$
is given in [Pan16] §3.3.

######  Lemma 5.1.

Let G be a unitary group, a special orthogonal group or an orthogonal group.
Then $\rho\in{\mathcal{E}}(G)$ is pseudo-unipotent if and only if
$\rho\chi_{\text{\bf G}}$ is unipotent, i.e., $\rho$ is pseudo-unipotent if
and only if it belongs to the Lusztig series ${\mathcal{E}}(G)_{-1}$.

###### Proof.

For such a classical group G, $-1$ is in the center of $G^{*}$, and hence is
in every rational maximal torus $\text{\bf T}^{*}$ of $\text{\bf G}^{*}$. It
is known that $R_{\text{\bf T}_{w},\theta_{w}}=\chi_{\text{\bf G}}R_{\text{\bf
T}_{w},1}$ for any $w\in\text{\bf W}_{\text{\bf G}}$ (cf. [DM91] proposition
13.30), and the mapping

${\mathcal{E}}(G)_{-1}\rightarrow{\mathcal{E}}(G)_{1}\quad\text{given by
}\rho\mapsto\rho\chi_{\text{\bf G}}$

is a bijection. Therefore, $\langle\rho,R_{\text{\bf
T}_{w},\theta_{w}}\rangle_{G}\neq 0$ if and only if
$\langle\rho\chi_{\text{\bf G}},R_{\text{\bf T}_{w},1}\rangle_{G}\neq 0$. ∎

From the previous lemma we see that a pseudo-unipotent cuspidal character of a
unitary group or an orthogonal group $G$ is exactly of the form
$\zeta\chi_{G}$ where $\zeta$ is a unipotent cuspidal character of $G$.

######  Lemma 5.2.

Let $\text{\bf G}={\rm Sp}_{2n}$. Then $\rho\in{\mathcal{E}}(G)$ is pseudo-
unipotent if and only if it is in the Lusztig series ${\mathcal{E}}(G)_{s}$
where $s\in G^{*}$ such that $C_{\text{\bf G}^{*}}(s)\simeq{\rm
O}_{2n}^{\epsilon}$ where $\epsilon=\pm$.

###### Proof.

Let $\rho$ be an irreducible pseudo-unipotent character of $G={\rm
Sp}_{2n}(q)$. Then we have $\langle\rho,R_{\text{\bf
T}_{w},\theta_{w}}\rangle_{G}\neq 0$ for some $(\text{\bf T}_{w},\theta_{w})$.
Let $(\text{\bf T}^{*},s)$ be the pair corresponding to $(\text{\bf
T}_{w},\theta_{w})$ where $\text{\bf T}^{*}$ is a rational maximal torus of
${\rm SO}_{2n+1}$ and $s\in T^{*}$. From the definition of $\theta_{w}$ we
know that each $\lambda_{i}$ of $s=(\lambda_{1},\ldots,\lambda_{n})\in T^{*}$
is $-1$. Moreover, $\text{\bf T}^{*}$ is either a maximal torus of ${\rm
O}_{2n}^{+}\subset{\rm SO}_{2n+1}$ or a maximal torus of ${\rm
O}_{2n}^{-}\subset{\rm SO}_{2n+1}$. Therefore, $C_{\text{\bf
G}^{*}}(s)\simeq{\rm O}_{2n}^{\epsilon}$ for $\epsilon=\pm$. ∎

###### Proof of Proposition 1.6.

Let $\text{\bf G}={\rm Sp}_{2n}$. Consider the semisimple $s=s^{\epsilon}$ in
$G^{*}$ such that $C_{\text{\bf G}^{*}}(s)\simeq{\rm O}_{2n}^{\epsilon}$. By
Proposition 2.9 for a pseudo-unipotent character $\rho\in{\mathcal{E}}({\rm
Sp}_{2n}(q))_{s}$ to be cuspidal, the unipotent character
${\mathfrak{L}}_{s}(\rho)$ of ${\rm O}^{\epsilon}_{2n}(q)$ must be cuspidal,
i.e., we must have $n=m^{2}$ and $\epsilon=(-1)^{m}$ for some integer $m$.
When $m>0$, from Subsection 3.1, we know that ${\rm O}_{2m^{2}}^{(-1)^{m}}(q)$
has two unipotent cuspidal characters. Therefore ${\rm Sp}_{2m^{2}}(q)$ has
exactly two pseudo-unipotent cuspidal characters. ∎

######  Remark 5.3.

Let $\rho_{1},\rho_{2}$ denote the two pseudo-unipotent cuspidal characters of
${\rm Sp}_{2m^{2}}(q)$ given by Proposition 1.6. Because the two unipotent
cuspidal characters of ${\rm O}^{(-1)^{m}}_{2m^{2}}$ are different by the sign
character, we have $\rho_{2}=\rho_{1}^{c}$ (cf. (4.11)).

