# On the Unicity and the Ambiguity of Lusztig Parametrizations for Finite
Classical Groups

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

The Lusztig correspondence is a bijective mapping between the Lusztig series
indexed by the conjugacy class of a semisimple element $s$ in the connected
component $(G^{*})^{0}$ of the dual group of $G$ and the set of irreducible
unipotent characters of the centralizer of $s$ in $G^{*}$. In this article we
discuss the unicity and ambiguity of such a bijective correspondence. In
particular, we show that the Lusztig correspondence for a classical group can
be made to be unique if we require it to be compatible with the parabolic
induction and the finite theta correspondence.

###### Key words and phrases:

rank, unipotent character, reductive dual pair

###### 2010 Mathematics Subject Classification:

Primary: 20C33; Secondary: 22E50

###### Contents

  1. 1 Introduction
    1. 1.1
    2. 1.2
    3. 1.3
    4. 1.4
  2. 2 Symbols and Special symbols
    1. 2.1 Irreducible characters of Weyl groups
    2. 2.2 Lusztig’s symbols
    3. 2.3 Special symbols
  3. 3 Lusztig Parametrization of Unipotent Characters
    1. 3.1 Unipotent characters of ${\rm GL}_{n}$ or ${\rm U}_{n}$
    2. 3.2 Uniform almost characters
    3. 3.3 Lusztig’s parametrization of unipotent characters
    4. 3.4 Cells in a family of unipotent characters
  4. 4 Uniqueness of the Lusztig Parametrizations
    1. 4.1 Unipotent characters of ${\rm Sp}_{2n}$
    2. 4.2 Unipotent characters of ${\rm SO}_{2n+1}$
    3. 4.3 Unipotent characters of ${\rm O}^{\epsilon}_{2n}$
  5. 5 Finite Theta Correspondence of Unipotent Characters
    1. 5.1 Finite theta correspondence on unipotent characters
    2. 5.2 Finite theta correspondence on unipotent characters
    3. 5.3 On the uniqueness of Lusztig parametrization for even orthogonal groups
  6. 6 Lusztig Correspondence and Finite Theta Correspondence
    1. 6.1 Lusztig correspondences
    2. 6.2 Lusztig correspondence and parabolic induction
    3. 6.3 Modified Lusztig correspondence
    4. 6.4 Theta correspondence and modified Lusztig correspondence
  7. 7 Lusztig Correspondences for ${\rm SO}_{2n+1}$
    1. 7.1 Lusztig correspondence for ${\rm SO}_{2n+1}$
  8. 8 Lusztig Correspondence for ${\rm O}^{\epsilon}_{2n}$
    1. 8.1 Lusztig correspondence for ${\rm O}^{\epsilon}_{2n}$
    2. 8.2 Basic characters of ${\rm O}^{\epsilon}_{2n}$
    3. 8.3 A uniqueness choice of ${\mathcal{L}}_{s}$
  9. 9 Lusztig Correspondences for ${\rm Sp}_{2n}$
    1. 9.1 Lusztig correspondence for ${\rm Sp}_{2n}$
    2. 9.2 Basic characters of ${\rm Sp}_{2n}$
    3. 9.3 The choice of ${\mathcal{L}}_{s}$ with respect to $({\rm Sp}_{2n},{\rm SO}_{2n^{\prime}+1})$
    4. 9.4 The choice of ${\mathcal{L}}_{s}$ with respect to $({\rm Sp}_{2n},{\rm O}^{\epsilon}_{2n^{\prime}})$

##  1\. Introduction

### 1.1.

Let G be a classical group defined over a finite field ${\mathbf{F}}_{q}$ of
odd characteristic, and let $F$ be the corresponding Frobenius endomorphism.
Let $G=\text{\bf G}^{F}$ denote the finite subgroup of rational points, and
let ${\mathcal{E}}(\text{\bf G})$ denote the set of irreducible characters
(i.e., the characters of irreducible representations) of $G$.

Let $R^{\text{\bf G}}_{\text{\bf T}^{*},s}$ denote the _Deligne-Lusztig
virtual characters_ (cf. [Car85], [GM20]) indexed by conjugacy class of pair
$(\text{\bf T}^{*},s)$ where $\text{\bf T}^{*}$ is a rational maximal torus in
the dual group $\text{\bf G}^{*}$ and $s$ is a rational semisimple element
contained in $\text{\bf T}^{*}$. Let ${\mathcal{V}}(\text{\bf G})$ be the
space of (complex valued) class function on $G$, and let
${\mathcal{V}}(\text{\bf G})^{\sharp}$ denote the subspace spanned by Deligne-
Lusztig virtual characters. Note that ${\mathcal{V}}(\text{\bf G})$ is an
inner product space with the inner product $\langle,\rangle_{\text{\bf G}}$
given by

$\langle f_{1},f_{2}\rangle_{\text{\bf G}}=\frac{1}{|G|}\sum_{g\in
G}f_{1}(g)\overline{f_{2}(g)}$

for $f_{1},f_{2}\in{\mathcal{V}}(\text{\bf G})$, and $\overline{f_{2}(g)}$
denotes the complex conjugate of the value $f_{2}(g)$. For
$f\in{\mathcal{V}}(\text{\bf G})$ the orthogonal projection of $f$ onto
${\mathcal{V}}(\text{\bf G})^{\sharp}$ is denoted by $f^{\sharp}$ and called
the _uniform projection_. A class function $f\in{\mathcal{V}}(\text{\bf G})$
is called _uniform_ if $f^{\sharp}=f$.

A natural question is how much an irreducible character $\rho$ of a classical
group $G$ can be determined by its uniform projection $\rho^{\sharp}$?

If G is a general linear group ${\rm GL}_{n}$ or a unitary group ${\rm
U}_{n}$, then every irreducible character is uniform (i.e.,
$\rho=\rho^{\sharp}$) and the above question is trivial. However, if G is a
symplectic group or an orthogonal group, the question is more subtle. Recall
that $\rho\in{\mathcal{E}}(\text{\bf G})$ is called _unipotent_ if
$\langle\rho,R^{\text{\bf G}}_{\text{\bf T}^{*},1}\rangle_{\text{\bf G}}\neq
0$ for some $\text{\bf T}^{*}$. If $\rho$ is unipotent and G is connected, it
is known that $\rho$ is uniquely determined by its uniform projection, i.e.,
$\rho^{\prime\sharp}=\rho^{\sharp}$ if and only if $\rho^{\prime}=\rho$ (cf.
[DM90] proposition 6.3 and [GM20] theorem 4.4.23). If $\rho$ is unipotent and
$\text{\bf G}={\rm O}^{\epsilon}_{2n}$, it is also known that
$\rho^{\prime\sharp}=\rho^{\sharp}$ if and only if $\rho^{\prime}=\rho$ or
$\rho^{\prime}=\rho\cdot{\rm sgn}$ (cf. [Pan19b] proposition 3.6).

In this paper, the above question will be answered completely (cf. Corollary
7.5, Corollary 9.4 and Corollary 8.4) for classical groups:

######  Theorem 1.1.

Let G be a symplectic group or an orthogonal group, and let
$\rho,\rho^{\prime}\in{\mathcal{E}}(\text{\bf G})$. Then
$\rho^{\prime\sharp}=\rho^{\sharp}$ if and only of

(1.2) $\rho^{\prime}=\begin{cases}\rho,&\text{if\/ $\text{\bf G}={\rm
SO}_{2n+1}$};\\\ \rho,\rho^{c}&\text{if\/ $\text{\bf G}={\rm Sp}_{2n}$};\\\
\rho,\rho^{c},\rho\cdot{\rm sgn},\rho^{c}\cdot{\rm sgn}&\text{if\/ $\text{\bf
G}={\rm O}^{\epsilon}_{2n}$}.\end{cases}$

Here “${\rm sgn}$” denotes the sign character of an orthogonal group, and
“$\rho^{c}$” denotes the character obtained from $\rho$ by conjugating an
element in the corresponding similitude group (cf. [Wal04] §4.3, §4.10). Note
that if $\rho$ is unipotent then $\rho^{c}=\rho$, and then (1.2) is reduced to
the above known result.

### 1.2.

In fact, the above theorem is a consequence of a more precise result on the
ambiguity of the Lusztig parametrization of irreducible characters of finite
classical groups. From now on, we assume that G is a symplectic group or an
orthogonal group. It is known by Lusztig that there is a partition

${\mathcal{E}}(\text{\bf G})=\bigcup_{(s)}{\mathcal{E}}(\text{\bf G},s)$

where the union $\bigcup_{(s)}$ runs over conjugacy classes of semisimple
elements in the connected component of the dual group $G^{*}$ and

${\mathcal{E}}(\text{\bf G},s)=\\{\,\rho\in{\mathcal{E}}(\text{\bf
G})\mid\langle\rho,R^{\text{\bf G}}_{\text{\bf T}^{*},s}\rangle_{\text{\bf
G}}\neq 0\text{ for some }\text{\bf T}^{*}\text{ containing }s\,\\}.$

Each ${\mathcal{E}}(\text{\bf G},s)$ is called a _Lusztig series_. In
particular, the subset ${\mathcal{E}}(\text{\bf G},1)$ consists of unipotent
characters.

Now we first focus on the set of unipotent characters. Let
${\mathcal{V}}(\text{\bf G},1)$ denote the subspace spanned by elements in
${\mathcal{E}}(\text{\bf G},1)$, and let ${\mathcal{V}}(\text{\bf
G},1)^{\sharp}$ denote the uniform projection of ${\mathcal{V}}(\text{\bf
G},1)$. Following from Lusztig (cf. [Lus81], [Lus82]) we can define a set
${\mathcal{S}}_{\text{\bf G}}^{\sharp}$ (cf. (3.7)) and class functions
$R_{\Sigma}^{\text{\bf G}}$ (cf. (3.5)) for $\Sigma\in{\mathcal{S}}_{\text{\bf
G}}^{\sharp}$ such that the set $\\{\,R_{\Sigma}^{\text{\bf
G}}\mid\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp}\,\\}$ forms an
orthonormal basis for ${\mathcal{V}}(\text{\bf G},1)^{\sharp}$. Lusztig
constructs a set ${\mathcal{S}}_{\text{\bf G}}$ (cf. (2.7)) of similar classes
of _symbols_ and a bijective mapping ${\mathcal{L}}_{1}^{\text{\bf
G}}\colon{\mathcal{S}}_{\text{\bf G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$
denoted by $\Lambda\mapsto\rho_{\Lambda}$ such that the value
$\langle\rho_{\Lambda},R^{\text{\bf G}}_{\Sigma}\rangle_{\text{\bf G}}$ is
specified (cf. Proposition 3.12). Such a mapping ${\mathcal{L}}_{1}^{\text{\bf
G}}$ is called a _Lusztig parametrization_ of unipotent characters of G.
Because the uniform projection $\rho_{\Lambda}^{\sharp}$ can be obtained by
the values $\langle\rho_{\Lambda},R^{\text{\bf G}}_{\Sigma}\rangle_{\text{\bf
G}}$, the problem of the uniqueness of ${\mathcal{L}}_{1}^{\text{\bf G}}$ is
equivalent to the problem whether a unipotent character $\rho_{\Lambda}$ can
be uniquely determined by its uniform projection $\rho_{\Lambda}^{\sharp}$.

As described in Subsection 1.1, ${\mathcal{L}}_{1}^{\text{\bf G}}$ is known to
be unique if $\text{\bf G}={\rm SO}_{2n+1}$ or ${\rm Sp}_{2n}$. However, due
to the disconnectedness of ${\rm O}^{\epsilon}_{2n}$, the mapping
${\mathcal{L}}_{1}^{{\rm O}^{\epsilon}_{2n}}$ is not uniquely determined. By
using the result of “ _cells_ ” by Lusztig, we determine how ambiguous a
Lusztig parametrization ${\mathcal{L}}_{1}^{{\rm O}^{\epsilon}_{2n}}$ could be
and we show in Proposition 5.9 that ${\mathcal{L}}_{1}^{{\rm
O}^{\epsilon}_{2n}}$ can be chosen to be unique if we require it to be

  * •

_compatible with parabolic induction_ and

  * •

_compatible with theta correspondence_ on cuspidal characters or ${\bf
1}_{{\rm O}^{+}_{2}},{\rm sgn}_{{\rm O}^{+}_{2}}$.

In particular, ${\mathcal{L}}_{1}^{{\rm O}^{\epsilon}_{2n}}$ (and
${\mathcal{L}}_{1}^{{\rm Sp}_{2n^{\prime}}}$) can be given so that
$(\rho_{\Lambda},\rho^{\prime}_{\Lambda^{\prime}})\in\Theta_{\text{\bf
G},\text{\bf G}^{\prime}}$ if and only if
$(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}_{\text{\bf G},\text{\bf
G}^{\prime}}$ for $(\text{\bf G},\text{\bf G}^{\prime})=({\rm
O}^{\epsilon}_{2n},{\rm Sp}_{2n^{\prime}})$ where ${\mathcal{B}}_{\text{\bf
G},\text{\bf G}^{\prime}}$ is a relation between ${\mathcal{S}}_{\text{\bf
G}}$ and ${\mathcal{S}}_{\text{\bf G}^{\prime}}$ defined in Subsection 5.2.

### 1.3.

For a general Lusztig series ${\mathcal{E}}(\text{\bf G},s)$, Lusztig shows
(cf. [Lus84]) that there exists a bijection

${\mathfrak{L}}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)$

satisfying

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

where $\epsilon_{\text{\bf G}}=(-1)^{\kappa(\text{\bf G})}$, $\kappa(\text{\bf
G})$ denotes the rational rank of G, and $C_{\text{\bf G}^{*}}(s)$ denotes the
centralizer of $s$ in $\text{\bf G}^{*}$. Such a bijection
${\mathfrak{L}}_{s}$ will be called a _Lusztig correspondence_ in this article
(it is called a _Jordan decomposition_ in [GM20]).

Now the question is to understand whether the Lusztig correspondence
${\mathfrak{L}}_{s}$ is uniquely determined by (1.3). If the answer is
negative, then we want to know what kind of conditions need to be enforced to
make ${\mathfrak{L}}_{s}$ unique. Some discussion on this problem can be
founded in [GM20] appendix A.5. If G is a connected group with connected
center, it is shown in [DM90] theorem 7.1 that ${\mathfrak{L}}_{s}$ can be
uniquely determined by (1.3) and some extra conditions.

For $s\in G^{*}$, we have a decomposition $s=s^{(0)}\times s^{(1)}\times
s^{(2)}$ 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 $-1$. Then we can define groups $\text{\bf G}^{(0)}(s)$,
$\text{\bf G}^{(-)}(s)$ and $\text{\bf G}^{(+)}(s)$ (cf. (6.12)) so that there
is a bijective mapping

${\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)\rightarrow{\mathcal{E}}(\text{\bf
G}^{(0)}(s)\times\text{\bf G}^{(-)}(s)\times\text{\bf G}^{(+)}(s),1).$

Combining Lusztig parametrization ${\mathcal{L}}_{1}$ of unipotent characters
for $\text{\bf G}^{(0)}$, $\text{\bf G}^{(-)}(s)$, $\text{\bf G}^{(+)}(s)$,
and above bijection and the inverse ${\mathfrak{L}}_{s}^{-1}$ of a Lusztig
correspondence, we obtain a bijective mapping

${\mathcal{L}}_{s}\colon{\mathcal{S}}_{\text{\bf
G}^{(0)}(s)}\times{\mathcal{S}}_{\text{\bf
G}^{(-)}(s)}\times{\mathcal{S}}_{\text{\bf
G}^{(+)}(s)}\rightarrow{\mathcal{E}}(\text{\bf G},s)$

denoted by
$(x,\Lambda_{1},\Lambda_{2})\mapsto\rho_{x,\Lambda_{1},\Lambda_{2}}$ which is
called a _modified Lusztig correspondence_. Then we prove the following
results on the unicity and ambiguity of ${\mathcal{L}}_{s}$ (or equivalently,
the unicity and ambiguity of ${\mathfrak{L}}_{s}$) for classical groups:

  1. (1)

Suppose that $\text{\bf G}={\rm SO}_{2n+1}$. There is a unique modified
Lusztig correspondence ${\mathcal{L}}_{s}$ (cf. Theorem 7.1). Note that ${\rm
SO}_{2n+1}$ is a connected group with connected center, so this case is
covered by [DM90] theorem 7.1. However, for this group, no extra condition
other than (1.3) is needed to ensure the uniqueness of ${\mathcal{L}}_{s}$.

  2. (2)

Suppose that $\text{\bf G}={\rm O}^{\epsilon}_{2n}$ where $\epsilon=+$ or $-$.
There exists a unique modified Lusztig correspondence ${\mathcal{L}}_{s}$
which is compatible with the parabolic induction and some other conditions (on
_basic characters_). (cf. Theorem 8.12).

  3. (3)

Suppose that $\text{\bf G}={\rm Sp}_{2n}$. Here we provide two choices of the
modified Lusztig correspondence ${\mathcal{L}}_{s}$:

    1. (a)

There exists a unique modified Lusztig correspondence ${\mathcal{L}}_{s}$
which is compatible with the parabolic induction and compatible with the theta
correspondence for the dual pair $(\text{\bf G},\text{\bf G}^{\prime})=({\rm
Sp}_{2n},{\rm SO}_{2n^{\prime}+1})$, i.e., we show that
$(\rho_{x,\Lambda_{1},\Lambda_{2}},\rho_{x^{\prime},\Lambda_{1}^{\prime},\Lambda_{2}^{\prime}})\in\Theta_{\text{\bf
G},\text{\bf G}^{\prime}}^{\psi}$ if and only if

       * •

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

       * •

$\Lambda_{2}=\Lambda^{\prime}_{1}$, and

       * •

$(\Lambda_{1},\Lambda^{\prime}_{2})\in{\mathcal{B}}_{\text{\bf
G}^{(-)}(s),\text{\bf G}^{(+)}(s^{\prime})}$

where $\rho_{x^{\prime},\Lambda_{1}^{\prime},\Lambda_{2}^{\prime}}$ is given
by the unique modified Lusztig correspondence in (1) (cf. Theorem 9.7 and
Theorem 9.9).

    2. (b)

There exists a unique modified Lusztig correspondence ${\mathcal{L}}_{s}$
which is compatible with the parabolic induction and compatible with the theta
correspondence for the dual pair $(\text{\bf G},\text{\bf G}^{\prime})=({\rm
Sp}_{2n},{\rm O}^{\epsilon}_{2n^{\prime}})$, i.e.,
$(\rho_{x,\Lambda_{1},\Lambda_{2}},\rho_{x^{\prime},\Lambda_{1}^{\prime},\Lambda_{2}^{\prime}})\in\Theta_{\text{\bf
G},\text{\bf G}^{\prime}}^{\psi}$ if and only if

       * •

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

       * •

$\Lambda_{1}=\Lambda^{\prime}_{1}$, and

       * •

$(\Lambda_{2},\Lambda^{\prime}_{2})\in{\mathcal{B}}_{\text{\bf
G}^{(+)}(s),\text{\bf G}^{(+)}(s^{\prime})}$

where $\rho_{x^{\prime},\Lambda_{1}^{\prime},\Lambda_{2}^{\prime}}$ is given
by the unique modified Lusztig correspondence in (2) (cf. Theorem 9.11 and
Theorem 9.12).

It seems that two choices in (a) and (b) of the modified Lusztig
correspondence ${\mathcal{L}}_{s}$ for ${\rm Sp}_{2n}$ should be the same.
However, we do not know how to obtain the conclusion yet.

### 1.4.

The contents of this article are as follows. In Section 2, we recall the
notion and some basic result of “ _symbols_ ” and “ _special symbols_ ” by
Lusztig from [Lus77]. In Section 3 we first recall the notion of “ _almost
characters_ ” by Lusztig from [Lus81] and [Lus82]. Then we record some results
of cells from [Pan19a]. In Section 4 we show the uniqueness of
${\mathcal{L}}_{1}^{\text{\bf G}}$ for $\text{\bf G}={\rm Sp}_{2n}$ and ${\rm
SO}_{2n+1}$ by using the results in the previous section. Moreover, we also
discuss the ambiguity of ${\mathcal{L}}_{1}^{\text{\bf G}}$ for $\text{\bf
G}={\rm O}^{\epsilon}_{2n}$. In Section 5 we discuss the relation between the
theta correspondence $\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$ on
unipotent characters for $(\text{\bf G},\text{\bf G}^{\prime})=({\rm
Sp}_{2n},{\rm O}^{\epsilon}_{2n^{\prime}})$ and Lusztig parametrizations
${\mathcal{L}}_{1}^{\text{\bf G}}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ for $\text{\bf G}={\rm
Sp}_{2n},{\rm O}^{\epsilon}_{2n}$. In Section 6 we discuss the relation
between the theta correspondence $\Theta^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}$ on certain Lusztig series for $(\text{\bf G},\text{\bf
G}^{\prime})=({\rm Sp}_{2n},{\rm SO}_{2n^{\prime}+1})$ or $({\rm Sp}_{2n},{\rm
O}^{\epsilon}_{2n^{\prime}})$ and the Lusztig correspondence
${\mathfrak{L}}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)$. In Section 7 we
show that the Lusztig correspondence
${\mathfrak{L}}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)$ is unique for
$\text{\bf G}={\rm SO}_{2n+1}$. In Section 8 we show that ${\mathfrak{L}}_{s}$
can be chosen to be unique for $\text{\bf G}={\rm O}^{\epsilon}_{2n}$ if we
require ${\mathfrak{L}}_{s}$ to be compatible with the parabolic induction and
some other conditions on “basic characters”. In the final section we discuss
the uniqueness of the Lusztig correspondence ${\mathfrak{L}}_{s}$ for
$\text{\bf G}={\rm Sp}_{2n}$. In particular, we show that a unique
${\mathfrak{L}}_{s}$ can be chosen to be compatible with the theta
correspondence for the dual pair $({\rm Sp}_{2n},{\rm SO}_{2n+1})$ or for the
dual pair $({\rm Sp}_{2n},{\rm O}^{\epsilon}_{2n})$.