Next we want to investigate all the first occurrences $(\rho,\rho^{\prime})$
of unipotent or pseudo-unipotent cuspidal characters for symplectic-orthogonal
dual pairs.

###  5.2. Correspondence for $({\rm Sp}_{2n},{\rm
O}^{\epsilon}_{2n^{\prime}})$

Suppose that $\rho$ is a unipotent cuspidal character of ${\rm
Sp}_{2m(m+1)}(q)$, and $(\rho,\rho^{\prime})$ is a first occurrence of
cuspidal characters for the dual pair $({\rm Sp}_{2m(m+1)},{\rm
O}_{2n^{\prime}}^{\epsilon})$. Then from [AM93] theorem 5.2 we know that
either

  1. (1)

$n^{\prime}=m^{2}$ and $\epsilon=(-1)^{m}$; or

  2. (2)

$n^{\prime}=(m+1)^{2}$ and $\epsilon=(-1)^{m+1}$.

Moreover, $\rho^{\prime}$ is a unipotent cuspidal character of ${\rm
O}_{2n^{\prime}}^{\epsilon}(q)$.

Now we consider the first occurrence of pseudo-unipotent cuspidal characters.

######  Proposition 5.4.

Suppose that $\rho$ is a pseudo-unipotent cuspidal character of ${\rm
Sp}_{2m^{2}}(q)$ and $(\rho,\rho^{\prime})$ is a first occurrence of cuspidal
characters for $({\rm Sp}_{2m^{2}},{\rm O}_{2n^{\prime}}^{\epsilon})$. Then
$n^{\prime}=m^{2}$ or $m^{2}+1$ (depending on $\epsilon$). Moreover, if
$n^{\prime}=m^{2}$, $\rho^{\prime}$ is also pseudo-unipotent.

###### Proof.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm Sp}_{2m^{2}},{\rm
O}^{\epsilon}_{2n^{\prime}})$. Suppose $\rho\in{\mathcal{E}}(G)_{s}$ for some
$s$ and write $\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$ and
$\rho^{\prime}\in{\mathcal{E}}(G^{\prime})_{s^{\prime}}$ and
$\Xi_{s^{\prime}}(\rho^{\prime})=\rho^{\prime(0)}\otimes\rho^{\prime(1)}\otimes\rho^{\prime(2)}$.
By Lemma 5.2 and the definition in Subsection 2.3, we know that $\text{\bf
G}^{(0)}$ is trivial, $\text{\bf G}^{(1)}={\rm
O}_{2m^{2}}^{\epsilon^{\prime}}$ and $\epsilon^{\prime}=(-1)^{m}$, $\text{\bf
G}^{(2)}={\rm Sp}_{0}$, and $\rho^{(1)}$ is a unipotent cuspidal character of
${\rm O}_{2m^{2}}^{\epsilon^{\prime}}(q)$.

By Proposition 2.16 we know that both $\text{\bf G}^{\prime(0)}$ is trivial,
$\text{\bf G}^{\prime(1)}={\rm O}_{2m^{2}}^{\epsilon^{\prime}}$,
$\rho^{(1)}=\rho^{\prime(1)}$, and $(\rho^{(2)},\rho^{\prime(2)})$ is a first
occurrence of unipotent cuspidal characters for the dual pair $(\text{\bf
G}^{(2)},\text{\bf G}^{\prime(2)})$. Then by [AM93] theorem 5.3, we know that
$\text{\bf G}^{\prime(2)}$ is either ${\rm O}_{0}^{+}$ or ${\rm O}_{2}^{-}$.

  1. (1)

Suppose $\text{\bf G}^{\prime(2)}={\rm O}_{0}^{+}$. Then $s^{\prime}$ is the
element $-1\in G^{\prime*}={\rm O}_{2m^{2}}^{\epsilon}(q)$ with
$\epsilon=\epsilon^{\prime}=(-1)^{m}$. Hence $n^{\prime}=m^{2}$ and
$\rho^{\prime}$ is pseudo-unipotent.