##  2\. Symbols and Special symbols

###  2.1. Irreducible characters of Weyl groups

Let ${\mathcal{P}}(n)$ denote the set partitions of $n$. It is known that the
set of irreducible characters ${\mathcal{E}}(S_{n})$ of the symmetric group
$S_{n}$ is parametrized by ${\mathcal{P}}(n)$. For
$\lambda\in{\mathcal{P}}(n)$, we write $|\lambda|=n$, and the corresponding
irreducible character of $S_{n}$ is denoted by $\varphi_{\lambda}$.

Let $W_{n}$ denote the Coxeter group of type $B_{n}$, i.e., $W_{n}$ consists
of all permutations on $\\{1,2,\ldots,n,n^{*},(n-1)^{*},\ldots,1^{*}\\}$ which
commutes with the involution

$(1,1^{*})(2,2^{*})\cdots(n,n^{*})$

where $(i,j)$ denote the transposition of $i,j$. For $i=1,\ldots,n-1$, let

$s_{i}=(i,i+1)(i^{*},(i+1)^{*})\quad\text{and}\quad\sigma_{n}=(n,n^{*}).$

It is known that $W_{n}$ is generated by
$\\{s_{1},\ldots,s_{n-1},\sigma_{n}\\}$. Each element of $W_{n}$ induces a
permutation of $\\{1,2,\ldots,n\\}$. So we have a surjective homomorphism
$W_{n}\rightarrow S_{n}$ with kernel isomorphic to
$({\mathbb{Z}}/2{\mathbb{Z}})^{n}$. Therefore, $\varphi_{\lambda}$ can be
regarded as an irreducible character of $W_{n}$.

An ordered pair $\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}$ of two partitions is
called a _bi-partitions_. Let ${\mathcal{P}}_{2}(n)$ denote the set of
bipartitions of $n$, i.e.,

$\textstyle{\mathcal{P}}_{2}(n)=\left\\{\,\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}\mid|\mu|+|\nu|=n\,\right\\}.$

For a bi-partition $\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}$, we define its
_transpose_ by $\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}^{\rm
t}=\genfrac{[}{]}{0.0pt}{}{\nu}{\mu}$. A bi-partition
$\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}$ is called _degenerate_ if $\mu=\nu$, and
it is called _non-degenerate_ otherwise. For
$\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}\in{\mathcal{P}}_{2}(n)$ such that
$\mu\in{\mathcal{P}}(k)$ and $\nu\in{\mathcal{P}}(l)$ where $k+l=n$, we define

(2.1) $\varphi_{\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}}={\rm
Ind}^{W_{n}}_{W_{k}\times
W_{l}}(\varphi_{\mu}\otimes(\varepsilon_{l}\varphi_{\nu}))$

where $\varepsilon_{l}\colon W_{l}\rightarrow\\{\pm 1\\}$ is given by
$s_{i}\mapsto 1$ and $\sigma_{l}\mapsto-1$. It is known that
$\varphi_{\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}}$ is an irreducible character of
$W_{n}$, and the mapping ${\mathcal{P}}_{2}(n)\rightarrow{\mathcal{E}}(W_{n})$
by
$\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}\mapsto\varphi_{\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}}$
gives a parametrization of ${\mathcal{E}}(W_{n})$ such that

  * •

$\varphi_{\genfrac{[}{]}{0.0pt}{}{n}{-}}={\bf 1}_{W_{n}}$

  * •

$\varphi_{\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}}\cdot\varepsilon_{n}=\varphi_{\genfrac{[}{]}{0.0pt}{}{\nu}{\mu}}$,
in particular $\varphi_{\genfrac{[}{]}{0.0pt}{}{-}{n}}=\varepsilon_{n}$

(cf. [GP00] theorem 5.5.6).

The kernel $W_{n}^{+}$ of $\varepsilon_{n}$ is a subgroup of index two
generated by $\\{s_{1},\ldots,s_{n-1},\sigma_{n}s_{n-1}\sigma_{n}\\}$. Let
$W_{n}^{-}=W_{n}\smallsetminus W_{n}^{+}$. It is well known that if
$\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}$ is non-degenerate, then
$\varphi_{\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}}|_{W_{n}^{+}}=\varphi_{\genfrac{[}{]}{0.0pt}{}{\nu}{\mu}}|_{W_{n}^{+}}$
which is an irreducible character of $W_{n}^{+}$; if
$\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}$ is degenerate, then
$\varphi_{\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}}|_{W_{n}^{+}}$ is a sum of two
irreducible characters of $W_{n}^{+}$.

###  2.2. Lusztig’s symbols

In this subsection, we recall some basic notations of “symbols” from [Lus77].
A _symbol_ $\Lambda$ is an ordered pair

(2.2)
$\Lambda=\binom{A}{B}=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}$

of two finite sets $A,B$ (possibly empty) of non-negative integers. The sets
$A,B$ are also denoted by $\Lambda^{*},\Lambda_{*}$ and called the _first row_
, the _second row_ of $\Lambda$ respectively. A symbol $\Lambda$ is called
_reduced_ if $0\not\in A\cap B$. If $\Lambda=\binom{A}{B}$, then we define its
_transpose_ by $\Lambda^{\rm t}=\binom{B}{A}$. We denote
$\Lambda_{1}\subset\Lambda_{2}$ and call $\Lambda_{1}$ a _subsymbol_ of
$\Lambda_{2}$ if both $\Lambda_{1}^{*}\subset\Lambda_{2}^{*}$ and
$(\Lambda_{1})_{*}\subset(\Lambda_{2})_{*}$. If
$\Lambda_{1}\subset\Lambda_{2}$, their difference is defined by
$\Lambda_{2}\smallsetminus\Lambda_{1}=\binom{\Lambda_{2}^{*}\smallsetminus\Lambda_{1}^{*}}{(\Lambda_{2})_{*}\smallsetminus(\Lambda_{1})_{*}}$.
If both $\Lambda_{1}^{*}\cap\Lambda_{2}^{*}=\emptyset$ and
$(\Lambda_{1})_{*}\cap(\Lambda_{2})_{*}=\emptyset$, we define
$\Lambda_{1}\cup\Lambda_{2}=\binom{\Lambda_{1}^{*}\cup\Lambda_{2}^{*}}{(\Lambda_{1})_{*}\cup(\Lambda_{2})_{*}}$.

For a symbol $\Lambda$ given in (2.2), its _rank_ and _defect_ are defined by

(2.3) $\displaystyle\begin{split}{\rm
rk}(\Lambda)&=\sum_{i=1}^{m_{i}}a_{i}+\sum_{j=1}^{m_{2}}b_{j}-\left\lfloor\left(\frac{|A|+|B|-1}{2}\right)^{2}\right\rfloor,\\\
{\rm def}(\Lambda)&=|A|-|B|.\end{split}$

From the definition, it is not difficult to check that

(2.4) ${\rm rk}(\Lambda)\geq\left\lfloor\left(\frac{{\rm
def}(\Lambda)}{2}\right)^{2}\right\rfloor.$

A symbol $\Lambda$ is called _degenerate_ if $\Lambda^{\rm t}=\Lambda$. If
$\Lambda$ is degenerate, then ${\rm rk}(\Lambda)$ is even and ${\rm
def}(\Lambda)=0$.

We define an equivalence relation “$\sim$” on symbols generated by

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

If $\Lambda_{1}\sim\Lambda_{2}$, two symbols $\Lambda_{1},\Lambda_{2}$ are
called _similar_. It is not difficult to see that two symbols in the same
similar class have the same rank and the same defect, and each similar class
contains a unique reduced symbol. Let ${\mathcal{S}}$ denote the set of
similar classes of symbols, and let
${\mathcal{S}}_{n,\delta}\subset{\mathcal{S}}$ denote the subset of similar
classes of symbols of rank $n$ and defect $\delta$.

A symbol $\Lambda$ is called _cuspidal_ if (2.4) is an equality. It is not
difficult to see that a symbol is cuspidal if and only if it is similar to a
symbol of the forms $\binom{k,k-1,\ldots,0}{-}$ or $\binom{-}{k,k-1,\ldots,0}$
for some non-negative integer $k$. Note that $\binom{A}{-}$ means that the
second row of the symbol is the empty set.

A mapping $\Upsilon$ from symbols to bi-partitions is given by

(2.5)
$\Upsilon\colon\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}\mapsto\genfrac{[}{]}{0.0pt}{}{a_{1}-(m_{1}-1),a_{2}-(m_{1}-2),\ldots,a_{m_{1}-1}-1,a_{m_{1}}}{b_{1}-(m_{2}-1),b_{2}-(m_{2}-2),\ldots,b_{m_{2}}-1,b_{m_{2}}}.$

If $\Upsilon(\Lambda)=\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}$, we write
$\Upsilon(\Lambda)^{*}=\mu$ and $\Upsilon(\Lambda)_{*}=\nu$ to denote the
first row and the second row of the bi-partition $\Upsilon(\Lambda)$. We can
check that $\Upsilon(\Lambda_{1})=\Upsilon(\Lambda_{2})$ if and only if
$\Lambda_{1}\sim\Lambda_{2}$, and then $\Upsilon$ gives a bijection

(2.6)
${\mathcal{S}}_{n,\delta}\rightarrow{\mathcal{P}}_{2}\left(n-\bigl{\lfloor}\bigl{(}\tfrac{\delta}{2}\bigr{)}^{2}\bigr{\rfloor}\right).$

Modified from Lusztig, we define

(2.7) $\displaystyle\begin{split}{\mathcal{S}}_{{\rm
O}^{+}_{2n}}&=\\{\,\Lambda\in{\mathcal{S}}\mid{\rm rk}(\Lambda)=n,\ {\rm
def}(\Lambda)\equiv 0\pmod{4}\,\\},\\\ {\mathcal{S}}_{{\rm
Sp}_{2n}}&=\\{\,\Lambda\in{\mathcal{S}}\mid{\rm rk}(\Lambda)=n,\ {\rm
def}(\Lambda)\equiv 1\pmod{4}\,\\},\\\ {\mathcal{S}}_{{\rm
O}^{-}_{2n}}&=\\{\,\Lambda\in{\mathcal{S}}\mid{\rm rk}(\Lambda)=n,\ {\rm
def}(\Lambda)\equiv 2\pmod{4}\,\\},\\\ {\mathcal{S}}_{{\rm
SO}_{2n+1}}&=\\{\,\Lambda\in{\mathcal{S}}\mid{\rm rk}(\Lambda)=n,\ {\rm
def}(\Lambda)\equiv 3\pmod{4}\,\\}.\end{split}$

Note that $\Lambda\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{2n}}$ if and only if
$\Lambda^{\rm t}\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{2n}}$ where $\epsilon=+$
or $-$. Then we define

(2.8) ${\mathcal{S}}_{{\rm
SO}^{\epsilon}_{2n}}=\\{\,\Lambda\in{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\mid\Lambda\neq\Lambda^{\rm t}\,\\}/\\{\Lambda,\Lambda^{\rm
t}\\}\cup\\{\,\Lambda^{\rm I},\Lambda^{\rm
II}\mid\Lambda\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{2n}},\
\Lambda=\Lambda^{\rm t}\,\\},$

i.e., in ${\mathcal{S}}_{{\rm SO}^{\epsilon}_{2n}}$ a non-degenerate symbol is
identified with its transpose, and a degenerated symbol $\Lambda$ occurs with
multiplicity $2$ and the two copies are denoted by $\Lambda^{\rm
I},\Lambda^{\rm II}$ respectively. Note that ${\mathcal{S}}_{{\rm
O}^{-}_{2n}}$ does not contain any degenerate symbols and so
${\mathcal{S}}_{{\rm SO}^{-}_{2n}}={\mathcal{S}}_{{\rm
O}^{-}_{2n}}/\\{\Lambda,\Lambda^{\rm t}\\}$.

###  2.3. Special symbols

Let $\text{\bf G}={\rm Sp}_{2n}$, ${\rm SO}_{2n+1}$, ${\rm O}^{\epsilon}_{2n}$
or ${\rm SO}^{\epsilon}_{2n}$ where $\epsilon=+$ or $-$. A symbol
$Z=\binom{a_{1},a_{2},\ldots,a_{m+1}}{b_{1},b_{2},\ldots,b_{m}}$ of defect $1$
is called _special_ if $a_{1}\geq b_{1}\geq a_{2}\geq b_{2}\geq\cdots\geq
a_{m}\geq b_{m}\geq a_{m+1}$; similarly, a symbol
$Z=\binom{a_{1},a_{2},\ldots,a_{m}}{b_{1},b_{2},\ldots,b_{m}}$ of defect $0$
is called _special_ if $a_{1}\geq b_{1}\geq a_{2}\geq b_{2}\geq\cdots\geq
a_{m}\geq b_{m}$. Define

$\delta_{0}=\begin{cases}1,&\text{if $\text{\bf G}={\rm Sp}_{2n}$ or ${\rm
SO}_{2n+1}$};\\\ 0,&\text{if $\text{\bf G}={\rm SO}^{\epsilon}_{2n}$ or ${\rm
O}^{\epsilon}_{2n}$}.\end{cases}$

For a special symbol $Z$ of defect $\delta_{0}$, we define

$\displaystyle{\mathcal{S}}_{Z}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}\mid\Lambda^{*}\cup\Lambda_{*}=Z^{*}\cup
Z_{*},\ \Lambda^{*}\cap\Lambda_{*}=Z^{*}\cap Z_{*}\,\\},$
$\displaystyle{\mathcal{S}}_{Z,\delta_{0}}$
$\displaystyle=\\{\,\Lambda\in{\mathcal{S}}_{Z}\mid{\rm
def}(\Lambda)=\delta_{0}\,\\},$ $\displaystyle{\mathcal{S}}_{Z}^{\text{\bf
G}}$ $\displaystyle={\mathcal{S}}_{Z}\cap{\mathcal{S}}_{\text{\bf G}},$

i.e., ${\mathcal{S}}_{Z}$ is the subset of ${\mathcal{S}}$ consisting of the
symbols of the exactly the same entries of $Z$. Note that in the above
definition of ${\mathcal{S}}_{Z}^{\text{\bf G}}$, the special symbol $Z$ is
not required to be in ${\mathcal{S}}_{\text{\bf G}}$. It is clear that

(2.9) ${\mathcal{S}}_{\text{\bf G}}=\bigcup_{Z}{\mathcal{S}}_{Z}^{\text{\bf
G}}$

where $\bigcup_{Z}$ runs over

  * •

special symbols of rank $n$ and defect $1$ if $\text{\bf G}={\rm Sp}_{2n}$ or
${\rm SO}_{2n+1}$;

  * •

special symbols of rank $n$ and defect $0$ if $\text{\bf G}={\rm SO}^{+}_{2n}$
or ${\rm O}^{+}_{2n}$;

  * •

non-degenerate special symbols of rank $n$ and defect $0$ if $\text{\bf
G}={\rm SO}^{-}_{2n}$ or ${\rm O}^{-}_{2n}$.

######  Example 2.10.

  1. (1)

Suppose that $Z=\binom{1}{0}\in{\mathcal{S}}_{1,0}$. Then we have

$\textstyle{\mathcal{S}}_{Z}^{{\rm
O}^{+}_{2}}=\left\\{\binom{1}{0},\binom{0}{1}\right\\},\qquad{\mathcal{S}}_{Z}^{{\rm
O}^{-}_{2}}=\left\\{\binom{-}{1,0},\binom{1,0}{-}\right\\}.$

  2. (2)

Suppose that $Z=\binom{2,0}{1}\in{\mathcal{S}}_{2,1}$. Then we have

$\textstyle{\mathcal{S}}_{Z}^{{\rm
Sp}_{4}}=\left\\{\binom{2,0}{1},\binom{2,1}{0},\binom{1,0}{2},\binom{-}{2,1,0}\right\\},\qquad{\mathcal{S}}_{Z}^{{\rm
SO}_{5}}=\left\\{\binom{1}{2,0},\binom{0}{2,1},\binom{2}{1,0},\binom{2,1,0}{-}\right\\}.$

For a special symbol $Z$, let $Z_{\rm I}$ denote the subsymbol consisting of “
_singles_ ”, i.e.,

$Z_{\rm I}=Z\smallsetminus\binom{Z^{*}\cap Z_{*}}{Z^{*}\cap Z_{*}},$

and we define the _degree_ of $Z$ by

(2.11) $\deg(Z)=\frac{|Z_{\rm I}|-{\rm def}(Z)}{2}$

where $|Z_{\rm I}|$ denotes the number of entries in $Z_{\rm I}$, i.e.,
$|Z_{\rm I}|=|(Z_{\rm I})^{*}|+|(Z_{\rm I})_{*}|$. Note that $\deg(Z)$ is
always a non-negative integer. For a subsymbol $M\subset Z_{\rm I}$, we define
a symbol $\Lambda_{M}\in{\mathcal{S}}_{Z}$ by

(2.12) $\Lambda_{M}=(Z\smallsetminus M)\cup M^{\rm t}.$

It is not difficult to see that

${\mathcal{S}}_{Z}^{\text{\bf G}}=\begin{cases}\\{\,\Lambda_{M}\mid M\subset
Z_{\rm I},\ |M|\ \text{even}\,\\},&\text{if $\text{\bf G}={\rm Sp}_{2n}$ and
${\rm def}(Z)=1$};\\\ \\{\,\Lambda_{M}\mid M\subset Z_{\rm I},\ |M|\
\text{odd}\,\\},&\text{if $\text{\bf G}={\rm SO}_{2n+1}$ and ${\rm
def}(Z)=1$};\\\ \\{\,\Lambda_{M}\mid M\subset Z_{\rm I},\ |M|\
\text{even}\,\\},&\text{if $\text{\bf G}={\rm O}^{+}_{2n}$ and ${\rm
def}(Z)=0$};\\\ \\{\,\Lambda_{M}\mid M\subset Z_{\rm I},\ |M|\
\text{odd}\,\\},&\text{if $\text{\bf G}={\rm O}^{-}_{2n}$ and ${\rm
def}(Z)=0$}.\end{cases}$

Note that

${\mathcal{S}}_{Z}^{{\rm SO}^{\epsilon}_{2n}}=\begin{cases}\\{Z^{\rm I},Z^{\rm
II}\\},&\text{if $Z$ is degenerate};\\\ {\mathcal{S}}_{Z}^{{\rm
O}^{\epsilon}_{2n}}/\\{\Lambda,\Lambda^{\rm t}\\},&\text{if $Z$ is non-
degenerate}\end{cases}$

where $Z^{\rm I},Z^{\rm II}$ are both equal to $Z$ but are regarded as two
elements. Then we have

$|{\mathcal{S}}_{Z}^{\text{\bf G}}|=\begin{cases}2^{2\deg(Z)},&\text{if
$\text{\bf G}={\rm Sp}_{2n}$ or ${\rm SO}_{2n+1}$};\\\
2^{2\deg(Z)-1},&\text{if $\text{\bf G}={\rm O}^{\epsilon}_{2n}$ and
$\deg(Z)>0$};\\\ 2^{2\deg(Z)-2},&\text{if $\text{\bf G}={\rm
SO}^{\epsilon}_{2n}$ and $\deg(Z)>0$};\\\ 1,&\text{if $\text{\bf G}={\rm
O}^{+}_{2n}$ and $\deg(Z)=0$};\\\ 2,&\text{if $\text{\bf G}={\rm SO}^{+}_{2n}$
and $\deg(Z)=0$}.\end{cases}$

Note that the family ${\mathcal{S}}_{Z}^{\text{\bf G}}$ is empty if $\text{\bf
G}={\rm O}^{-}_{2n},{\rm SO}^{-}_{2n}$ and $\deg(Z)=0$.

Finally we define a pairing $\langle,\rangle\colon{\mathcal{S}}_{Z}^{\text{\bf
G}}\times{\mathcal{S}}_{Z,\delta_{0}}\rightarrow{\mathbf{F}}_{2}$ by

(2.13) $\langle\Lambda_{M_{1}},\Lambda_{M_{2}}\rangle=|M_{1}\cap
M_{2}|\pmod{2}.$

######  Lemma 2.14.

Let $Z$ be a special symbol of defect $\delta_{0}$. Then for any
$\Lambda\in{\mathcal{S}}_{Z}^{\text{\bf G}}$ and
$\Sigma\in{\mathcal{S}}_{Z,\delta_{0}}$, we have

$\displaystyle\langle\Lambda,\Sigma\rangle$ $\displaystyle=\langle\Lambda^{\rm
t},\Sigma\rangle,$ $\displaystyle\langle\Lambda,\Sigma\rangle$
$\displaystyle\begin{cases}=\langle\Lambda,\Sigma^{\rm t}\rangle,&\text{if\/
$\text{\bf G}={\rm SO}^{+}_{2n},{\rm O}^{+}_{2n}$};\\\
\neq\langle\Lambda,\Sigma^{\rm t}\rangle,&\text{if\/ $\text{\bf G}={\rm
SO}^{-}_{2n},{\rm O}^{-}_{2n}$}.\end{cases}$

###### Proof.

Suppose that $\Lambda=\Lambda_{M_{1}}$ and $\Sigma=\Lambda_{M_{2}}$ for some
$M_{1},M_{2}\subset Z_{\rm I}$. It is clear that $\Lambda^{\rm
t}=\Lambda_{Z_{\rm I}\smallsetminus M_{1}}$. The assumption
$\Sigma\in{\mathcal{S}}_{Z,\delta_{0}}$ implies that $|M_{2}|$ is even. Then

$|M_{1}\cap M_{2}|+|(Z_{\rm I}\smallsetminus M_{1})\cap M_{2}|=|M_{2}|\equiv
0\pmod{2}.$

Hence the first equality is obtained.

Now suppose that $\text{\bf G}={\rm SO}^{\epsilon}_{2n}$ or ${\rm
O}^{\epsilon}_{2n}$. Note that $|M_{1}|$ is even if $\epsilon=+$, and
$|M_{1}|$ is odd if $\epsilon=-$. Then the remaining assertion is obtained by
the analogous argument. ∎

##  3\. Lusztig Parametrization of Unipotent Characters

###  3.1. Unipotent characters of ${\rm GL}_{n}$ or ${\rm U}_{n}$

In this subsection, let G be a general linear group ${\rm GL}_{n}$ or a
unitary group ${\rm U}_{n}$. It is well known that the Weyl group
$W_{\text{\bf G}}=S_{n}$, and ${\mathcal{E}}(S_{n})$ is parametrized by
${\mathcal{P}}(n)$. For $\lambda\in{\mathcal{P}}(n)$, we define

$R_{\lambda}^{\text{\bf G}}=\frac{1}{|S_{n}|}\sum_{w\in
S_{n}}\varphi_{\lambda}(w)R^{\text{\bf G}}_{\text{\bf T}_{w},1}$

where $\varphi_{\lambda}$ denotes the irreducible character of $S_{n}$
corresponding to $\lambda$. It is known that $R_{\lambda}^{\text{\bf G}}$ is
an irreducible unipotent character of $G$, i.e., $R_{\lambda}^{\text{\bf
G}}\in{\mathcal{E}}(\text{\bf G},1)$. Let ${\mathcal{S}}_{{\rm
GL}_{n}}={\mathcal{S}}_{{\rm U}_{n}}={\mathcal{P}}(n)$. Then the mapping

${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)\quad\text{ given by
}\lambda\mapsto\rho_{\lambda}:=R_{\lambda}^{\text{\bf G}}$

is a bijection. To distinguish ${\mathcal{S}}_{{\rm U}_{n}}$ from
${\mathcal{S}}_{{\rm GL}_{n}}$, an element in ${\mathcal{S}}_{{\rm U}_{n}}$
will be denoted by $[\bar{\lambda}_{1},\ldots,\bar{\lambda}_{m}]$ for
$[\lambda_{1},\ldots,\lambda_{m}]\in{\mathcal{P}}(n)$. An element of the form
$[\bar{k},\bar{k-1},\ldots,\bar{1}]\in{\mathcal{S}}_{{\rm U}_{k(k+1)/2}}$ for
some $k$ is called _cuspidal_. For a unitary group, it is well known that
$\lambda$ is cuspidal if and only if $\rho_{\lambda}$ is cuspidal.

It is known that the parametrization ${\mathcal{L}}_{1}$ above is compatible
with the parabolic induction on unipotent characters. More precisely, let
$\text{\bf G}_{n}={\rm GL}_{n}$, ${\rm U}_{2n}$ or ${\rm U}_{2n+1}$. For
$\rho\in{\mathcal{E}}(\text{\bf G}_{n},1)$, define

(3.1) $\Omega(\rho)=\left\\{\,\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}_{n+1},1)\mid\left\langle\rho^{\prime},R^{\text{\bf G}_{n+1}}_{\text{\bf
G}_{n}\times{\rm GL}^{\dagger}_{1}}(\rho\otimes{\bf
1})\right\rangle_{\\!\text{\bf G}_{n+1}}\neq 0\,\right\\}$

where $R^{\text{\bf G}_{n+1}}_{\text{\bf G}_{n}\times{\rm GL}^{\dagger}_{1}}$
is the standard parabolic induction, ${\rm GL}^{\dagger}_{1}={\rm GL}_{1}$
defined over ${\mathbf{F}}_{q}$ if $\text{\bf G}_{n}={\rm GL}_{n}$; and ${\rm
GL}^{\dagger}_{1}$ is the restriction to ${\mathbf{F}}_{q}$ of ${\rm GL}_{1}$
defined over a quadratic extension of ${\mathbf{F}}_{q}$ if $\text{\bf
G}_{n}={\rm U}_{2n}$ or ${\rm U}_{2n+1}$. For
$\lambda\in{\mathcal{S}}_{\text{\bf G}_{n}}$, we define $\Omega(\lambda)$ to
be a subset of ${\mathcal{S}}_{\text{\bf G}_{n+1}}$ consisting of partitions
of the following types:

  * •

If $\text{\bf G}_{n}={\rm GL}_{n}$ and $\lambda\in{\mathcal{S}}_{\text{\bf
G}_{n}}$, then $\Omega(\lambda)$ consists of all partitions
$\lambda^{\prime}\in{\mathcal{S}}_{\text{\bf G}_{n+1}}$ whose Young diagrams
are obtained by adding a box to the Young diagram of $\lambda$.

  * •

If $\text{\bf G}_{n}={\rm U}_{2n}$ or ${\rm U}_{2n+1}$ and
$\lambda\in{\mathcal{S}}_{\text{\bf G}_{n}}$, then $\Omega(\lambda)$ consists
of all partitions $\lambda^{\prime}\in{\mathcal{S}}_{\text{\bf G}_{n+1}}$
whose Young diagrams are obtained by adding two boxes in the same row or in
the same column to the Young diagram of $\lambda$.

######  Example 3.2.

  1. (1)

Suppose that $\lambda=[3,1,1]\in{\mathcal{S}}_{{\rm GL}_{5}}$. Then

$\Omega(\lambda)=\\{[4,1,1],[3,2,1],[3,1,1,1]\\}\subset{\mathcal{S}}_{{\rm
GL}_{6}}.$

  2. (2)

Suppose that $\lambda=[\bar{3},\bar{1},\bar{1}]\in{\mathcal{S}}_{{\rm
U}_{5}}$. Then

$\Omega(\lambda)=\\{[\bar{5},\bar{1},\bar{1}],[\bar{3},\bar{3},\bar{1}],[\bar{3},\bar{2},\bar{2}],[\bar{3},\bar{1},\bar{1},\bar{1},\bar{1}]\\}\subset{\mathcal{S}}_{{\rm
U}_{7}}.$

Now the parametrization ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ by $\lambda\mapsto\rho_{\lambda}$
is said to be _compatible with parabolic induction_ if the following diagram

(3.3) $\begin{CD}{\mathcal{S}}_{\text{\bf
G}_{n}}@>{\Omega}>{}>{\mathcal{S}}_{\text{\bf G}_{n+1}}\\\
@V{{\mathcal{L}}_{1}}V{}V@V{}V{{\mathcal{L}}_{1}}V\\\ {\mathcal{E}}(\text{\bf
G}_{n},1)@>{\Omega}>{}>{\mathcal{E}}(\text{\bf G}_{n+1},1)\end{CD}$

commutes, i.e.,
$\Omega(\rho_{\lambda})=\\{\,\rho_{\lambda^{\prime}}\mid\lambda^{\prime}\in\Omega(\lambda)\,\\}$
for any $\lambda\in{\mathcal{S}}_{\text{\bf G}_{n}}$.

###  3.2. Uniform almost characters

First suppose G is a connected classical group ${\rm SO}_{2n+1}$, ${\rm
Sp}_{2n}$, or ${\rm SO}^{\epsilon}_{2n}$. For a rational maximal torus
$\text{\bf T}^{*}$ in the dual group $\text{\bf G}^{*}$ and a rational element
$s\in T^{*}$, the _Deligne-Lusztig virtual characters_ $R^{\text{\bf
G}}_{\text{\bf T}^{*},s}$ is defined in [DL76] (see also [Car85]). For the
disconnected group ${\rm O}^{\epsilon}_{2n}$, $R^{{\rm
O}^{\epsilon}_{2n}}_{\text{\bf T}^{*},s}$ is defined by

(3.4) $R^{{\rm O}^{\epsilon}_{2n}}_{\text{\bf T}^{*},s}={\rm Ind}_{{\rm
SO}^{\epsilon}_{2n}}^{{\rm O}^{\epsilon}_{2n}}R^{{\rm
SO}^{\epsilon}_{2n}}_{\text{\bf T}^{*},s}.$

For rational maximal tori in G, the following is well known (cf. [Sri79] §3):

  * •

If $\text{\bf G}={\rm Sp}_{2n}$ or ${\rm SO}_{2n+1}$, it is known that any
rational maximal torus in G is conjugate (under $G$) to $\text{\bf T}_{w}$ for
some $w\in W_{n}$. Moreover, $\text{\bf T}_{w}$ and $\text{\bf
T}_{w^{\prime}}$ are conjugate if and only if $w,w^{\prime}$ are conjugate
under $W_{n}$.

  * •

If $\text{\bf G}={\rm SO}^{\epsilon}_{2n}$ or ${\rm O}^{\epsilon}_{2n}$, it is
known that any rational maximal torus is conjugate (under $G$) to $\text{\bf
T}_{w}$ for some $w\in W_{n}^{\epsilon}$ (cf. Subsection 2.1). Moreover,
$\text{\bf T}_{w}$ and $\text{\bf T}_{w^{\prime}}$ are conjugate if and only
if $w,w^{\prime}$ are conjugate under $W_{n}$.

We recall some definitions from [Lus78] §3.17 and §3.19 (see also [Pan19b]
§2.3). For $\Sigma\in{\mathcal{S}}_{n,\delta_{0}}$ (cf. Subsection 2.3), we
define a uniform unipotent class function $R_{\Sigma}^{\text{\bf
G}}\in{\mathcal{V}}(\text{\bf G},1)^{\sharp}$ by

(3.5) $\displaystyle\begin{split}R^{\text{\bf
G}}_{\Sigma}&=\begin{cases}\displaystyle\frac{1}{\sqrt{2}|W_{n}^{+}|}\sum_{w\in
W_{n}^{+}}\varphi_{\Upsilon(\Sigma)}(w)R^{\text{\bf G}}_{\text{\bf
T}_{w},1},&\text{if $\text{\bf G}={\rm SO}^{+}_{2n}$ and $\Sigma$
degenerate};\\\ \displaystyle\frac{1}{|W_{n}^{\epsilon}|}\sum_{w\in
W_{n}^{\epsilon}}\varphi_{\Upsilon(\Sigma)}(w)R^{\text{\bf G}}_{\text{\bf
T}_{w},1},&\text{if $\text{\bf G}={\rm SO}^{\epsilon}_{2n}$ and $\Sigma$ non-
degenerate};\\\ \displaystyle\frac{1}{|W_{n}|}\sum_{w\in
W_{n}}\varphi_{\Upsilon(\Sigma)}(w)R^{\text{\bf G}}_{\text{\bf
T}_{w},1},&\text{if $\text{\bf G}={\rm Sp}_{2n}$};\\\
\displaystyle\frac{1}{|W_{n}|}\sum_{w\in W_{n}}\varphi_{\Upsilon(\Sigma^{\rm
t})}(w)R^{\text{\bf G}}_{\text{\bf T}_{w},1},&\text{if $\text{\bf G}={\rm
SO}_{2n+1}$}.\end{cases}\\\ R_{\Sigma}^{{\rm
O}^{\epsilon}_{2n}}&=\frac{1}{\sqrt{2}}{\rm Ind}_{{\rm
SO}^{\epsilon}_{2n}}^{{\rm O}^{\epsilon}_{2n}}(R^{{\rm
SO}^{\epsilon}_{2n}}_{\Sigma}).\end{split}$

Note that $\varphi_{\Upsilon(\Sigma)}$ is the irreducible character of $W_{n}$
given in (2.1) and $\Upsilon$ is the bijection
${\mathcal{S}}_{n,\delta_{0}}\rightarrow{\mathcal{P}}_{2}(n)$ given in (2.6).
The class function $R^{\text{\bf G}}_{\Sigma}$ is called an _almost character_
of $G$.

######  Lemma 3.6.

Suppose that $\text{\bf G}={\rm O}^{\epsilon}_{2n}$ where $\epsilon=+$ or $-$.
Then $R^{\text{\bf G}}_{\Sigma^{\rm t}}=\epsilon R^{\text{\bf G}}_{\Sigma}$
for any $\Sigma\in{\mathcal{S}}_{n,0}$.

###### Proof.

From (3.5), we know that

$R^{{\rm O}^{\epsilon}_{2n}}_{\Sigma^{\rm
t}}=\begin{cases}\displaystyle\frac{1}{2|W_{n}^{+}|}\sum_{w\in
W_{n}^{+}}\varphi_{\Upsilon(\Sigma^{\rm t})}(w)R^{{\rm
O}^{\epsilon}_{2n}}_{\text{\bf T}_{w},1},&\text{if $\epsilon=+$ and $\Sigma$
degenerate};\\\ \displaystyle\frac{1}{\sqrt{2}|W_{n}^{\epsilon}|}\sum_{w\in
W_{n}^{\epsilon}}\varphi_{\Upsilon(\Sigma^{\rm t})}(w)R^{{\rm
O}^{\epsilon}_{2n}}_{\text{\bf T}_{w},1},&\text{if $\Sigma$ non-
degenerate}.\end{cases}$

Moreover, from Subsection 2.1, we have $\varphi_{\Upsilon(\Sigma^{\rm
t})}(w)=\epsilon\varphi_{\Upsilon(\Sigma)}(w)$ for $w\in W^{\epsilon}_{n}$.
Then the lemma is proved. ∎

Now we define a set ${\mathcal{S}}_{\text{\bf G}}^{\sharp}$ as follows.

(3.7) ${\mathcal{S}}_{\text{\bf
G}}^{\sharp}=\begin{cases}{\mathcal{S}}_{n,1},&\text{if $\text{\bf G}={\rm
Sp}_{2n},{\rm SO}_{2n+1}$};\\\ {\mathcal{S}}_{n,0}/\\{\Sigma,\Sigma^{\rm
t}\\},&\text{if $\text{\bf G}={\rm SO}^{+}_{2n},{\rm O}^{+}_{2n}$};\\\
\\{\,\Sigma\in{\mathcal{S}}_{n,0}\mid\Sigma\neq\Sigma^{\rm
t}\,\\}/\\{\Sigma,\Sigma^{\rm t}\\},&\text{if $\text{\bf G}={\rm
SO}^{-}_{2n},{\rm O}^{-}_{2n}$}.\end{cases}$

Here ${\mathcal{S}}_{n,0}/\\{\Sigma,\Sigma^{\rm t}\\}$ means that we choose a
representative for each subset $\\{\Sigma,\Sigma^{\rm t}\\}$ in
${\mathcal{S}}_{n,0}$.

######  Example 3.8.

We know that
${\mathcal{S}}_{2,0}=\left\\{\binom{2}{0},\binom{0}{2},\binom{2,1}{1,0},\binom{1,0}{2,1},\binom{1}{1}\right\\}$.
The following are all four possible choices of ${\mathcal{S}}_{{\rm
SO}^{-}_{4}}^{\sharp}$ (or ${\mathcal{S}}_{{\rm O}^{-}_{4}}^{\sharp}$):

$\textstyle\left\\{\binom{2}{0},\binom{2,1}{1,0}\right\\},\quad\left\\{\binom{2}{0}\binom{1,0}{2,1}\right\\},\quad\left\\{\binom{0}{2},\binom{2,1}{1,0}\right\\},\quad\left\\{\binom{0}{2},\binom{1,0}{2,1}\right\\}.$

Moreover, for each above choice ${\mathcal{S}}_{{\rm SO}^{-}_{4}}^{\sharp}$,
we see that ${\mathcal{S}}_{2,0}\smallsetminus{\mathcal{S}}_{{\rm
SO}^{-}_{4}}^{\sharp}$ is a choice for ${\mathcal{S}}_{{\rm
SO}^{+}_{4}}^{\sharp}$ (or ${\mathcal{S}}_{{\rm O}^{+}_{4}}^{\sharp}$).

Similar to (2.9), we have the decomposition

${\mathcal{S}}_{\text{\bf
G}}^{\sharp}=\bigcup_{Z}{{\mathcal{S}}_{Z}^{\text{\bf G}}}^{\sharp}$

where the union $\bigcup_{Z}$ is as in (2.9) and

(3.9) ${{\mathcal{S}}_{Z}^{\text{\bf
G}}}^{\sharp}=\begin{cases}{\mathcal{S}}_{Z}\cap{\mathcal{S}}_{n,1},&\text{if
$\text{\bf G}={\rm Sp}_{2n},{\rm SO}_{2n+1}$};\\\
({\mathcal{S}}_{Z}\cap{\mathcal{S}}_{n,0})/\\{\Sigma,\Sigma^{\rm
t}\\},&\text{if $\text{\bf G}={\rm SO}^{\epsilon}_{2n},{\rm
O}^{\epsilon}_{2n}$}.\end{cases}$

######  Lemma 3.10.

Let $\text{\bf G}={\rm Sp}_{2n}$, ${\rm SO}_{2n+1}$, ${\rm
SO}^{\epsilon}_{2n}$ or ${\rm O}^{\epsilon}_{2n}$. Then the set
$\\{\,R_{\Sigma}^{\text{\bf G}}\mid\Sigma\in{\mathcal{S}}_{\text{\bf
G}}^{\sharp}\,\\}$ forms an orthonormal basis for ${\mathcal{V}}(\text{\bf
G},1)^{\sharp}$.

###### Proof.

First we show that the set $\\{\,R_{\Sigma}^{\text{\bf
G}}\mid\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp}\,\\}$ is orthonormal.