  2. (2)

Suppose $\text{\bf G}^{\prime(2)}={\rm O}_{2}^{-}$. Then $s^{\prime}$ is the
element $(-1,1)\in C_{\text{\bf G}^{*}}(s^{\prime})={\rm
O}_{2m^{2}}^{\epsilon^{\prime}}\times{\rm O}^{-}_{2}\subset\text{\bf G}^{*}$
with $\epsilon=-\epsilon^{\prime}=(-1)^{m+1}$. Hence $n^{\prime}=m^{2}+1$ and
$\rho^{\prime}$ is not pseudo-unipotent (and nor unipotent).

∎

###  5.3. Correspondence for $({\rm Sp}_{2n},{\rm O}_{2n^{\prime}+1})$

######  Proposition 5.5.

Suppose $\rho$ is the unipotent cuspidal character of ${\rm Sp}_{2m(m+1)}(q)$,
and $(\rho,\rho^{\prime})$ is a first occurrence for $({\rm Sp}_{2m(m+1)},{\rm
O}_{2n^{\prime}+1})$. Then $n^{\prime}=m(m+1)$ and $\rho^{\prime}$ is a
pseudo-unipotent cuspidal character of ${\rm O}_{2m(m+1)+1}(q)$.

###### Proof.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm Sp}_{2m^{2}},{\rm
O}_{2n^{\prime}+1})$, $\text{\bf G}^{\prime 0}={\rm SO}_{2n^{\prime}+1}$,
$\rho^{\prime 0}=\rho^{\prime}|_{G^{\prime 0}}$. Then $\rho^{\prime 0}$ is an
irreducible character. As in the proof of Proposition 5.4, suppose that $\rho$
is in ${\mathcal{E}}(G)_{s}$ and write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$,
$\rho^{\prime}_{0}\in{\mathcal{E}}(G^{\prime 0})_{s^{\prime}}$,
$\Xi_{s^{\prime}}(\rho^{\prime
0})=\rho^{\prime(0)}\otimes\rho^{\prime(1)}\otimes\rho^{\prime(2)}$,
$C_{\text{\bf G}^{*}}(s)={\rm O}_{2m^{2}}^{\epsilon}$, and
$\epsilon=(-1)^{m}$. Now both $\text{\bf G}^{(0)}$ and $\text{\bf G}^{(1)}$
are trivial, $\text{\bf G}^{(2)}={\rm Sp}_{2m(m+1)}$, and $\rho^{(2)}=\rho$.
By Proposition 2.17 we know that $\text{\bf G}^{\prime(0)}$ is trivial,
$\text{\bf G}^{\prime(1)}\simeq\text{\bf G}^{(2)}={\rm Sp}_{2m(m+1)}$, and
$(\rho^{(1)},\rho^{\prime(2)})$ is a first occurrence of unipotent cuspidal
characters for the dual pair $(\text{\bf G}^{(1)},\text{\bf G}^{\prime(2)})$.
This means that $\text{\bf G}^{\prime(2)}={\rm Sp}_{0}$ and $\rho^{\prime(2)}$
is the trivial character. Therefore, $n^{\prime}=m(m+1)$ and $\rho^{\prime}$
is a pseudo-unipotent cuspidal character of ${\rm O}_{2m(m+1)+1}(q)$. ∎

######  Proposition 5.6.

Suppose that $\rho$ is a pseudo-unipotent cuspidal character of ${\rm
Sp}_{2m^{2}}(q)$, and $(\rho,\rho^{\prime})$ is a first occurrence of cuspidal
characters for $({\rm Sp}_{2m^{2}},{\rm O}_{2n^{\prime}+1})$. Then
$n^{\prime}=m(m-1)$ or $m(m+1)$, and $\rho^{\prime}$ is a unipotent cuspidal
character of ${\rm O}_{2n^{\prime}+1}(q)$.