  1. (1)

Suppose that $\text{\bf G}={\rm Sp}_{2n}$ or ${\rm SO}_{2n+1}$. From [Car85]
proposition 7.3.6, it is not difficult to see that

$\langle R^{\text{\bf G}}_{\Sigma},R^{\text{\bf
G}}_{\Sigma^{\prime}}\rangle_{\text{\bf
G}}=\langle\varphi_{\Upsilon(\Sigma)},\varphi_{\Upsilon(\Sigma^{\prime})}\rangle_{W_{n}}=\begin{cases}1,&\text{if
$\Sigma=\Sigma^{\prime}$};\\\ 0,&\text{if
$\Sigma\neq\Sigma^{\prime}$}.\end{cases}$

  2. (2)

Suppose that $\text{\bf G}={\rm SO}^{+}_{2n}$. Then

$\langle R^{\text{\bf G}}_{\Sigma},R^{\text{\bf
G}}_{\Sigma^{\prime}}\rangle_{\text{\bf
G}}=\begin{cases}\langle\varphi_{\Upsilon(\Sigma)},\varphi_{\Upsilon(\Sigma^{\prime})}\rangle_{W_{n}^{+}},&\text{if
both $\Sigma,\Sigma^{\prime}$ are non-degenerate};\\\
\frac{1}{\sqrt{2}}\langle\varphi_{\Upsilon(\Sigma)},\varphi_{\Upsilon(\Sigma^{\prime})}\rangle_{W_{n}^{+}},&\text{if
exactly one of $\Sigma,\Sigma^{\prime}$ is degenerate};\\\
\frac{1}{2}\langle\varphi_{\Upsilon(\Sigma)},\varphi_{\Upsilon(\Sigma^{\prime})}\rangle_{W_{n}^{+}},&\text{if
both $\Sigma,\Sigma^{\prime}$ are degenerate}.\end{cases}$

We know that

$\langle\varphi_{\Upsilon(\Sigma)},\varphi_{\Upsilon(\Sigma)}\rangle_{W_{n}^{+}}=\begin{cases}1,&\text{if
$\Sigma$ non-degenerate};\\\ 2,&\text{if $\Sigma$ degenerate}.\end{cases}$

Therefore, we conclude that

$\langle R^{\text{\bf G}}_{\Sigma},R^{\text{\bf
G}}_{\Sigma^{\prime}}\rangle_{\text{\bf G}}=\begin{cases}1,&\text{if
$\Sigma=\Sigma^{\prime}$};\\\ 0,&\text{if
$\Sigma\neq\Sigma^{\prime},\Sigma^{\prime{\rm t}}$}.\end{cases}$

  3. (3)

Suppose that $\text{\bf G}={\rm SO}^{-}_{2n}$. Fix an element $\sigma_{0}\in
W_{n}^{-}$, so an element in $W_{n}^{-}$ can be written as $\sigma_{0}w$ for
some $w\in W_{n}^{+}$. We know that $\varphi_{\Upsilon(\Sigma)}(\sigma_{0}w)$
is either equal to $\varphi_{\Upsilon(\Sigma)}(w)$ or
$-\varphi_{\Upsilon(\Sigma)}(w)$. Therefore, by the proof in (2) we again
conclude that

$\langle R^{\text{\bf G}}_{\Sigma},R^{\text{\bf
G}}_{\Sigma^{\prime}}\rangle_{\text{\bf G}}=\begin{cases}1,&\text{if
$\Sigma=\Sigma^{\prime}$};\\\ 0,&\text{if
$\Sigma\neq\Sigma^{\prime},\Sigma^{\prime{\rm t}}$}.\end{cases}$

  4. (4)

Suppose that $\text{\bf G}={\rm O}^{\epsilon}_{2n}$. From (3.5), we know that

$R^{{\rm O}^{\epsilon}_{2n}}_{\Sigma}(g)=\begin{cases}\sqrt{2}R^{{\rm
SO}^{\epsilon}_{2n}}_{\Sigma}(g),&\text{if $g\in{\rm
SO}^{\epsilon}_{2n}(q)$};\\\ 0,&\text{if $g\in{\rm
O}^{\epsilon}_{2n}(q)\smallsetminus{\rm SO}^{\epsilon}_{2n}(q)$}\end{cases}$

for any $\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp}$. Because $|{\rm
O}^{\epsilon}_{2n}(q)|=2|{\rm SO}^{\epsilon}_{2n}(q)|$, we have

$\langle R^{{\rm O}^{\epsilon}_{2n}}_{\Sigma},R^{{\rm
O}^{\epsilon}_{2n}}_{\Sigma^{\prime}}\rangle_{{\rm O}^{\epsilon}_{2n}}=\langle
R^{{\rm SO}^{\epsilon}_{2n}}_{\Sigma},R^{{\rm
SO}^{\epsilon}_{2n}}_{\Sigma^{\prime}}\rangle_{{\rm SO}^{\epsilon}_{2n}}.$

Next we show that the set $\\{\,R_{\Sigma}^{\text{\bf
G}}\mid\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp}\,\\}$ spans
${\mathcal{V}}(\text{\bf G},1)^{\sharp}$. It is well known that

(3.11)
$\sum_{\varphi\in{\mathcal{E}}(W_{n})}\overline{\varphi(w)}\varphi(w^{\prime})=\begin{cases}\frac{|W_{n}|}{|{\mathcal{O}}_{w}|},&\text{if
$w,w^{\prime}$ are conjugate};\\\ 0,&\text{if $w,w^{\prime}$ are not
conjugate}\end{cases}$

where ${\mathcal{O}}_{w}$ denotes the conjugacy class of $w$ (cf. [Ser77]
proposition 7).

  1. (1)

Suppose that $\text{\bf G}={\rm Sp}_{2n}$. For $w\in W_{n}$, by (3.5) and
(3.11) we have

$\displaystyle\sum_{\Sigma\in{\mathcal{S}}_{\text{\bf
G}}^{\sharp}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}R^{\text{\bf
G}}_{\Sigma}$ $\displaystyle=\frac{1}{|W_{n}|}\sum_{w^{\prime}\in
W_{n}}\left[\sum_{\Sigma\in{\mathcal{S}}_{n,1}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}\varphi_{\Upsilon(\Sigma)}(w^{\prime})\right]R^{\text{\bf
G}}_{\text{\bf T}_{w^{\prime}},1}$
$\displaystyle=\frac{1}{|{\mathcal{O}}_{w}|}\sum_{w^{\prime}\in W_{n},\
w^{\prime}\text{ conjugate to }w}R^{\text{\bf G}}_{\text{\bf
T}_{w^{\prime}},1}$ $\displaystyle=R_{\text{\bf T}_{w},1}^{\text{\bf G}}.$

The proof for $\text{\bf G}={\rm SO}_{2n+1}$ is similar.

  2. (2)

Suppose that $\text{\bf G}={\rm SO}^{-}_{2n}$. For $w\in W_{n}^{-}$, by (3.5)
and (3.11) we have

$\displaystyle\sum_{\Sigma\in{\mathcal{S}}_{\text{\bf
G}}^{\sharp}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}R^{\text{\bf
G}}_{\Sigma}$
$\displaystyle=\frac{1}{2}\sum_{\Sigma\in{\mathcal{S}}_{n,0}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}R^{\text{\bf
G}}_{\Sigma}$ $\displaystyle=\frac{1}{2|W_{n}^{-}|}\sum_{w^{\prime}\in
W_{n}^{-}}\left[\sum_{\Sigma\in{\mathcal{S}}_{n,0}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}\varphi_{\Upsilon(\Sigma)}(w^{\prime})\right]R^{\text{\bf
G}}_{\text{\bf T}_{w^{\prime}},1}$
$\displaystyle=\frac{1}{|{\mathcal{O}}_{w}|}\sum_{w^{\prime}\in W_{n}^{-},\
w^{\prime}\text{ conjugate to }w}R^{\text{\bf G}}_{\text{\bf
T}_{w^{\prime}},1}$ $\displaystyle=R_{\text{\bf T}_{w},1}^{\text{\bf G}}.$

Note that for $w\in W_{n}^{-}$, the whole conjugacy class ${\mathcal{O}}_{w}$
is contained in $W_{n}^{-}$. The proof for $\text{\bf G}={\rm O}^{-}_{2n}$ is
similar.

  3. (3)

Suppose that $\text{\bf G}={\rm SO}^{+}_{2n}$. For $w\in W_{n}^{+}$, we have

$\displaystyle\sum_{\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp},\
\Sigma\text{ non-
degenerate}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}R^{\text{\bf
G}}_{\Sigma}+\sum_{\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp},\
\Sigma\text{
degenerate}}\frac{1}{\sqrt{2}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}R^{\text{\bf
G}}_{\Sigma}$ $\displaystyle=\frac{1}{2|W_{n}^{+}|}\sum_{w^{\prime}\in
W_{n}^{+}}\left[\sum_{\Sigma\in{\mathcal{S}}_{n,0}}\overline{\varphi_{\Upsilon(\Sigma)}(w)}\varphi_{\Upsilon(\Sigma)}(w^{\prime})\right]R^{\text{\bf
G}}_{\text{\bf T}_{w^{\prime}},1}$ $\displaystyle=R_{\text{\bf
T}_{w},1}^{\text{\bf G}}.$

The proof for $\text{\bf G}={\rm O}^{+}_{2n}$ is similar.

Therefore for all cases, we show that $\\{\,R_{\Sigma}^{\text{\bf
G}}\mid\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp}\,\\}$ forms an
orthonormal basis for ${\mathcal{V}}(\text{\bf G},1)^{\sharp}$. ∎

###  3.3. Lusztig’s parametrization of unipotent characters

######  Proposition 3.12 (Lusztig).

Let $\text{\bf G}={\rm Sp}_{2n}$, ${\rm SO}_{2n+1}$, ${\rm
SO}^{\epsilon}_{2n}$, or ${\rm O}^{\epsilon}_{2n}$ where $\epsilon=+$ or $-$.
There exists a bijection ${\mathcal{L}}_{1}={\mathcal{L}}_{1}^{\text{\bf
G}}\colon{\mathcal{S}}_{\text{\bf G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$
by $\Lambda\mapsto\rho_{\Lambda}$ satisfying

(3.13) $\langle\rho_{\Lambda},R_{\Sigma}^{\text{\bf G}}\rangle_{\text{\bf
G}}=\begin{cases}\frac{1}{c_{Z}}(-1)^{\langle\Lambda,\Sigma\rangle},&\text{if
$\Lambda\in{\mathcal{S}}_{Z}^{\text{\bf
G}},\Sigma\in{{\mathcal{S}}_{Z}^{\text{\bf G}}}^{\sharp}$ for some special
$Z$};\\\ 0,&\text{otherwise},\end{cases}$

where

$c_{Z}=\begin{cases}2^{\deg(Z)},&\text{if\/ $\text{\bf G}={\rm Sp}_{2n},{\rm
SO}_{2n+1}$};\\\ 2^{\deg(Z)},&\text{if\/ $\text{\bf G}={\rm O}^{+}_{2n}$ and
$Z$ degenerate};\\\ 2^{\deg(Z)+1/2},&\text{if\/ $\text{\bf G}={\rm
SO}^{+}_{2n}$ and $Z$ degenerate};\\\ 2^{\deg(Z)-1/2},&\text{if\/ $\text{\bf
G}={\rm SO}^{\epsilon}_{2n},{\rm O}^{\epsilon}_{2n}$ and $Z$ non-
degenerate}.\end{cases}$

###### Proof.

If $\text{\bf G}={\rm Sp}_{2n}$ ${\rm SO}_{2n+1}$ or ${\rm
SO}^{\epsilon}_{2n}$, the result is from [Lus81] theorem 5.8 and [Lus82]
theorem 3.15. If $\text{\bf G}={\rm O}^{\epsilon}_{2n}$, a proof can be found
in [Pan21] proposition 3.6. Note that the definition of $R^{\text{\bf
G}}_{\Sigma}$ here is slightly different from that in [Pan21] §3.4. ∎

A bijective mapping ${\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ satisfying (3.13) is called a
_Lusztig parametrization_ of unipotent characters of $G$.

######  Remark 3.14.

For $\text{\bf G}={\rm Sp}_{2n}$, our definition ${\mathcal{S}}_{\text{\bf
G}}$ in (2.7) is slightly different from the original definition by Lusztig in
[Lus77] p.134. The definition here is more convenient for finite theta
correspondence on unipotent characters. Moreover, it is easy to see that two
definitions are equivalent (cf. [Pan21] (3.5)).

######  Lemma 3.15.

Suppose that $\text{\bf G}={\rm Sp}_{2n}$, ${\rm SO}_{2n+1}$, ${\rm
SO}^{\epsilon}_{2n}$, or ${\rm O}^{\epsilon}_{2n}$ where $\epsilon=+$ or $-$.
Let
${\mathcal{L}}_{1},{\mathcal{L}}^{\prime}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ be two Lusztig parametrizations
of unipotent characters of $G$. Then
${\mathcal{L}}_{1}(\Lambda)^{\sharp}={\mathcal{L}}^{\prime}_{1}(\Lambda)^{\sharp}$.

###### Proof.

Write ${\mathcal{L}}_{1}(\Lambda)=\rho_{\Lambda}$ and
${\mathcal{L}}^{\prime}_{1}(\Lambda)=\rho^{\prime}_{\Lambda}$ for
$\Lambda\in{\mathcal{S}}_{\text{\bf G}}$. From (3.13), we see that

$\langle\rho_{\Lambda},R^{\text{\bf G}}_{\Sigma}\rangle_{\text{\bf
G}}=\langle\rho^{\prime}_{\Lambda},R^{\text{\bf G}}_{\Sigma}\rangle_{\text{\bf
G}}$

for any $\Sigma\in{\mathcal{S}}_{\text{\bf G}}^{\sharp}$. Moreover, by Lemma
3.10 we see that

(3.16) $\rho_{\Lambda}^{\sharp}=\sum_{\Sigma\in{\mathcal{S}}_{\text{\bf
G}}^{\sharp}}\langle\rho_{\Lambda},R^{\text{\bf G}}_{\Sigma}\rangle_{\text{\bf
G}}R^{\text{\bf G}}_{\Sigma}.$

Then the lemma is proved. ∎

Suppose that $\text{\bf G}=\prod_{i=1}^{k}\text{\bf G}_{i}$ where each
$\text{\bf G}_{i}$ is a classical group. It is clear that

$\displaystyle{\mathcal{E}}(\text{\bf G},1)$
$\displaystyle=\\{\,\rho_{1}\otimes\cdots\otimes\rho_{k}\mid\rho_{i}\in{\mathcal{E}}(\text{\bf
G}_{i},1)\,\\},$ $\displaystyle{\mathcal{V}}(\text{\bf G},1)$
$\displaystyle=\bigotimes_{i=1}^{k}{\mathcal{V}}(\text{\bf G}_{i},1),$
$\displaystyle{\mathcal{V}}(\text{\bf G},1)^{\sharp}$
$\displaystyle=\bigotimes_{i=1}^{k}{\mathcal{V}}(\text{\bf
G}_{i},1)^{\sharp}.$

Then we define

${\mathcal{S}}_{\text{\bf G}}=\prod_{i=1}^{k}{\mathcal{S}}_{\text{\bf
G}_{i}},\qquad{\mathcal{S}}_{\text{\bf
G}}^{\sharp}=\prod_{i=1}^{k}{\mathcal{S}}_{\text{\bf G}_{i}}^{\sharp},$

and then any Lusztig parametrization
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ is of the form

$\displaystyle{\mathcal{L}}_{1}\colon\prod_{i=1}^{k}{\mathcal{S}}_{\text{\bf
G}_{i}}$
$\displaystyle\rightarrow{\mathcal{E}}\Biggl{(}\prod_{i=1}^{k}\text{\bf
G}_{i},1\Biggr{)}$ $\displaystyle(\Lambda_{1},\ldots,\Lambda_{k})$
$\displaystyle\mapsto\rho_{\Lambda_{1}}\otimes\cdots\otimes\rho_{\Lambda_{k}}$

where $\Lambda_{i}\mapsto\rho_{\Lambda_{i}}$ is given by a Lusztig
parametrization ${\mathcal{S}}_{\text{\bf
G}_{i}}\rightarrow{\mathcal{E}}(\text{\bf G}_{i},1)$.

The parametrization ${\mathcal{L}}_{1}$ given by Lusztig in Proposition 3.12
is compatible with parabolic induction as follows. Let $\text{\bf G}_{n}={\rm
Sp}_{2n}$, ${\rm SO}_{2n+1}$, ${\rm SO}^{\epsilon}_{2n}$ or ${\rm
O}^{\epsilon}_{2n}$. For $\rho\in{\mathcal{E}}(\text{\bf G}_{n},1)$, let
$\Omega(\rho)\subset{\mathcal{E}}(\text{\bf G}_{n+1},1)$ be defined similarly
as in (3.1). For $\Lambda\in{\mathcal{S}}_{\text{\bf G}_{n}}$, then
$\Omega(\Lambda)$ consists of all symbols
$\Lambda^{\prime}\in{\mathcal{S}}_{\text{\bf G}_{n+1}}$ such that

  * •

${\rm def}(\Lambda^{\prime})={\rm def}(\Lambda)$, and

  * •

$\Upsilon(\Lambda^{\prime})$ is obtained from $\Upsilon(\Lambda)$ by adding a
box to the Young diagram of $\Upsilon(\Lambda)^{*}$ or
$\Upsilon(\Lambda)_{*}$.

######  Example 3.17.

Suppose that $\text{\bf G}_{n}={\rm Sp}_{4}$ and $\Lambda=\binom{2,0}{1}$,
then $\Upsilon(\Lambda)=\genfrac{[}{]}{0.0pt}{}{1}{1}$, and so
$\Upsilon(\Lambda^{\prime})$ is equal to $\genfrac{[}{]}{0.0pt}{}{2}{1}$,
$\genfrac{[}{]}{0.0pt}{}{1,1}{1}$, $\genfrac{[}{]}{0.0pt}{}{1}{2}$, or
$\genfrac{[}{]}{0.0pt}{}{1}{1,1}$. Therefore

$\Omega(\Lambda)=\left\\{\textstyle\binom{3,0}{1},\binom{2,1}{1},\binom{2,0}{2},\binom{3,1,0}{2,1}\right\\}\subset{\mathcal{S}}_{{\rm
Sp}_{6}}.$

For $\Sigma\in{\mathcal{S}}_{\text{\bf G}_{n}}^{\sharp}$, it is known that
(cf. [GP00] §6.1.9)

${\rm Ind}_{W_{n}\times S_{1}}^{W_{n+1}}(\varphi_{\Upsilon(\Sigma)}\otimes{\bf
1})=\sum_{\Sigma^{\prime}\in\Omega(\Sigma)}\varphi_{\Upsilon(\Sigma^{\prime})}.$

By direct computation (cf. [Lus81] (4.6.3)), we have

$R_{\text{\bf G}_{n}\times{\rm GL}_{1}}^{\text{\bf
G}_{n+1}}(R_{\Sigma}^{\text{\bf
G}_{n}})=\sum_{\Sigma^{\prime}\in\Omega(\Sigma)}R_{\Sigma^{\prime}}^{\text{\bf
G}_{n+1}}.$

We say that the parametrization
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ is _compatible with parabolic
induction_ if the diagram analogous to (3.3) commutes, i.e.,

$\Omega(\rho_{\Lambda})=\\{\,\rho_{\Lambda^{\prime}}\mid\Lambda^{\prime}\in\Omega(\Lambda)\,\\}.$

Note that ${\rm def}(\Lambda^{\prime})={\rm def}(\Lambda)$ for any
$\Lambda^{\prime}\in\Omega(\Lambda)$. This means that under the
parametrization by Lusztig the defects of symbols are preserved by parabolic
induction on unipotent characters. Therefore, if ${\rm
def}(\Lambda^{\prime})\neq 0$ and $\Lambda^{\prime}\in\Omega(\Lambda)$, then
$\Lambda^{\prime{\rm t}}\not\in\Omega(\Lambda)$.

######  Lemma 3.18.

Let $\Lambda\in{\mathcal{S}}_{{\rm O}^{+}_{2n}}$ such that ${\rm
def}(\Lambda)=0$ and $\Lambda\neq\Lambda^{\rm t}$. Suppose that $n\geq 2$.
Then there exists $\Lambda_{1}\in{\mathcal{S}}_{{\rm O}^{+}_{2(n-1)}}$ such
that $\Lambda\in\Omega(\Lambda_{1})$ and $\Lambda^{\rm
t}\not\in\Omega(\Lambda_{1})$.

###### Proof.

Write $\Lambda=\binom{a_{1},\ldots,a_{m}}{b_{1},\ldots,b_{m}}$ where
$a_{m},b_{m}$ are not both zero. Let $i$ be the largest index such that
$a_{i}\neq b_{i}$. Such an index $i$ exists because we assume that
$\Lambda\neq\Lambda^{\rm t}$. Now we consider the following cases:

  * •

Suppose that $i=m=1$. We know that $\Lambda\neq\binom{1}{0},\binom{0}{1}$
because $\binom{1}{0},\binom{0}{1}\in{\mathcal{S}}_{{\rm O}^{+}_{2}}$ which
contradicts to the assumption $n\geq 2$.

    * –

If either $a_{1}\geq 2$ and $b_{1}=0$; or $b_{1}>a_{1}\geq 1$, let
$\Lambda_{1}=\binom{a_{1}-1}{b_{1}}$.

    * –

If either $a_{1}=0$ and $b_{1}\geq 2$; or $a_{1}>b_{1}\geq 1$, let
$\Lambda_{1}=\binom{a_{1}}{b_{1}-1}$.

  * •

Suppose that $i=m>1$.

    * –

If $a_{m}>b_{m}$ and $b_{m-1}\geq a_{m-1}$, then we have $b_{m-1}>b_{m}+1$ and
let
$\Lambda_{1}=\binom{a_{1},\ldots,a_{m}}{b_{1},\ldots,b_{m-2},b_{m-1}-1,b_{m}}$.

    * –

If $a_{m}>b_{m}$ and $a_{m-1}>b_{m-1}$, let
$\Lambda_{1}=\binom{a_{1},\ldots,a_{m-1},a_{m}-1}{b_{1},\ldots,b_{m}}$.

    * –

If $b_{m}>a_{m}$ and $a_{m-1}\geq b_{m-1}$, let
$\Lambda_{1}=\binom{a_{1},\ldots,a_{m-2},a_{m-1}-1,a_{m}}{b_{1},\ldots,b_{m}}$.

    * –

If $b_{m}>a_{m}$ and $b_{m-1}>a_{m-1}$, let
$\Lambda_{1}=\binom{a_{1},\ldots,a_{m}}{b_{1},\ldots,b_{m-1},b_{m}-1}$.

  * •

Suppose that $i<m$.

    * –

If $a_{i}>b_{i}$, then $a_{i}>b_{i}>b_{i+1}=a_{i+1}$ and let
$\Lambda_{1}=\binom{a_{1},\ldots,a_{i-1},a_{i}-1,a_{i+1},\ldots,a_{m}}{b_{1},\ldots,b_{m}}$.

    * –

If $b_{i}>a_{i}$, then $b_{i}>a_{i}>a_{i+1}=b_{i+1}$ and let
$\Lambda_{1}=\binom{a_{1},\ldots,a_{m}}{b_{1},\ldots,b_{i-1},b_{i}-1,b_{i+1},\ldots,b_{m}}$.

For all cases, it is not difficult to check $\Lambda_{1}\in{\mathcal{S}}_{{\rm
O}^{+}_{2(n-1)}}$, $\Lambda\in\Omega(\Lambda_{1})$ and $\Lambda^{\rm
t}\not\in\Omega(\Lambda_{1})$. ∎

######  Remark 3.19.

It is obvious that the statement in above lemma is not true without the
assumption $n\geq 2$. Note that $\Lambda=\binom{1}{0}\in{\mathcal{S}}_{{\rm
O}^{+}_{2}}$, $\Lambda\neq\Lambda^{\rm t}$. However, ${\mathcal{S}}_{{\rm
O}^{+}_{0}}=\bigl{\\{}\binom{-}{-}\bigr{\\}}$ and
$\Omega(\binom{-}{-})=\bigl{\\{}\binom{1}{0},\binom{0}{1}\bigr{\\}}$.

###  3.4. Cells in a family of unipotent characters

In this subsection, we recall some result on _cells_ by Lusztig (cf. [Lus81],
[Lus82]). Some details and examples can be found in [Pan19a] §4. Let $Z$ be a
special symbol of rank $n$. An _arrangement_ $\Phi$ of $Z$ is defined as
follows:

  * •

if ${\rm def}(Z)=1$, $\Phi$ is a partition of $Z_{\rm I}$ into $\deg(Z)$ pairs
and one isolated element such that each pair contains one element in $(Z_{\rm
I})^{*}$ and one element in $(Z_{\rm I})_{*}$;

  * •

if ${\rm def}(Z)=0$, $\Phi$ is a partition of $Z_{\rm I}$ into $\deg(Z)$ pairs
such that each pair contains one element in $(Z_{\rm I})^{*}$ and one element
in $(Z_{\rm I})_{*}$.

A subset of pairs $\Psi$ in an arrangement $\Phi$ is denoted by
$\Psi\leq\Phi$. For such an arrangement $\Phi$ and a subset of pairs $\Psi$,
we define a subset $C_{\Phi,\Psi}^{\text{\bf G}}=C_{\Phi,\Psi}$ of
${\mathcal{S}}_{\text{\bf G}}$ as follows:

  * •

if ${\rm def}(Z)=1$, we define

$C_{\Phi,\Psi}=\\{\,\Lambda_{M}\mid M\subset Z_{\rm I},\ |M|\text{ even},\
|M\cap\Psi^{\prime}|\equiv|(\Phi\smallsetminus\Psi)\cap\Psi^{\prime*}|\pmod{2}\text{
for all }\Psi^{\prime}\leq\Phi\,\\};$

  * •

if ${\rm def}(Z)=0$, we define

$C_{\Phi,\Psi}=\\{\,\Lambda_{M}\mid M\subset Z_{\rm I},\
|M\cap\Psi^{\prime}|\equiv|(\Phi\smallsetminus\Psi)\cap\Psi^{\prime*}|\pmod{2}\text{
for all }\Psi^{\prime}\leq\Phi\,\\}$

It is not difficult to see that for ,

$C_{\Phi,\Psi}\subset\begin{cases}{\mathcal{S}}_{Z}^{{\rm Sp}_{2n}},&\text{if
${\rm def}(Z)=1$};\\\ {\mathcal{S}}_{Z}^{{\rm O}^{+}_{2n}},&\text{if ${\rm
def}(Z)=0$ and $\\#(\Phi\smallsetminus\Psi)$ even};\\\ {\mathcal{S}}_{Z}^{{\rm
O}^{-}_{2n}},&\text{if ${\rm def}(Z)=0$ and $\\#(\Phi\smallsetminus\Psi)$
odd}.\end{cases}$

Here $\\#(\Phi\smallsetminus\Psi)$ denotes the number of pairs in
$\Phi\smallsetminus\Psi$. For $\text{\bf G}={\rm O}^{\epsilon}_{2n}$, a
special symbol $Z$ of rank $n$ and defect $0$, and an arrangement $\Phi$ of
$Z$, a subset of pairs $\Psi$ is called _admissible_ if
$\\#(\Phi\smallsetminus\Psi)$ is even when $\epsilon=+$; and
$\\#(\Phi\smallsetminus\Psi)$ is odd when $\epsilon=-$.

The following lemmas are from [Pan19a] lemma 4.17, proposition 4.18, lemma
4.34, and proposition 4.35:

######  Lemma 3.20.

Let $\text{\bf G}={\rm Sp}_{2n}$, and let $\Lambda\mapsto\rho_{\Lambda}$ be a
Lusztig parametrization of unipotent characters. Let $Z$ be a special symbol
of rank $n$ and defect $1$, $\Phi$ an arrangement of $Z$, $\Psi$ a subset of
pairs of $\Phi$.

  1. (i)

The class function $\sum_{\Lambda\in C_{\Phi,\Psi}}\rho_{\Lambda}$ is uniform.

  2. (ii)

For any two distinct symbols
$\Lambda_{1},\Lambda_{2}\in{\mathcal{S}}_{Z}^{\text{\bf G}}$, there exists an
arrangement $\Phi$ of $Z$ with two subsets of pairs $\Psi_{1},\Psi_{2}$ such
that $\Lambda_{i}\in C_{\Phi,\Psi_{i}}$ for $i=1,2$ and $C_{\Phi,\Psi_{1}}\cap
C_{\Phi,\Psi_{2}}=\emptyset$.

######  Lemma 3.21.

Let $\text{\bf G}={\rm O}^{\epsilon}_{2n}$ where $\epsilon=+$ or $-$, and let
$\Lambda\mapsto\rho_{\Lambda}$ be a Lusztig parametrization of unipotent
characters. Let $Z$ be a special symbol of rank $n$ and defect $0$, $\Phi$ an
arrangement of $Z$, $\Psi$ an admissible subset of pairs of $\Phi$.

  1. (i)

The class function $\sum_{\Lambda\in C_{\Phi,\Psi}}\rho_{\Lambda}$ is uniform.

  2. (ii)

$\Lambda\in C_{\Phi,\Psi}$ if and only if $\Lambda^{\rm t}\in C_{\Phi,\Psi}$.

  3. (iii)

For any two symbols $\Lambda_{1},\Lambda_{2}\in{\mathcal{S}}_{Z}^{\text{\bf
G}}$ such that $\Lambda_{1}\neq\Lambda_{2},\Lambda_{2}^{\rm t}$, there exists
an arrangement $\Phi$ of $Z$ with subsets of pairs $\Psi_{1},\Psi_{2}$ such
that $\Lambda_{i}\in C_{\Phi,\Psi_{i}}$ for $i=1,2$ and $C_{\Phi,\Psi_{1}}\cap
C_{\Phi,\Psi_{2}}=\emptyset$.

##  4\. Uniqueness of the Lusztig Parametrizations

###  4.1. Unipotent characters of ${\rm Sp}_{2n}$

In this subsection, let $\text{\bf G}={\rm Sp}_{2n}$. The following lemma is
[Pan19b] proposition 3.3. We provide a proof for the sake of completion.

######  Lemma 4.1.

Suppose that $\text{\bf G}={\rm Sp}_{2n}$, and let
$\rho_{1},\rho_{2}\in{\mathcal{E}}(\text{\bf G},1)$. If
$\rho_{1}^{\sharp}=\rho_{2}^{\sharp}$, then $\rho_{1}=\rho_{2}$.

###### Proof.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ be a Lusztig parametrization
given in Proposition 3.12, and write
$\rho_{1}=\rho_{\Lambda_{1}},\rho_{2}=\rho_{\Lambda_{2}}$ for some
$\Lambda_{1},\Lambda_{2}\in{\mathcal{S}}_{\text{\bf G}}$. From (3.13), the
assumption $\rho_{\Lambda_{1}}^{\sharp}=\rho_{\Lambda_{2}}^{\sharp}$ means
that $\Lambda_{1},\Lambda_{2}\in{\mathcal{S}}_{Z}^{\text{\bf G}}$ for some
special symbol $Z$ of rank $n$ and defect $1$. For any arrangement $\Phi$ of
$Z$ and any subset of pairs $\Psi\leq\Phi$, we know that $\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda}$ is uniform by Lemma 3.20, and so

$\displaystyle\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{1}}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}=\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{1}}^{\sharp}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}=\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{2}}^{\sharp}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}=\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{2}}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}.$

Now if $\Lambda_{1}\neq\Lambda_{2}$, by Lemma 3.20 we can find an arrangement
$\Phi$ of $Z$ and a subset of pairs $\Psi\leq\Phi$ such that $\Lambda_{1}\in
C_{\Phi,\Psi}$ and $\Lambda_{2}\not\in C_{\Phi,\Psi}$, and hence

$\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{1}}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}=1\quad\text{and}\quad\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{2}}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}=0.$

So we must have $\Lambda_{1}=\Lambda_{2}$, i.e., $\rho_{1}=\rho_{2}$. ∎

######  Proposition 4.2.

Let G be ${\rm Sp}_{2n}$. Then there is a unique bijection
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ satisfying (3.13).

###### Proof.

Suppose that we have two parametrizations $\Lambda\mapsto\rho_{\Lambda}$ and
$\Lambda\mapsto\rho^{\prime}_{\Lambda}$ from ${\mathcal{S}}_{\text{\bf G}}$ to
${\mathcal{E}}(\text{\bf G},1)$ satisfying (3.13). From (3.16), we see that
condition (3.13) implies that
$(\rho_{\Lambda})^{\sharp}=(\rho^{\prime}_{\Lambda})^{\sharp}$. Then by Lemma
4.1, we conclude that $\rho_{\Lambda}=\rho^{\prime}_{\Lambda}$, i.e., two
parametrizations coincide. ∎

######  Corollary 4.3.

Let $\text{\bf G}_{n}={\rm Sp}_{2n}$. Then the bijection
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}_{n}}\rightarrow{\mathcal{E}}(\text{\bf G}_{n},1)$ given in Proposition 4.2
is compatible with the parabolic induction, i.e., the diagram analogous to
(3.3) commutes.

###### Proof.

The original construction of the bijection ${\mathcal{S}}_{\text{\bf
G}_{n}}\rightarrow{\mathcal{E}}(\text{\bf G}_{n},1)$ by Lusztig is compatible
with the parabolic induction (cf. [Lus77]). By Proposition 4.2, Lusztig’s
original construction is the only bijection satisfying (3.13) and hence the
corollary is obtained. ∎

For a non-negative integer $k$, we define the symbol

(4.4) $\Lambda_{k}^{\rm Sp}=\begin{cases}\binom{2k,2k-1,\ldots,0}{-},&\text{if
$k$ is even};\\\ \binom{-}{2k,2k-1,\ldots,0},&\text{if $k$ is
odd}.\end{cases}$

The following are easy to check:

  * •

${\rm rk}(\Lambda^{\rm Sp}_{k})=k(k+1)$ and ${\rm def}(\Lambda^{\rm
Sp}_{k})\equiv 1\pmod{4}$, i.e., $\Lambda_{k}^{\rm Sp}\in{\mathcal{S}}_{{\rm
Sp}_{2k(k+1)}}$,

  * •

if $\Lambda\in{\mathcal{S}}_{{\rm Sp}_{2n}}$ with $n<k(k+1)$, then $|{\rm
def}(\Lambda)|<|{\rm def}(\Lambda^{\rm Sp}_{k})|$,

  * •

if $\Lambda\in{\mathcal{S}}_{{\rm Sp}_{2k(k+1)}}$ and $\Lambda\neq\Lambda^{\rm
Sp}_{k}$, then $|{\rm def}(\Lambda)|<|{\rm def}(\Lambda^{\rm Sp}_{k})|$.

Because the defects are preserved by the parabolic induction, we have the
following corollary:

######  Corollary 4.5.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
Sp}_{2n}}\rightarrow{\mathcal{E}}({\rm Sp}_{2n},1)$ be the parametrization in
Proposition 4.2. Then the unique cuspidal unipotent character $\zeta_{k}^{\rm
Sp}$ of ${\rm Sp}_{2k(k+1)}(q)$ is parametrized by the symbol $\Lambda^{\rm
Sp}_{k}$, i.e., $\zeta^{\rm Sp}_{k}=\rho_{\Lambda^{\rm Sp}_{k}}$.

######  Lemma 4.6.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
Sp}_{2n}}\rightarrow{\mathcal{E}}({\rm Sp}_{2n},1)$ be the parametrization in
Proposition 4.2. Then

  1. (i)

${\bf 1}_{{\rm Sp}_{2n}}=\rho_{\binom{n}{-}}$,

  2. (ii)

${\rm St}_{{\rm Sp}_{2n}}=\rho_{\Lambda}$ where
$\Lambda=\binom{n,n-1,\ldots,1,0}{n,n-1,\ldots,1}$.

###### Proof.

From [Car85] corollary 7.6.5, we know that $R^{{\rm
Sp}_{2n}}_{\binom{n}{-}}={\bf 1}_{{\rm Sp}_{2n}}$. Because now $\binom{n}{-}$
is a special symbol of degree $0$, we have ${\mathcal{S}}_{\binom{n}{-}}^{{\rm
Sp}_{2n}}=\bigl{\\{}\binom{n}{-}\bigr{\\}}$, $\rho_{\binom{n}{-}}=R^{{\rm
Sp}_{2n}}_{\binom{n}{-}}$ by (3.13), and so (i) is proved.

Write $\Lambda=\binom{n,n-1,\ldots,1,0}{n,n-1,\ldots,1}$. From [Car85]
corollary 7.6.6, we see that $R^{{\rm Sp}_{2n}}_{\Lambda}={\rm St}_{{\rm
Sp}_{2n}}$. Again, now $\Lambda$ is a special symbol of degree $0$, we have
$\rho_{\Lambda}=R^{{\rm Sp}_{2n}}_{\Lambda}$, and (ii) is proved. ∎

######  Example 4.7.

Let $\text{\bf G}={\rm Sp}_{4}$. We know that

$\displaystyle{\mathcal{S}}_{{\rm Sp}_{4}}$
$\displaystyle=\bigl{\\{}\textstyle\binom{2}{-}\bigr{\\}}\cup\bigl{\\{}\binom{2,1,0}{2,1}\bigr{\\}}\cup\bigl{\\{}\binom{2,1}{0},\binom{2,0}{1},\binom{1,0}{2},\binom{-}{2,1,0}\bigr{\\}},$
$\displaystyle{\mathcal{S}}_{{\rm Sp}_{4}}^{\sharp}$
$\displaystyle=\bigl{\\{}\textstyle\binom{2}{-}\bigr{\\}}\cup\bigl{\\{}\binom{2,1,0}{2,1}\bigr{\\}}\cup\bigl{\\{}\binom{2,1}{0},\binom{2,0}{1},\binom{1,0}{2}\bigr{\\}}.$

The character values of irreducible characters in
${\mathcal{E}}(W_{2})=\left\\{\,\varphi_{\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}}\mid\genfrac{[}{]}{0.0pt}{}{\mu}{\nu}\in{\mathcal{P}}_{2}(2)\,\right\\}$
are given by the following table:

$(w)$ | $\\{1\\}$ | $\\{\sigma_{2},s_{1}\sigma_{2}s_{1}\\}$ | $\\{s_{1}\sigma_{2}s_{1}\sigma_{2}\\}$ | $\\{s_{1},\sigma_{2}s_{1}\sigma_{2}\\}$ | $\\{s_{1}\sigma_{2},\sigma_{2}s_{1}\\}$
---|---|---|---|---|---
$\varphi_{\genfrac{[}{]}{0.0pt}{}{2}{0}}$ | $1$ | $1$ | $1$ | $1$ | $1$
$\varphi_{\genfrac{[}{]}{0.0pt}{}{1,1}{0}}$ | $1$ | $1$ | $1$ | $-1$ | $-1$
$\varphi_{\genfrac{[}{]}{0.0pt}{}{1}{1}}$ | $2$ | $0$ | $-2$ | $0$ | $0$
$\varphi_{\genfrac{[}{]}{0.0pt}{}{0}{2}}$ | $1$ | $-1$ | $1$ | $1$ | $-1$
$\varphi_{\genfrac{[}{]}{0.0pt}{}{0}{1,1}}$ | $1$ | $-1$ | $1$ | $-1$ | $1$
$\text{\bf T}_{w}$ | $\text{\bf T}_{1}$ | $\text{\bf T}_{2}$ | $\text{\bf T}_{3}$ | $\text{\bf T}_{4}$ | $\text{\bf T}_{5}$

where $s_{1},\sigma_{2}$ are defined in Subsection 2.1. We can check that

$\displaystyle R_{\text{\bf T}_{1},1}$
$\displaystyle=2\theta_{9}+\theta_{11}+\theta_{12}+\theta_{13}+\theta_{0},$
$\displaystyle R_{\text{\bf T}_{2},1}$
$\displaystyle=\theta_{11}-\theta_{12}-\theta_{13}+\theta_{0},$ $\displaystyle
R_{\text{\bf T}_{3},1}$
$\displaystyle=-2\theta_{10}-\theta_{11}-\theta_{12}+\theta_{13}+\theta_{0},$
$\displaystyle R_{\text{\bf T}_{4},1}$
$\displaystyle=-\theta_{11}+\theta_{12}-\theta_{13}+\theta_{0},$
$\displaystyle R_{\text{\bf T}_{5},1}$
$\displaystyle=-\theta_{9}+\theta_{10}+\theta_{13}+\theta_{0}$

where $\theta_{i}$ is the notion from [Sri68]. Therefore by (3.5), we have

$\displaystyle R^{\text{\bf G}}_{\binom{2}{-}}$
$\displaystyle=\frac{1}{8}\left[R_{\text{\bf T}_{1},1}+2R_{\text{\bf
T}_{2},1}+R_{\text{\bf T}_{3},1}+2R_{\text{\bf T}_{4},1}+2R_{\text{\bf
T}_{5},1}\right]=\theta_{0},$ $\displaystyle R^{\text{\bf
G}}_{\binom{2,0}{1}}$ $\displaystyle=\frac{1}{4}\left[R_{\text{\bf
T}_{1},1}-R_{\text{\bf
T}_{3},1}\right]=\frac{1}{2}(\theta_{9}+\theta_{10}+\theta_{11}+\theta_{12}),$
$\displaystyle R^{\text{\bf G}}_{\binom{2,1}{0}}$
$\displaystyle=\frac{1}{8}\left[R_{\text{\bf T}_{1},1}+2R_{\text{\bf
T}_{2},1}+R_{\text{\bf T}_{3},1}-2R_{\text{\bf T}_{4},1}-2R_{\text{\bf
T}_{5},1}\right]=\frac{1}{2}(\theta_{9}-\theta_{10}+\theta_{11}-\theta_{12}),$
$\displaystyle R^{\text{\bf G}}_{\binom{1,0}{2}}$
$\displaystyle=\frac{1}{8}\left[R_{\text{\bf T}_{1},1}-2R_{\text{\bf
T}_{2},1}+R_{\text{\bf T}_{3},1}+2R_{\text{\bf T}_{4},1}-2R_{\text{\bf
T}_{5},1}\right]=\frac{1}{2}(\theta_{9}-\theta_{10}-\theta_{11}+\theta_{12}),$
$\displaystyle R^{\text{\bf G}}_{\binom{2,1,0}{2,1}}$
$\displaystyle=\frac{1}{8}\left[R_{\text{\bf T}_{1},1}-2R_{\text{\bf
T}_{2},1}+R_{\text{\bf T}_{3},1}-2R_{\text{\bf T}_{4},1}+2R_{\text{\bf
T}_{5},1}\right]=\theta_{13}.$

It is known that $\theta_{0}={\bf 1}_{{\rm Sp}_{4}}=\rho_{\binom{2}{-}}$ and
$\theta_{13}={\rm St}_{{\rm Sp}_{4}}=\rho_{\binom{2,1,0}{2,1}}$ by Lemma 4.6.
Let $Z=\binom{2,0}{1}$. The table for $(-1)^{\langle\Sigma,\Lambda\rangle}$
for $\Sigma\in{\mathcal{S}}_{Z,1}$ and $\Lambda\in{\mathcal{S}}_{Z}^{\text{\bf
G}}$ is

| $\binom{2,0}{1}$ | $\binom{2,1}{0}$ | $\binom{1,0}{2}$ | $\binom{-}{2,1,0}$
---|---|---|---|---
$\binom{2,0}{1}$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$
$\binom{2,1}{0}$ | $\phantom{-}1$ | $\phantom{-}1$ | $-1$ | $-1$
$\binom{1,0}{2}$ | $\phantom{-}1$ | $-1$ | $\phantom{-}1$ | $-1$

Then we have

$\displaystyle{\rho_{\binom{2,0}{1}}}^{\sharp}$
$\displaystyle=\frac{1}{2}\left[R_{\binom{2,0}{1}}+R_{\binom{2,1}{0}}+R_{\binom{1,0}{2}}\right],$
$\displaystyle{\rho_{\binom{2,1}{0}}}^{\sharp}$
$\displaystyle=\frac{1}{2}\left[R_{\binom{2,0}{1}}+R_{\binom{2,1}{0}}-R_{\binom{1,0}{2}}\right],$
$\displaystyle{\rho_{\binom{1,0}{2}}}^{\sharp}$
$\displaystyle=\frac{1}{2}\left[R_{\binom{2,0}{1}}-R_{\binom{2,1}{0}}+R_{\binom{1,0}{2}}\right],$
$\displaystyle{\rho_{\binom{-}{2,1,0}}}^{\sharp}$
$\displaystyle=\frac{1}{2}\left[R_{\binom{2,0}{1}}-R_{\binom{2,1}{0}}-R_{\binom{1,0}{2}}\right].$

We know that

$\displaystyle R^{{\rm Sp}_{4}}_{{\rm Sp}_{2}\times{\rm
GL}_{1}}(\rho_{\binom{1}{-}}\otimes{\bf 1})$ $\displaystyle={\bf 1}_{{\rm
Sp}_{4}}+\theta_{11}+\theta_{9},$ $\displaystyle R^{{\rm Sp}_{4}}_{{\rm
Sp}_{2}\times{\rm GL}_{1}}(\rho_{\binom{1,0}{1}}\otimes{\bf 1})$
$\displaystyle=\theta_{9}+\theta_{12}+{\rm St}_{{\rm Sp}_{4}},$
$\displaystyle\textstyle\Omega(\binom{1}{-})$
$\displaystyle=\textstyle\left\\{\binom{2}{-},\binom{2,1}{0},\binom{2,0}{1}\right\\},$
$\displaystyle\textstyle\Omega(\binom{1,0}{1})$
$\displaystyle=\textstyle\left\\{\binom{2,0}{1},\binom{1,0}{2},\binom{2,1,0}{2,1}\right\\}.$

Then by Corollary 4.3, we conclude that $\theta_{9}=\rho_{\binom{2,0}{1}}$,
$\theta_{10}=\rho_{\binom{-}{2,1,0}}$, $\theta_{11}=\rho_{\binom{2,1}{0}}$,
and $\theta_{12}=\rho_{\binom{1,0}{2}}$.