###### Proof.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm Sp}_{2m^{2}},{\rm
O}_{2n^{\prime}+1})$, $\text{\bf G}^{\prime 0}={\rm SO}_{2n^{\prime}+1}$,
$\rho^{\prime 0}=\rho^{\prime}|_{G^{\prime 0}}$. As in the proof of
Proposition 5.5, suppose that $\rho$ is in ${\mathcal{E}}(G)_{s}$
$\rho^{\prime 0}\in{\mathcal{E}}(G^{\prime 0})_{s^{\prime}}$, and write
$\Xi_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$,
$\Xi_{s^{\prime}}(\rho^{\prime
0})=\rho^{\prime(0)}\otimes\rho^{\prime(1)}\otimes\rho^{\prime(2)}$,
$C_{\text{\bf G}^{*}}(s)={\rm O}_{2m^{2}}^{\epsilon}$, and
$\epsilon=(-1)^{m}$. Now both $\text{\bf G}^{(0)}$ and $\text{\bf G}^{(2)}$
are trivial, $\text{\bf G}^{(1)}={\rm O}_{2m^{2}}^{\epsilon}$, and
$\rho^{(1)}$ is a unipotent cuspidal character of ${\rm
O}_{2m^{2}}^{\epsilon}(q)$. By Proposition 2.17 we know that both $\text{\bf
G}^{\prime(0)}$ and $\text{\bf G}^{\prime(1)}$ are trivial, and
$(\rho^{(1)},\rho^{\prime(2)})$ is a first occurrence of unipotent cuspidal
characters for the dual pair $(\text{\bf G}^{(1)},\text{\bf G}^{\prime(2)})$.
Then by [AM93] theorem 5.3, we know that $\text{\bf G}^{\prime(2)}$ is ${\rm
Sp}_{2m(m-1)}$ or ${\rm Sp}_{2m(m+1)}$. We see that $s^{\prime}$ is the
identity element in $(\text{\bf G}^{\prime 0})^{*}={\rm Sp}_{2n^{\prime}}$ and
this implies that $n^{\prime}$ is $m(m-1)$ or $m(m+1)$, and $\rho^{\prime}$ is
unipotent. ∎

## References

  * [AM93] J. Adams and A. Moy, _Unipotent representations and reductive dual pairs over finite fields_ , Trans. Amer. Math. Soc. 340 (1993), 309–321.
  * [AMR96] A.-M. Aubert, J. Michel, and R. Rouquier, _Correspondance de Howe pour les groupes réductifs sur les corps finis_ , Duke Math. J. 83 (1996), 353–397.
  * [DM91] F. Digne and J. Michel, _Representations of finite groups of Lie type_ , Cambridge University Press, Cambridge, 1991.
  * [FS90] P. Fong and B. Srinivasan, _Brauer trees in classical groups_ , J. Algebra 131 (1990), 179–225.
  * [Gér77] P. Gérardin, _Weil representations associated to finite fields_ , J. Algebra 46 (1977), 54–101.
  * [GH20] S. Gurevich and R. Howe, _Rank and duality in representation theory_ , Jpn. J. Math. 15 (2020), 223–309.
  * [How] R. Howe, _Invariant theory and duality for classical groups over finite fields with applications to their singular representation theory_ , preprint.
  * [Lus77] G. Lusztig, _Irreducible representations of finite classical groups_ , Invent. Math. 43 (1977), 125–175.
  * [Lus84] by same author, _Characters of reductive groups over a finite field_ , Ann. Math. Stud., no. 107, Princeton University Press, Princeton, 1984.
  * [LW20] D. Liu and Z. Wang, _Remarks on the theta correspondence over finite fields_ , Pacific J. Math. 306 (2020), 587–609.
  * [Pan] S.-Y. Pan, _On finite theta correspondences of Witt towers_ , in preparation.
  * [Pan02] by same author, _Local theta correspondence of representations of depth zero and theta dichotomy_ , J. Math. Soc. Japan 54 (2002), 793–845.
  * [Pan16] by same author, _Weil representation of finite symplectic groups and finite odd orthogonal groups_ , J. Algebra 453 (2016), 291–324.
  * [Pan19a] by same author, _Howe correspondence of unipotent characters for a finite symplectic/even-orthogonal dual pair_ , arXiv:1901.00623 (2019).
  * [Pan19b] by same author, _Lusztig correspondence and Howe correspondence for finite reductive dual pairs_ , arXiv:1906.01158 (2019).
  * [Pan19c] by same author, _Supercuspidal representations and preservation principle of theta correspondence_ , J. Reine Angew. Math. 750 (2019), 1–52.
  * [Pan21a] by same author, _On theta ranks of irreducible characters of a finite classical group_ , arXiv:2102.09220 (2021).
  * [Pan21b] by same author, _Uniform projection of the Weil character_ , Adv. Math. 381 (2021), 107624, 80 pp.
  * [Pan22] by same author, _On the unicity and the ambiguity of Lusztig parametrizations for finite classical groups_ , arXiv:2206.04900 (2022).
  * [Wal04] J.-L. Waldspurger, _Une conjecture de Lusztig pour les groupes classiques_ , Mém. Soc. Math. Fr. (N.S.) (2004), no. 96, vi+166 pp.