###  4.2. Unipotent characters of ${\rm SO}_{2n+1}$

######  Lemma 4.8.

Suppose that $\text{\bf G}={\rm SO}_{2n+1}$, and let
$\rho_{1},\rho_{2}\in{\mathcal{E}}(\text{\bf G},1)$. If
$\rho_{1}^{\sharp}=\rho_{2}^{\sharp}$, then $\rho_{1}=\rho_{2}$.

###### Proof.

The proof is similar to that of Lemma 4.1. ∎

######  Proposition 4.9.

Suppose that $\text{\bf G}={\rm SO}_{2n+1}$. Then there is a unique bijection
${\mathcal{S}}_{\text{\bf G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$
satisfying (3.13).

###### Proof.

The proof is similar to that of Proposition 4.2. ∎

######  Corollary 4.10.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
SO}_{2n+1}}\rightarrow{\mathcal{E}}({\rm SO}_{2n+1},1)$ be the parametrization
given in Proposition 4.9. Then

  * (i)

${\bf 1}_{{\rm SO}_{2n+1}}=\rho_{\binom{-}{n}}$;

  * (ii)

${\rm St}_{{\rm SO}_{2n+1}}=\rho_{\Lambda}$ where
$\Lambda=\binom{n,n-1,\ldots,1}{n,n-1,\ldots,1,0}$;

  * (iii)

if $n=k(k+1)$ for some non-negative integer $k$, then the unique cuspidal
unipotent character $\zeta^{{\rm SO}_{\rm odd}}_{k}$ of ${\rm SO}_{2n+1}(q)$
is parametrized by the symbol

$\Lambda^{{\rm SO}_{\rm
odd}}_{k}=\begin{cases}\binom{-}{2k,2k-1,\ldots,0},&\text{if $k$ is even};\\\
\binom{2k,2k-1,\ldots,0}{-},&\text{if $k$ is odd}.\end{cases}$

###### Proof.

For (i) and (ii), the proofs are analogous to that of Corollary 4.5, for (iii)
the proof is analogous to that of Lemma 4.6. ∎

###  4.3. Unipotent characters of ${\rm O}^{\epsilon}_{2n}$

From (3.4), we know that $R^{{\rm O}^{\epsilon}_{2n}}_{\text{\bf
T},1}\cdot{\rm sgn}_{{\rm O}^{\epsilon}_{2n}}=R^{{\rm
O}^{\epsilon}_{2n}}_{\text{\bf T},1}$ and then

$\langle\rho,R^{{\rm O}^{\epsilon}_{2n}}_{\text{\bf T},1}\rangle_{{\rm
O}^{\epsilon}_{2n}}=\langle\rho\cdot{\rm sgn}_{{\rm
O}^{\epsilon}_{2n}},R^{{\rm O}^{\epsilon}_{2n}}_{\text{\bf T},1}\rangle_{{\rm
O}^{\epsilon}_{2n}}$

for any $\rho\in{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$. Therefore,

(4.11) $\rho^{\sharp}=(\rho\cdot{\rm sgn}_{{\rm
O}^{\epsilon}_{2n}})^{\sharp},$

i.e., two irreducible characters $\rho,\rho\cdot{\rm sgn}_{{\rm
O}^{\epsilon}_{2n}}$ are not able to be distinguished by their uniform
projections.

The following lemma is [Pan19b] proposition 3.5. We also provide a proof here:

######  Lemma 4.12.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$ by
$\Lambda\mapsto\rho_{\Lambda}$ be a Lusztig parametrization of unipotent
characters. Then $(\rho_{\Lambda_{1}})^{\sharp}=(\rho_{\Lambda_{2}})^{\sharp}$
if and only if $\Lambda_{1}=\Lambda_{2}$ or $\Lambda_{1}=\Lambda_{2}^{\rm t}$.

###### Proof.

For $\Lambda\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{2n}}$ and
$\Sigma\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{2n}}^{\sharp}$, we have
$\langle\Lambda,\Sigma\rangle=\langle\Lambda^{\rm t},\Sigma\rangle$ by Lemma
2.14. Then we have $(\rho_{\Lambda^{\rm
t}})^{\sharp}=(\rho_{\Lambda})^{\sharp}$. Now if
$\Lambda_{1}\neq\Lambda_{2},\Lambda_{2}^{\rm t}$, by Lemma 3.21 we can find an
arrangement $\Phi$ of $Z$ and an admissible subset of pairs $\Psi\leq\Phi$
such that $\Lambda_{1}\in C_{\Phi,\Psi}$ and $\Lambda_{2}\not\in
C_{\Phi,\Psi}$, and hence

$\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{1}}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}=1\quad\text{and}\quad\Biggl{\langle}\sum_{\Lambda\in
C_{\Phi,\Psi}}\rho_{\Lambda},\rho_{\Lambda_{2}}\Biggr{\rangle}_{\\!\\!\text{\bf
G}}=0.$

As in the proof of Lemma 4.1, we get a contradiction. ∎

######  Corollary 4.13.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$ by
$\Lambda\mapsto\rho_{\Lambda}$ be a Lusztig parametrization of unipotent
characters. Then $\rho_{\Lambda^{\rm t}}=\rho_{\Lambda}\cdot{\rm sgn}_{{\rm
O}^{\epsilon}_{2n}}$.

###### Proof.

If $\Lambda$ is degenerate, then clearly $\rho_{\Lambda^{\rm
t}}=\rho_{\Lambda}=R^{{\rm O}^{\epsilon}_{2n}}_{\Lambda}=R^{{\rm
O}^{\epsilon}_{2n}}_{\Lambda}\cdot{\rm sgn}_{{\rm
O}^{\epsilon}_{2n}}=\rho_{\Lambda}\cdot{\rm sgn}_{{\rm O}^{\epsilon}_{2n}}$.
If $\Lambda$ is non-degenerate, from (4.11) we know that
$(\rho_{\Lambda}\cdot{\rm sgn}_{{\rm
O}^{\epsilon}_{2n}})^{\sharp}=(\rho_{\Lambda})^{\sharp}$, and by Lemma 4.12 we
conclude that $\rho_{\Lambda}\cdot{\rm sgn}_{{\rm
O}^{\epsilon}_{2n}}=\rho_{\Lambda^{\rm t}}$. ∎

######  Corollary 4.14.

Let $\rho_{1},\rho_{2}\in{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$. If
$\rho_{1}^{\sharp}=\rho_{2}^{\sharp}$, then either $\rho_{1}=\rho_{2}$ or
$\rho_{1}=\rho_{2}\cdot{\rm sgn}_{{\rm O}^{\epsilon}_{2n}}$.

######  Corollary 4.15.

Let $\text{\bf G}={\rm O}^{\epsilon}_{2n}$, and let
$\Lambda\mapsto\rho_{\Lambda}$ be a Lusztig parametrization of unipotent
characters. Then any bijective mapping ${\mathcal{S}}_{\text{\bf
G}}\mapsto{\mathcal{E}}(\text{\bf G},1)$ such that $\\{\Lambda,\Lambda^{\rm
t}\\}\rightarrow\\{\rho_{\Lambda},\rho_{\Lambda^{\rm t}}\\}$ is also a Lusztig
parametrization of unipotent characters.

###### Proof.

Suppose that ${\mathcal{S}}_{\text{\bf G}}\mapsto{\mathcal{E}}(\text{\bf
G},1)$ given by $\Lambda\mapsto\rho^{\prime}_{\Lambda}$ is a bijection such
that $\\{\Lambda,\Lambda^{\rm
t}\\}\rightarrow\\{\rho_{\Lambda},\rho_{\Lambda^{\rm t}}\\}$, i.e.,
$\\{\rho^{\prime}_{\Lambda},\rho^{\prime}_{\Lambda^{\rm
t}}\\}=\\{\rho_{\Lambda},\rho_{\Lambda^{\rm t}}\\}$. This implies that
$(\rho^{\prime}_{\Lambda})^{\sharp}=(\rho^{\prime}_{\Lambda^{\rm
t}})^{\sharp}=(\rho_{\Lambda})^{\sharp}=(\rho_{\Lambda^{\rm t}})^{\sharp}$ and
hence the mapping $\Lambda\mapsto\rho^{\prime}_{\Lambda}$ satisfies (3.13),
i.e., $\Lambda\mapsto\rho^{\prime}_{\Lambda}$ is also a Lusztig
parametrization of unipotent characters for ${\rm O}^{\epsilon}_{2n}$. ∎

######  Corollary 4.16.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$ be a
Lusztig parametrization of unipotent characters.

  1. (i)

If $\epsilon=+$, then

    1. (a)

${\mathcal{L}}_{1}\colon\left\\{\binom{n}{0},\binom{0}{n}\right\\}\rightarrow\\{{\bf
1}_{{\rm O}^{+}_{2n}},{\rm sgn}_{{\rm O}^{+}_{2n}}\\}$,

    2. (b)

${\mathcal{L}}_{1}$ maps
$\bigl{\\{}\binom{n,n-1,\ldots,1}{n-1,n-2,\ldots,0},\binom{n-1,n-2,\ldots,0}{n,n-1,\ldots,1}\bigr{\\}}$
to the two Steinberg characters of ${\rm O}^{+}_{2n}$.

  2. (ii)

If $\epsilon=-$, then

    1. (a)

${\mathcal{L}}_{1}\colon\bigl{\\{}\binom{-}{n,0},\binom{n,0}{-}\bigr{\\}}\rightarrow\\{{\bf
1}_{{\rm O}^{-}_{2n}},{\rm sgn}_{{\rm O}^{-}_{2n}}\\}$

    2. (b)

${\mathcal{L}}_{1}$ maps
$\bigl{\\{}\binom{n,n-1,\ldots,1,0}{n-1,n-2,\ldots,1},\binom{n-1,n-2,\ldots,1}{n,n-1,\ldots,1,0}\bigr{\\}}$
to the two Steinberg characters of ${\rm O}^{-}_{2n}$.

###### Proof.

First suppose that $\epsilon=+$. From [Car85] corollary 7.6.5, we know that
$R_{\binom{n}{0}}^{{\rm SO}_{2n}^{+}}={\bf 1}_{{\rm SO}^{+}_{2n}}$. Therefore,

$R_{\binom{n}{0}}^{{\rm O}_{2n}^{+}}=R_{\binom{0}{n}}^{{\rm
O}_{2n}^{+}}=\frac{1}{\sqrt{2}}({\bf 1}_{{\rm O}^{+}_{2n}}+{\rm sgn}_{{\rm
O}^{+}_{2n}})$

and (i.a) is proved from (3.13). Write
$\Lambda=\binom{n-1,n-2,\ldots,0}{n,n-1,\ldots,1}$. From [Car85] corollary
7.6.6, we see that $R_{\Lambda}^{{\rm SO}^{+}_{2n}}={\rm St}_{{\rm
SO}^{+}_{2n}}$. Then $\sqrt{2}R_{\Lambda}^{{\rm
O}^{+}_{2n}}=\sqrt{2}R_{\Lambda^{\rm t}}^{{\rm O}^{+}_{2n}}$ is the sum of two
Steinberg characters of ${\rm O}^{+}_{2n}$ and (i.b) is proved.

The proof of (ii) is similar. ∎

It is known that ${\rm O}^{\epsilon_{k}}_{2k^{2}}$ where
$\epsilon_{k}=(-1)^{k}$ has two cuspidal unipotent characters, denoted by
$\zeta_{k}^{\rm I}$ and $\zeta_{k}^{\rm II}$. Then from above we see that any
Lusztig parametrization ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon_{k}}_{2k^{2}}}\rightarrow{\mathcal{E}}({\rm
O}^{\epsilon_{k}}_{2k^{2}},1)$ maps

$\left\\{\textstyle\binom{2k-1,2k-2,\ldots,1,0}{-},\binom{-}{2k-1,2k-2,\ldots,1,0}\right\\}\rightarrow\\{\zeta_{k}^{\rm
I},\zeta_{k}^{\rm II}\\}$

bijectively.

######  Example 4.17.

We consider the unipotent characters of ${\rm O}^{+}_{4}$ or ${\rm
O}^{-}_{4}$. Let $\text{\bf T}_{i}$ be given as in Example 4.7. It is know
that $\text{\bf T}_{1},\text{\bf T}_{3},\text{\bf T}_{4}$ are maximal tori in
${\rm O}^{+}_{4}$, and $\text{\bf T}_{2},\text{\bf T}_{5}$ are maximal tori in
${\rm O}^{-}_{4}$. It is know that

$\displaystyle R^{{\rm O}^{+}_{4}}_{\text{\bf T}_{1},1}$ $\displaystyle={\bf
1}_{{\rm O}^{+}_{4}}+{\rm sgn}_{{\rm
O}^{+}_{4}}+2\chi^{+}_{2q}+\chi^{+}_{q^{2}}+\chi^{+}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{+}_{4}},$ $\displaystyle R^{{\rm O}^{+}_{4}}_{\text{\bf
T}_{3},1}$ $\displaystyle={\bf 1}_{{\rm O}^{+}_{4}}+{\rm sgn}_{{\rm
O}^{+}_{4}}-2\chi^{+}_{2q}+\chi^{+}_{q^{2}}+\chi^{+}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{+}_{4}},$ $\displaystyle R^{{\rm O}^{+}_{4}}_{\text{\bf
T}_{4},1}$ $\displaystyle={\bf 1}_{{\rm O}^{+}_{4}}+{\rm sgn}_{{\rm
O}^{+}_{4}}-\chi^{+}_{q^{2}}-\chi^{+}_{q^{2}}\cdot{\rm sgn}_{{\rm
O}^{+}_{4}},$ $\displaystyle R^{{\rm O}^{-}_{4}}_{\text{\bf T}_{2},1}$
$\displaystyle={\bf 1}_{{\rm O}^{-}_{4}}+{\rm sgn}_{{\rm
O}^{-}_{4}}+\chi^{-}_{q^{2}}+\chi^{-}_{q^{2}}\cdot{\rm sgn}_{{\rm
O}^{-}_{4}},$ $\displaystyle R^{{\rm O}^{-}_{4}}_{\text{\bf T}_{5},1}$
$\displaystyle={\bf 1}_{{\rm O}^{-}_{4}}+{\rm sgn}_{{\rm
O}^{-}_{4}}-\chi^{-}_{q^{2}}-\chi^{-}_{q^{2}}\cdot{\rm sgn}_{{\rm O}^{-}_{4}}$

where $\chi_{2q}^{+},\chi_{q^{2}}^{+}$ are irreducible characters of ${\rm
O}^{+}_{4}(q)$ of degrees $2q,q^{2}$ respectively; similarly
$\chi_{q^{2}}^{-}$ is an irreducible character of ${\rm O}^{-}_{4}(q)$ of
degree $q^{2}$. And so we have

$\displaystyle{\mathcal{E}}({\rm O}^{+}_{4},1)$ $\displaystyle=\\{{\bf
1}_{{\rm O}^{+}_{4}},{\rm sgn}_{{\rm
O}^{+}_{4}},\chi^{+}_{2q},\chi^{+}_{q^{2}},\chi^{+}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{+}_{4}}\\},$ $\displaystyle{\mathcal{E}}({\rm O}^{-}_{4},1)$
$\displaystyle=\\{{\bf 1}_{{\rm O}^{-}_{4}},{\rm sgn}_{{\rm
O}^{-}_{4}},\chi^{-}_{q^{2}},\chi^{-}_{q^{2}}\cdot{\rm sgn}_{{\rm
O}^{-}_{4}}\\}.$

We know that

$\textstyle{\mathcal{S}}_{{\rm
O}^{+}_{4}}=\left\\{\binom{2}{0},\binom{0}{2},\binom{2,1}{1,0},\binom{1,0}{2,1},\binom{1}{1}\right\\},\quad{\mathcal{S}}_{{\rm
O}^{-}_{4}}=\left\\{\binom{-}{2,0},\binom{2,0}{-},\binom{1}{2,1,0},\binom{2,1,0}{1}\right\\}.$

Suppose we choose ${\mathcal{S}}_{{\rm
O}^{+}_{4}}^{\sharp}=\left\\{\binom{2}{0},\binom{2,1}{1,0},\binom{1}{1}\right\\}$,
and ${\mathcal{S}}_{{\rm
O}^{-}_{4}}^{\sharp}=\left\\{\binom{0}{2},\binom{1,0}{2,1}\right\\}$. Then we
have

$\displaystyle R^{{\rm O}^{+}_{4}}_{\binom{2}{0}}$
$\displaystyle=\frac{1}{4\sqrt{2}}\left[R^{{\rm O}^{+}_{4}}_{\text{\bf
T}_{1},1}+R^{{\rm O}^{+}_{4}}_{\text{\bf T}_{3},1}+2R^{{\rm
O}^{+}_{4}}_{\text{\bf T}_{4},1}\right]=\frac{1}{\sqrt{2}}\left[{\bf 1}_{{\rm
O}^{+}_{4}}+{\rm sgn}_{{\rm O}^{+}_{4}}\right],$ $\displaystyle R^{{\rm
O}^{+}_{4}}_{\binom{1}{1}}$ $\displaystyle=\frac{1}{4}\left[R^{{\rm
O}^{+}_{4}}_{\text{\bf T}_{1},1}-R^{{\rm O}^{+}_{4}}_{\text{\bf
T}_{3},1}\right]=\chi^{+}_{2q},$ $\displaystyle R^{{\rm
O}^{+}_{4}}_{\binom{2,1}{1,0}}$
$\displaystyle=\frac{1}{4\sqrt{2}}\left[R^{{\rm O}^{+}_{4}}_{\text{\bf
T}_{1},1}+R^{{\rm O}^{+}_{4}}_{\text{\bf T}_{3},1}-2R^{{\rm
O}^{+}_{4}}_{\text{\bf
T}_{4},1}\right]=\frac{1}{\sqrt{2}}\left[\chi^{+}_{q^{2}}+\chi^{+}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{+}_{4}}\right],$ $\displaystyle R^{{\rm
O}^{-}_{4}}_{\binom{0}{2}}$ $\displaystyle=\frac{1}{2\sqrt{2}}\left[-R^{{\rm
O}^{-}_{4}}_{\text{\bf T}_{2},1}-R^{{\rm O}^{-}_{4}}_{\text{\bf
T}_{5},1}\right]=\frac{1}{\sqrt{2}}\left[-{\bf 1}_{{\rm O}^{-}_{4}}-{\rm
sgn}_{{\rm O}^{-}_{4}}\right],$ $\displaystyle R^{{\rm
O}^{-}_{4}}_{\binom{1,0}{2,1}}$
$\displaystyle=\frac{1}{2\sqrt{2}}\left[-R^{{\rm O}^{-}_{4}}_{\text{\bf
T}_{2},1}+R^{{\rm O}^{-}_{4}}_{\text{\bf
T}_{5},1}\right]=\frac{1}{\sqrt{2}}\left[-\chi^{-}_{q^{2}}-\chi^{-}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{-}_{4}}\right].$

From (2.13), we know that

$\displaystyle\textstyle\bigl{\langle}\binom{2}{0},\binom{2}{0}\bigr{\rangle}=\bigl{\langle}\binom{0}{2},\binom{2}{0}\bigr{\rangle}=\bigl{\langle}\binom{2,1}{1,0},\binom{2,1}{1,0}\bigr{\rangle}=\bigl{\langle}\binom{1,0}{2,1},\binom{2,1}{1,0}\bigr{\rangle}$
$\displaystyle\equiv 0\pmod{2},$
$\displaystyle\textstyle\bigl{\langle}\binom{-}{2,0},\binom{0}{2}\bigr{\rangle}=\bigl{\langle}\binom{2,0}{-},\binom{0}{2}\bigr{\rangle}=\bigl{\langle}\binom{1}{2,1,0},\binom{1,0}{2,1}\bigr{\rangle}=\bigl{\langle}\binom{2,1,0}{1},\binom{1,0}{2,1}\bigr{\rangle}$
$\displaystyle\equiv 1\pmod{2}.$

By (3.13), we have

$\displaystyle({\bf 1}_{{\rm O}^{+}_{4}})^{\sharp}=({\rm sgn}_{{\rm
O}^{+}_{4}})^{\sharp}$ $\displaystyle=\frac{1}{\sqrt{2}}R^{{\rm
O}^{+}_{4}}_{\binom{2}{0}}=\frac{1}{2}({\bf 1}_{{\rm O}^{+}_{4}}+{\rm
sgn}_{{\rm O}^{+}_{4}}),$
$\displaystyle(\chi^{+}_{q^{2}})^{\sharp}=(\chi^{+}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{+}_{4}})^{\sharp}$ $\displaystyle=\frac{1}{\sqrt{2}}R^{{\rm
O}^{+}_{4}}_{\binom{2,1}{1,0}}=\frac{1}{2}(\chi^{+}_{q^{2}}+\chi^{+}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{+}_{4}}),$ $\displaystyle\chi^{+}_{2q}$ $\displaystyle=R^{{\rm
O}^{+}_{4}}_{\binom{1}{1}},$ $\displaystyle({\bf 1}_{{\rm
O}^{-}_{4}})^{\sharp}=({\rm sgn}_{{\rm O}^{-}_{4}})^{\sharp}$
$\displaystyle=-\frac{1}{\sqrt{2}}R^{{\rm
O}^{-}_{4}}_{\binom{0}{2}}=\frac{1}{2}({\bf 1}_{{\rm O}^{-}_{4}}+{\rm
sgn}_{{\rm O}^{-}_{4}}),$
$\displaystyle(\chi^{-}_{q^{2}})^{\sharp}=(\chi^{-}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{-}_{4}})^{\sharp}$ $\displaystyle=-\frac{1}{\sqrt{2}}R^{{\rm
O}^{-}_{4}}_{\binom{1,0}{2,1}}=\frac{1}{2}(\chi^{-}_{q^{2}}+\chi^{-}_{q^{2}}\cdot{\rm
sgn}_{{\rm O}^{-}_{4}}).$

Therefore any bijection ${\mathcal{S}}_{{\rm
O}^{+}_{4}}\rightarrow{\mathcal{E}}({\rm O}^{+}_{4},1)$ such that

$\displaystyle\bigl{\\{}\textstyle\binom{2}{0},\binom{0}{2}\bigr{\\}}$
$\displaystyle\mapsto\\{{\bf 1}_{{\rm O}^{+}_{4}},{\rm sgn}_{{\rm
O}^{+}_{4}}\\},$ $\displaystyle\textstyle\binom{1}{1}$
$\displaystyle\mapsto\chi^{+}_{2q},$
$\displaystyle\bigl{\\{}\textstyle\binom{2,1}{1,0},\binom{1,0}{2,1}\bigr{\\}}$
$\displaystyle\mapsto\\{\chi^{+}_{q^{2}},\chi^{+}_{q^{2}}\cdot{\rm sgn}_{{\rm
O}^{+}_{4}}\\}$

is a Lusztig parametrization for ${\rm O}^{+}_{4}$, and any bijection
${\mathcal{S}}_{{\rm O}^{-}_{4}}\rightarrow{\mathcal{E}}({\rm O}^{-}_{4},1)$
such that

$\displaystyle\bigl{\\{}\textstyle\binom{-}{2,0},\binom{2,0}{-}\bigr{\\}}$
$\displaystyle\mapsto\\{{\bf 1}_{{\rm O}^{-}_{4}},{\rm sgn}_{{\rm
O}^{-}_{4}}\\},$
$\displaystyle\bigl{\\{}\textstyle\binom{1}{2,1,0},\binom{2,1,0}{1}\bigr{\\}}$
$\displaystyle\mapsto\\{\chi^{-}_{q^{2}},\chi^{-}_{q^{2}}\cdot{\rm sgn}_{{\rm
O}^{-}_{4}}\\}$

is a Lusztig parametrization for ${\rm O}^{-}_{4}$.

##  5\. Finite Theta Correspondence of Unipotent Characters

In this section we want to purpose several conditions to enforce the
parametrization ${\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ for $\text{\bf G}={\rm
O}^{\epsilon}_{2n}$ to be unique.

###  5.1. Finite theta correspondence on unipotent characters

For a nontrivial additive character $\psi$ of ${\mathbf{F}}_{q}$, let
$\omega^{\psi}_{{\rm Sp}(W)}$ denote the (character of the ) _Weil
representation_ of the finite symplectic group ${\rm Sp}(W)$ with respect to
$\psi$.

Let $(\text{\bf G},\text{\bf G}^{\prime})$ be a _reductive dual pair_ of the
form $({\rm Sp}_{2n},{\rm SO}_{2n^{\prime}+1})$ or $({\rm
O}^{\epsilon}_{2n},{\rm Sp}_{2n^{\prime}})$ where $\epsilon=+$ or $-$. The
restriction $\omega^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$ of the Weil
character to $G\times G^{\prime}$ gives a decomposition

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

where the multiplicity $m_{\rho,\rho^{\prime}}$ is $1$ or $0$. Then we have a
relation

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

between ${\mathcal{E}}(\text{\bf G})$ and ${\mathcal{E}}(\text{\bf
G}^{\prime})$ called the _finite theta correspondence_ (or _Howe duality_) for
the dual pair $(\text{\bf G},\text{\bf G}^{\prime})$. We say that an
irreducible character $\rho\in{\mathcal{E}}(\text{\bf G})$ _occurs in_
$\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$ if there exists
$\rho^{\prime}\in{\mathcal{E}}(\text{\bf G}^{\prime})$ such that
$(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$.

For a symplectic space $V$ over ${\mathbf{F}}_{q}$, we have the _symplectic
similitude group_

${\rm GSp}(V)=\\{\,g\in{\rm GL}(V)\mid\langle gv,gw\rangle=k_{g}\langle
v,w\rangle\text{ for some }k_{g}\in{\mathbf{F}}_{q}^{\times}\text{ and any
}v,w\in V\,\\}.$

Note that ${\rm GSp}(V)$ normalizes the symplectic group ${\rm Sp}(V)$. Choose
an element $h\in{\rm GSp}_{2n}(q)$ such that $k_{h}$ is a non-square element
in ${\mathbf{F}}_{q}^{\times}$. For $\rho\in{\mathcal{E}}({\rm Sp}_{2n})$, we
define the _conjugate character_ $\rho^{c}\in{\mathcal{E}}({\rm Sp}_{2n})$ by
$\rho^{c}(g)=\rho(hgh^{-1})$ for any $g\in{\rm Sp}_{2n}(q)$.

######  Lemma 5.2.

Suppose that $(\text{\bf G},\text{\bf G}^{\prime})=({\rm Sp}_{2n},{\rm
SO}_{2n^{\prime}+1})$. If $(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf
G},\text{\bf G}^{\prime}}$, then $(\rho^{c},\rho^{\prime})\in\Theta_{\text{\bf
G},\text{\bf G}^{\prime}}^{\psi_{a}}$ 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 ${\mathbf{F}}_{q}^{\times}$.

###### Proof.

Suppose that $G={\rm Sp}(V)$ and $G^{\prime}={\rm SO}(V^{\prime})$ for a
$2n$-dimensional symplectic space $V$ and a $(2n+1)$-dimensional orthogonal
space $V^{\prime}$, and write $\omega^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}$ as in (5.1). Choose $h\in{\rm GSp}(V)$ such that $k_{h}$ is a
non-square element in ${\mathbf{F}}_{q}^{\times}$, and let
$\widetilde{h}=\iota(h,1)\in{\rm GSp}(V\otimes V^{\prime})$ where

$\iota\colon{\rm GSp}(V)\times{\rm SO}(V^{\prime})\rightarrow{\rm
GSp}(V\otimes V^{\prime}).$

Now clearly $k_{\widetilde{h}}=k_{h}$, and then by [Sze98] proposition 11 we
have $\omega^{\psi}_{{\rm Sp}(V\otimes V^{\prime})}\circ{\rm
Ad}_{\widetilde{h}}=\omega^{\psi_{a}}_{{\rm Sp}(V\otimes V^{\prime})}$.
Therefore,

$\omega^{\psi_{a}}_{\text{\bf G},\text{\bf
G}^{\prime}}=\sum_{\rho\in{\mathcal{E}}(\text{\bf G}),\
\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}^{\prime})}m_{\rho,\rho^{\prime}}(\rho\circ{\rm
Ad}_{h})\otimes\rho^{\prime}=\sum_{\rho\in{\mathcal{E}}(\text{\bf G}),\
\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}^{\prime})}m_{\rho,\rho^{\prime}}\rho^{c}\otimes\rho^{\prime}.$

Thus the lemma is proved. ∎

For $\rho\in{\mathcal{E}}({\rm O}^{\epsilon}_{2n})$, we can define the
_conjugate character_ $\rho^{c}\in{\mathcal{E}}({\rm O}^{\epsilon}_{2n})$ as
we did for a symplectic group given above.

######  Lemma 5.3.

Suppose that $(\text{\bf G},\text{\bf G}^{\prime})=({\rm
O}^{\epsilon}_{2n},{\rm Sp}_{2n^{\prime}})$. If
$(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$,
then $(\rho^{c},\rho^{\prime c})\in\Theta_{\text{\bf G},\text{\bf
G}^{\prime}}^{\psi}$, and $(\rho^{c},\rho^{\prime}),(\rho,\rho^{\prime
c})\in\Theta_{\text{\bf G},\text{\bf G}^{\prime}}^{\psi_{a}}$ where $a$ is a
non-square element in ${\mathbf{F}}_{q}^{\times}$.

###### Proof.

Suppose that $G={\rm O}(V)$ and $G^{\prime}={\rm Sp}(V^{\prime})$ for a
$2n$-dimensional orthogonal space $V$ and a $2n^{\prime}$-dimensional
symplectic space $V^{\prime}$, and write $\omega^{\psi}_{\text{\bf
G},\text{\bf G}^{\prime}}$ as in (5.1). Choose $h\in{\rm GO}(V)$ (the
orthogonal similitude group) and $h^{\prime}\in{\rm GSp}(V^{\prime})$ such
that both $k_{h},k_{h^{\prime}}$ are non-square elements in
${\mathbf{F}}_{q}^{\times}$ and let $\widetilde{h}=\iota(h,h^{\prime})\in{\rm
GSp}(V\otimes V^{\prime})$ where

$\iota\colon{\rm GO}(V)\times{\rm GSp}(V^{\prime})\rightarrow{\rm
GSp}(V\otimes V^{\prime}).$

Now $k_{\widetilde{h}}=k_{h}k_{h^{\prime}}$ becomes a square element in
${\mathbf{F}}_{q}^{\times}$ and therefore

$\displaystyle\omega^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}=\omega^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}\circ{\rm
Ad}_{\widetilde{h}}$ $\displaystyle=\sum_{\rho\in{\mathcal{E}}(\text{\bf G}),\
\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}^{\prime})}m_{\rho,\rho^{\prime}}(\rho\circ{\rm
Ad}_{h})\otimes(\rho^{\prime}\circ{\rm Ad}_{h^{\prime}})$
$\displaystyle=\sum_{\rho\in{\mathcal{E}}(\text{\bf G}),\
\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}^{\prime})}m_{\rho,\rho^{\prime}}\rho^{c}\otimes\rho^{\prime c}.$

So we have shown that $(\rho,\rho^{\prime})\in\Theta^{\psi}_{\text{\bf
G},\text{\bf G}^{\prime}}$ implies that $(\rho^{c},\rho^{\prime
c})\in\Theta_{\text{\bf G},\text{\bf G}^{\prime}}^{\psi}$. The other
assertions can be proved by an analogous argument in the proof of Lemma 5.2. ∎

Let $\text{\bf G}^{\prime}_{n^{\prime}}$ denote ${\rm SO}_{2n^{\prime}+1}$,
${\rm Sp}_{2n^{\prime}}$, or ${\rm O}^{\epsilon}_{2n^{\prime}}$. For
$\rho\in{\mathcal{E}}(\text{\bf G})$, it is well known that if $\rho$ occurs
in $\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}_{n^{\prime}}}$, then it
also occurs in $\Theta^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}_{n^{\prime\prime}}}$ for any $n^{\prime\prime}\geq n^{\prime}$. We
say that $\rho$ _first occurs in_ $\Theta^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}_{n^{\prime}}}$ if it occurs in $\Theta^{\psi}_{\text{\bf
G},\text{\bf G}^{\prime}_{n^{\prime}}}$ and does not occur in
$\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}_{n^{\prime}-1}}$.

###  5.2. Finite theta correspondence on unipotent characters

If $(\text{\bf G},\text{\bf G}^{\prime})=({\rm O}^{\epsilon}_{2n},{\rm
Sp}_{2n^{\prime}})$, then the unipotent characters are preserved by
$\Theta_{\text{\bf G},\text{\bf G}^{\prime}}^{\psi}$ (cf. [AM93] theorem 3.5),
i.e., we can write

$\displaystyle\omega^{\psi}_{\text{\bf G},\text{\bf G}^{\prime},1}$
$\displaystyle=\sum_{\rho\in{\mathcal{E}}(\text{\bf G},1),\
\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}^{\prime},1)}m_{\rho,\rho^{\prime}}\rho\otimes\rho^{\prime}$
$\displaystyle\Theta^{\psi}_{\text{\bf G},\text{\bf G}^{\prime},1}$
$\displaystyle=\Theta^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime}}\cap({\mathcal{E}}(\text{\bf G},1)\times{\mathcal{E}}(\text{\bf
G}^{\prime},1))$

where $\omega^{\psi}_{\text{\bf G},\text{\bf G}^{\prime},1}$ denotes the
unipotent part of $\omega^{\psi}_{\text{\bf G},\text{\bf G}^{\prime}}$. For
the theta correspondence on unipotent characters, the following are well-known
(cf. [AM93] theorem 5.2):

  * •

${\bf 1}_{{\rm O}^{+}_{2}}$ first occurs in the correspondence for the pair
$({\rm O}^{+}_{2},{\rm Sp}_{0})$,

  * •

${\rm sgn}_{{\rm O}^{+}_{2}}$ first occurs in the correspondence for the pair
$({\rm O}^{+}_{2},{\rm Sp}_{2})$,

  * •

$\zeta_{k}^{\rm I}$ first occurs in the correspondence for the pair $({\rm
O}^{\epsilon_{k}}_{2k^{2}},{\rm Sp}_{2k(k-1)})$,

  * •

$\zeta_{k}^{\rm II}$ first occurs in the correspondence for the pair $({\rm
O}^{\epsilon_{k}}_{2k^{2}},{\rm Sp}_{2k(k+1)})$

where $\epsilon_{k}=(-1)^{k}$, and $\zeta_{k}^{\rm I},\zeta_{k}^{\rm II}$ are
the unipotent cuspidal characters of ${\rm O}^{\epsilon_{k}}_{2k^{2}}(q)$
given in Subsection 4.3.

Now we recall some results on $\Theta_{\text{\bf G},\text{\bf
G}^{\prime},1}^{\psi}$ from [Pan19a]. For any two partitions
$\lambda=[\lambda_{1},\lambda_{2},\ldots]$ (with
$\lambda_{1}\geq\lambda_{2}\geq\cdots$), $\mu=[\mu_{1},\mu_{2},\ldots]$ (with
$\mu_{1}\geq\mu_{2}\geq\cdots$), we define a relation

(5.4) $\lambda\preccurlyeq\mu\qquad\text{if
}\mu_{1}\geq\lambda_{1}\geq\mu_{2}\geq\lambda_{2}\geq\mu_{3}\geq\lambda_{3}\geq\cdots.$

And then we define a relation ${\mathcal{B}}_{\text{\bf G},\text{\bf
G}^{\prime}}$ between ${\mathcal{S}}_{\text{\bf G}}$ and
${\mathcal{S}}_{\text{\bf G}^{\prime}}$ for $(\text{\bf G},\text{\bf
G}^{\prime})=({\rm O}^{\epsilon}_{2n},{\rm Sp}_{2n^{\prime}})$ as follows:

  * •

If $\epsilon=+$, let ${\mathcal{B}}_{\text{\bf G},\text{\bf G}^{\prime}}$ be
the set consisting of pairs $(\Lambda,\Lambda^{\prime})$ of symbols such that

    * –

$\Upsilon(\Lambda)_{*}\preccurlyeq\Upsilon(\Lambda^{\prime})^{*}$ and
$\Upsilon(\Lambda^{\prime})_{*}\preccurlyeq\Upsilon(\Lambda)^{*}$,

    * –

${\rm def}(\Lambda^{\prime})=-{\rm def}(\Lambda)+1$;

  * •

if $\epsilon=-$, let ${\mathcal{B}}_{\text{\bf G},\text{\bf G}^{\prime}}$ be
the set consisting of pairs $(\Lambda,\Lambda^{\prime})$ of symbols such that

    * –

$\Upsilon(\Lambda)^{*}\preccurlyeq\Upsilon(\Lambda^{\prime})_{*}$ and
$\Upsilon(\Lambda^{\prime})^{*}\preccurlyeq\Upsilon(\Lambda)_{*}$,

    * –

${\rm def}(\Lambda^{\prime})=-{\rm def}(\Lambda)-1$.

We say that a symbol $\Lambda\in{\mathcal{S}}_{\text{\bf G}}$ _occurs in_
${\mathcal{B}}_{\text{\bf G},\text{\bf G}^{\prime}}$ if there is
$\Lambda^{\prime}\in{\mathcal{S}}_{\text{\bf G}^{\prime}}$ such that
$(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}_{\text{\bf G},\text{\bf
G}^{\prime}}$. For $\Lambda\in{\mathcal{S}}_{\text{\bf G}}$, it is not
difficult to see that if $\Lambda$ occurs in ${\mathcal{B}}_{\text{\bf
G},\text{\bf G}^{\prime}_{n^{\prime}}}$, then it also occurs in
${\mathcal{B}}_{\text{\bf G},\text{\bf G}^{\prime}_{n^{\prime\prime}}}$ for
any $n^{\prime\prime}\geq n^{\prime}$. We say that $\Lambda$ _first occurs in_
${\mathcal{B}}_{\text{\bf G},\text{\bf G}^{\prime}_{n^{\prime}}}$ if it occurs
in ${\mathcal{B}}_{\text{\bf G},\text{\bf G}^{\prime}_{n^{\prime}}}$ and does
not occur in ${\mathcal{B}}_{\text{\bf G},\text{\bf
G}^{\prime}_{n^{\prime}-1}}$.

######  Example 5.5.

  1. (1)

We have $\binom{1}{0},\binom{0}{1}\in{\mathcal{S}}_{{\rm O}^{+}_{2}}$,
$\binom{0}{-}\in{\mathcal{S}}_{{\rm Sp}_{0}}$, and
$\binom{1}{-}\in{\mathcal{S}}_{{\rm Sp}_{2}}$. Now
$\Upsilon(\binom{1}{0})=\genfrac{[}{]}{0.0pt}{}{1}{0}$,
$\Upsilon(\binom{0}{1})=\genfrac{[}{]}{0.0pt}{}{0}{1}$,
$\Upsilon(\binom{0}{-})=\genfrac{[}{]}{0.0pt}{}{0}{0}$, and
$\Upsilon(\binom{1}{-})=\genfrac{[}{]}{0.0pt}{}{1}{0}$, and so $\binom{1}{0}$
first occurs in ${\mathcal{B}}_{{\rm O}^{+}_{2},{\rm Sp}_{0}}$ and
$\binom{0}{1}$ first occurs in ${\mathcal{B}}_{{\rm O}^{+}_{2},{\rm Sp}_{2}}$.

  2. (2)

Suppose that $k$ is even, and let $\Lambda_{k}^{\rm
I}=\binom{2k-1,2k-2,\ldots,0}{-}$. Then $\Lambda_{k}^{\rm
I}\in{\mathcal{S}}_{{\rm O}^{+}_{2k^{2}}}$, ${\rm def}(\Lambda_{k}^{\rm
I})=2k$ and $\Upsilon(\Lambda_{k}^{\rm I})=\genfrac{[}{]}{0.0pt}{}{0}{0}$.
Therefore, if $\Lambda_{k}^{\rm I}$ occurs in ${\mathcal{B}}_{{\rm
O}^{+}_{2k^{2}},{\rm Sp}_{2n^{\prime}}}$, then ${\mathcal{S}}_{{\rm
Sp}_{2n^{\prime}}}$ contains a symbol of defect $-2k+1$. By (2.4), we have
$n^{\prime}\geq k(k-1)$, i.e., $\Lambda_{k}^{\rm I}$ first occurs in
${\mathcal{B}}_{{\rm O}^{+}_{2k^{2}},{\rm Sp}_{2(k^{2}-k)}}$.

  3. (3)

Suppose that $k$ is odd, and let $\Lambda_{k}^{\rm
I}=\binom{-}{2k-1,2k-2,\ldots,0}$. Then $\Lambda_{k}^{\rm
I}\in{\mathcal{S}}_{{\rm O}^{-}_{2k^{2}}}$, ${\rm def}(\Lambda_{k}^{\rm
I})=-2k$ and $\Upsilon(\Lambda_{k}^{\rm I})=\genfrac{[}{]}{0.0pt}{}{0}{0}$.
This means that if $\Lambda_{k}^{\rm I}$ occurs in ${\mathcal{B}}_{{\rm
O}^{-}_{2k^{2}},{\rm Sp}_{2n^{\prime}}}$, then ${\mathcal{S}}_{{\rm
Sp}_{2n^{\prime}}}$ contains a symbol of defect $2k-1$. By (2.4), we have
$n^{\prime}\geq k(k-1)$, i.e., $\Lambda_{k}^{\rm I}$ first occurs in
${\mathcal{B}}_{{\rm O}^{-}_{2k^{2}},{\rm Sp}_{2(k^{2}-k)}}$.

The following proposition is from [Pan19a] corollary 5.36:

######  Proposition 5.6.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm O}^{\epsilon}_{2n},{\rm
Sp}_{2n^{\prime}})$ where $\epsilon=+$ or $-$. Let
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ by $\Lambda\mapsto\rho_{\Lambda}$
and ${\mathcal{L}}^{\prime}_{1}\colon{\mathcal{S}}_{\text{\bf
G}^{\prime}}\rightarrow{\mathcal{E}}(\text{\bf G}^{\prime},1)$ by
$\Lambda^{\prime}\mapsto\rho_{\Lambda^{\prime}}$ be any Lusztig
parametrizations for $\text{\bf G},\text{\bf G}^{\prime}$ respectively. Then
$(\rho_{\Lambda},\rho_{\Lambda^{\prime}})$ or $(\rho_{\Lambda^{\rm
t}},\rho_{\Lambda^{\prime}})$ occurs in $\Theta_{\text{\bf G},\text{\bf
G}^{\prime},1}^{\psi}$ if and only if $(\Lambda,\Lambda^{\prime})$ or
$(\Lambda^{\rm t},\Lambda^{\prime})$ occurs in ${\mathcal{B}}_{\text{\bf
G},\text{\bf G}^{\prime}}$.

######  Remark 5.7.

Note that the parametrization ${\mathcal{L}}^{\prime}_{1}$ is unique by
Corollary 4.5 but ${\mathcal{L}}_{1}$ is not. So we want to enforce more
conditions on ${\mathcal{L}}_{1}$ so that ${\mathcal{L}}_{1}$ is unique and
eliminate the ambiguity in the above proposition.

###  5.3. On the uniqueness of Lusztig parametrization for even orthogonal
groups

Now we want to enforce extra conditions on the Lusztig parametrization
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$ to
make it be uniquely determined.

  1. (I)

We require that ${\mathcal{L}}_{1}$ by $\Lambda\mapsto\rho_{\Lambda}$ is
compatible with the parabolic induction on unipotent characters, i.e., we
require that
$\Omega(\rho_{\Lambda})=\\{\,\rho_{\Lambda^{\prime}}\mid\Lambda^{\prime}\in\Omega(\Lambda)\,\\}$
where $\Omega(\rho_{\Lambda})$ and $\Omega(\Lambda)$ are defined as in
Subsection 3.3.

  2. (II)

We require that

     * •

for $k\geq 1$, ${\mathcal{L}}_{1}(\Lambda_{k}^{\rm I})=\zeta_{k}^{\rm I}$ and
${\mathcal{L}}_{1}(\Lambda_{k}^{\rm II})=\zeta_{k}^{\rm II}$, i.e.,
$\zeta_{k}^{\rm I}=\rho_{\Lambda_{k}^{\rm I}}$ and $\zeta_{k}^{\rm
II}=\rho_{\Lambda_{k}^{\rm II}}$ where

(5.8) $\displaystyle\begin{split}\Lambda_{k}^{\rm
I}&=\begin{cases}\binom{2k-1,2k-2,\ldots,1,0}{-},&\text{if $k$ is even};\\\
\binom{-}{2k-1,2k-2,\ldots,1,0},&\text{if $k$ is odd},\end{cases}\\\
\Lambda_{k}^{\rm II}&=(\Lambda_{k}^{\rm I})^{\rm t}\end{split}$

and $\xi_{k}^{\rm I},\xi_{k}^{\rm II}$ the two cuspidal unipotent characters
of ${\rm O}^{\epsilon_{k}}_{2k^{2}}$ given in the previous subsection and
$\epsilon_{k}=(-1)^{k}$;

     * •

${\mathcal{L}}_{1}(\binom{1}{0})={\bf 1}_{{\rm O}^{+}_{2}}$ and
${\mathcal{L}}_{1}(\binom{0}{1})={\rm sgn}_{{\rm O}^{+}_{2}}$, i.e.,
$\rho_{\binom{1}{0}}={\bf 1}_{{\rm O}^{+}_{2}}$ and $\rho_{\binom{0}{1}}={\rm
sgn}_{{\rm O}^{+}_{2}}$.

Note that in addition to the specification of ${\mathcal{L}}_{1}$ on cuspidal
symbols, due to Remark 3.19 we also need to assign the image of
${\mathcal{L}}_{1}$ at $\binom{1}{0}$ or $\binom{0}{1}$.

By Corollary 4.15, a Lusztig parametrization
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$
satisfying both (I) and (II) clearly exists.

######  Proposition 5.9.

There is a unique bijective parametrization ${\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$ where
$\epsilon=+$ or $-$ satisfying (3.13), and (I), (II) above.

###### Proof.

The existence of such a bijection ${\mathcal{L}}_{1}$ is obvious, so now we
consider the uniqueness. Let
${\mathcal{L}}_{1},{\mathcal{L}}^{\prime}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$ be two
Lusztig parametrizations of unipotent characters for ${\rm
O}^{\epsilon}_{2n}$. Moreover, suppose that
${\mathcal{L}}_{1},{\mathcal{L}}^{\prime}_{1}$ both satisfy (I) and (II)
above. For $\Lambda\in{\mathcal{S}}_{{\rm O}^{\epsilon}_{2n}}$, we know that
${\mathcal{L}}_{1}(\Lambda)^{\sharp}={\mathcal{L}}_{1}^{\prime}(\Lambda)^{\sharp}$
by Lemma 3.15, and hence by Corollary 4.14 either

${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}_{1}^{\prime}(\Lambda)\quad\text{ or
}\quad{\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}_{1}^{\prime}(\Lambda)\cdot{\rm
sgn}_{{\rm O}^{\epsilon}_{2n}}={\mathcal{L}}^{\prime}_{1}(\Lambda^{\rm t}),$

i.e., if
${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}^{\prime}_{1}(\Lambda^{\prime})$,
then either $\Lambda^{\prime}=\Lambda$ or $\Lambda^{\prime}=\Lambda^{\rm t}$.
Now we suppose that
${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}_{1}^{\prime}(\Lambda^{\prime})$ and
consider the following three cases:

  1. (1)

Suppose that $\Lambda$ is degenerate, i.e., $\Lambda=\Lambda^{\rm t}$. Then
${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}_{1}^{\prime}(\Lambda^{\prime})$
implies that ${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}_{1}^{\prime}(\Lambda)$
immediately.

  2. (2)

Suppose that ${\rm def}(\Lambda)\neq 0$. Suppose that the unipotent character
${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}^{\prime}_{1}(\Lambda^{\prime})$
where $\Lambda^{\prime}=\Lambda$ or $\Lambda^{\rm t}$ is in the Harish-Chandra
series initiated by some unipotent cuspidal character $\zeta$. Because ${\rm
def}(\Lambda)\neq 0$, we have $\zeta\neq\zeta\cdot{\rm sgn}$. By the
requirement in (II), we have
${\mathcal{L}}_{1}(\Lambda_{0})=\zeta={\mathcal{L}}^{\prime}_{1}(\Lambda_{0})$
for some cuspidal symbol $\Lambda_{0}$ such that ${\rm def}(\Lambda_{0})\neq
0$. By (I), we must have ${\rm def}(\Lambda)={\rm def}(\Lambda^{\prime})$, and
then we conclude that $\Lambda^{\prime}=\Lambda$, i.e.,
${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}^{\prime}_{1}(\Lambda)$.

  3. (3)

Suppose that $\Lambda$ is non-degenerate and ${\rm def}(\Lambda)=0$, i.e,
$\Lambda\in{\mathcal{S}}_{{\rm O}^{+}_{2n}}$ for some $n$. Now we are going to
prove this case by induction on $n$. For $n=1$, the equality
${\mathcal{L}}_{1}(\Lambda)={\mathcal{L}}^{\prime}_{1}(\Lambda)$ is enforced
by (II) above. Now suppose that $n\geq 2$. Because now $\Lambda^{\rm
t}\neq\Lambda$, by Lemma 3.18, there exists $\Lambda_{1}\in{\mathcal{S}}_{{\rm
O}^{+}_{2(n-1)}}$ such that $\Lambda\in\Omega(\Lambda_{1})$ and $\Lambda^{\rm
t}\not\in\Omega(\Lambda_{1})$. By (I) and the induction hypothesis, we have

${\mathcal{L}}_{1}(\Lambda)\in\Omega({\mathcal{L}}_{1}(\Lambda_{1}))=\Omega({\mathcal{L}}_{1}^{\prime}(\Lambda_{1}))\not\ni{\mathcal{L}}^{\prime}_{1}(\Lambda^{\rm
t}).$

Now ${\mathcal{L}}^{\prime}_{1}(\Lambda^{\rm
t})\neq{\mathcal{L}}_{1}(\Lambda)$ implies that
${\mathcal{L}}^{\prime}_{1}(\Lambda)={\mathcal{L}}_{1}(\Lambda)$.

Hence the proposition is proved. ∎

######  Corollary 5.10.

Let $\text{\bf G}={\rm O}^{\epsilon}_{2n}$, and let
$\Lambda\mapsto\rho_{\Lambda}$ be the Lusztig parametrization in Proposition
5.9. Then

  1. (i)

${\bf 1}_{{\rm O}^{-}_{2n}}=\rho_{\binom{-}{n,0}}$ and ${\rm sgn}_{{\rm
O}^{-}_{2n}}=\rho_{\binom{n,0}{-}}$;

  2. (ii)

${\bf 1}_{{\rm O}^{+}_{2n}}=\rho_{\binom{n}{0}}$ and ${\rm sgn}_{{\rm
O}^{+}_{2n}}=\rho_{\binom{0}{n}}$

###### Proof.

Let $\Lambda\mapsto\rho_{\Lambda}$ be the parametrization for ${\rm
O}^{\epsilon}_{2n}$ satisfying (3.13) and (I), (II) above. We know that ${\bf
1}_{{\rm O}^{-}_{2}}$ (resp. ${\rm sgn}_{{\rm O}^{-}_{2}}$) first occurs in
the correspondence for the pair $({\rm O}^{-}_{2},{\rm Sp}_{0})$ (resp. $({\rm
O}^{-}_{2},{\rm Sp}_{4})$). Therefore by the requirement in (II) above, we
have ${\bf 1}_{{\rm O}^{-}_{2}}=\zeta^{\rm I}_{1}=\rho_{\binom{-}{1,0}}$
(resp. ${\rm sgn}_{{\rm O}^{-}_{2}}=\zeta^{\rm
II}_{1}=\rho_{\binom{1,0}{-}}$). Write ${\bf 1}_{{\rm
O}^{-}_{2n}}=\rho_{\Lambda}$ for some $\Lambda\in{\mathcal{S}}_{{\rm
O}^{-}_{2n}}$. By Corollary 4.16 we know that $\Lambda$ is either
$\binom{-}{n,0}$ or $\binom{n,0}{-}$. Because ${\bf 1}_{{\rm O}^{-}_{2n}}$ is
an irreducible constituent of $R^{{\rm O}^{-}_{2n}}_{{\rm O}^{-}_{2}\times{\rm
GL}_{1}^{n-1}}(\rho_{\binom{-}{1,0}}\otimes{\bf 1})$. By (I) above, we must
have ${\rm def}(\Lambda)={\rm def}(\binom{-}{1,0})=-2$, and so we conclude
that ${\bf 1}_{{\rm O}^{-}_{2n}}=\rho_{\binom{-}{n,0}}$ and hence ${\rm
sgn}_{{\rm O}^{-}_{2n}}=\rho_{\binom{n,0}{-}}$.

Now we are going to prove case (ii) by induction on $n$. By (II) above, we
have $\rho_{\binom{1}{0}}={\bf 1}_{{\rm O}^{+}_{2}}$ and
$\rho_{\binom{0}{1}}={\rm sgn}_{{\rm O}^{+}_{2}}$. Now by the induction
hypothesis, for $n\geq 2$, we assume that ${\bf 1}_{{\rm
O}^{+}_{2(n-1)}}=\rho_{\binom{n-1}{0}}$ and ${\rm sgn}_{{\rm
O}^{+}_{2(n-1)}}=\rho_{\binom{0}{n-1}}$. Suppose that ${\bf 1}_{{\rm
O}^{+}_{2n}}=\rho_{\Lambda}$ for some $\Lambda\in{\mathcal{S}}_{{\rm
O}^{+}_{2n}}$. Then we know that either $\Lambda=\binom{n}{0}$ or
$\Lambda=\binom{0}{n}$ by Corollary 4.16. Because ${\bf 1}_{{\rm
O}^{+}_{2n}}\in\Omega({\bf 1}_{{\rm O}^{+}_{2(n-1)}})$, by (II) we see that
$\Lambda\in\Omega(\binom{n-1}{0})$ and therefore $\Lambda$ must be
$\binom{n}{0}$, i.e., we conclude that ${\bf 1}_{{\rm
O}^{+}_{2n}}=\rho_{\binom{n}{0}}$ and ${\rm sgn}_{{\rm
O}^{+}_{2n}}=\rho_{\binom{0}{n}}$. ∎

######  Example 5.11.

Keep the notation in Example 4.17. Suppose that
$\rho_{1},\rho_{2}\in{\mathcal{E}}({\rm O}^{-}_{4},1)$ are the two irreducible
characters of degree $q^{2}$ satisfying

$R_{{\rm O}^{-}_{2}\times{\rm GL}_{1}}^{{\rm O}^{-}_{4}}({\bf 1}_{{\rm
O}^{-}_{2}}\otimes{\bf 1})={\bf 1}_{{\rm O}^{-}_{4}}+\rho_{1},\qquad R_{{\rm
O}^{-}_{2}\times{\rm GL}_{1}}^{{\rm O}^{-}_{4}}({\rm sgn}_{{\rm
O}^{-}_{2}}\otimes{\bf 1})={\rm sgn}_{{\rm O}^{-}_{4}}+\rho_{2}.$

Now (II) implies that ${\bf 1}_{{\rm O}^{-}_{2}}=\rho_{\binom{-}{1,0}}$ and
${\rm sgn}_{{\rm O}^{-}_{2}}=\rho_{\binom{1,0}{-}}$. Therefore the
parametrization ${\mathcal{S}}_{{\rm O}^{-}_{4}}\rightarrow{\mathcal{E}}({\rm
O}^{-}_{4},1)$ satisfying (I), (II) above must be

$\rho_{\binom{-}{2,0}}={\bf 1}_{{\rm
O}^{-}_{4}},\qquad\rho_{\binom{2,0}{-}}={\rm sgn}_{{\rm
O}^{-}_{4}},\qquad\rho_{\binom{1}{2,1,0}}=\rho_{1},\qquad\rho_{\binom{2,1,0}{1}}=\rho_{2}.$

The following proposition which justifies our choice of ${\mathcal{L}}_{1}$ in
Proposition 5.9 is from [Pan19a] theorem 1.8.

######  Proposition 5.12.

Let $(\text{\bf G},\text{\bf G}^{\prime})=({\rm
O}^{\epsilon}_{2n^{\prime}},{\rm Sp}_{2n})$ where $\epsilon=+$ or $-$. Let
${\mathcal{L}}_{1}\colon{\mathcal{S}}_{\text{\bf
G}}\rightarrow{\mathcal{E}}(\text{\bf G},1)$ and
${\mathcal{L}}^{\prime}_{1}\colon{\mathcal{S}}_{\text{\bf
G}^{\prime}}\rightarrow{\mathcal{E}}(\text{\bf G}^{\prime},1)$ be the unique
Lusztig parametrizations given in Proposition 5.9 and Proposition 4.2
respectively. Then the diagram

$\begin{CD}{\mathcal{S}}_{\text{\bf G}}@>{{\mathcal{B}}_{\text{\bf
G},\text{\bf G}^{\prime}}}>{}>{\mathcal{S}}_{\text{\bf G}^{\prime}}\\\
@V{{\mathcal{L}}_{1}}V{}V@V{}V{{\mathcal{L}}^{\prime}_{1}}V\\\
{\mathcal{E}}(\text{\bf G},1)@>{\Theta^{\psi}_{\text{\bf G},\text{\bf
G}^{\prime},1}}>{}>{\mathcal{E}}(\text{\bf G}^{\prime},1)\\\ \end{CD}$

commutes, i.e., $(\rho_{\Lambda},\rho_{\Lambda^{\prime}})$ occurs in
$\Theta_{\text{\bf G},\text{\bf G}^{\prime},1}^{\psi}$ if and only if
$(\Lambda,\Lambda^{\prime})\in{\mathcal{B}}_{\text{\bf G},\text{\bf
G}^{\prime}}$.

When both $\Lambda,\Lambda^{\prime}$ are cuspidal, the commutativity of the
above diagram can be seen by the requirement (II) (cf. Example 5.5). For
general $\Lambda,\Lambda^{\prime}$, the commutativity follows from the fact
that both the correspondence $\Theta_{\text{\bf G},\text{\bf
G}^{\prime}}^{\psi}$ and the parametrizations
${\mathcal{L}}_{1},{\mathcal{L}}^{\prime}_{1}$ are compatible with the
parabolic induction. Details of the proof can be found in [Pan19a].

######  Example 5.13.

Let ${\mathcal{L}}_{1}\colon{\mathcal{S}}_{{\rm
O}^{\epsilon}_{2n}}\rightarrow{\mathcal{E}}({\rm O}^{\epsilon}_{2n},1)$ by
$\Lambda\mapsto\rho_{\Lambda}$ be the parametrization in Proposition 5.9. Then
by Corollary 4.16 the two Steinberg characters of ${\rm O}^{\epsilon}_{2n}$
are parametrized by the symbols

$\begin{cases}\binom{n,n-1,\ldots,1}{n-1,n-2,\ldots,0},\binom{n-1,n-2,\ldots,0}{n,n-1,\ldots,1},&\text{if
$\epsilon=+$};\\\
\binom{n,n-1,\ldots,0}{n-1,n-2,\ldots,1},\binom{n-1,n-2,\ldots,1}{n,n-1,\ldots,0},&\text{if
$\epsilon=-$}.\end{cases}$

  1. (1)

If $\epsilon=+$, then one of the Steinberg characters of ${\rm O}^{+}_{2n}$
first occurs in the correspondence $\Theta^{\psi}_{{\rm O}^{+}_{2n},{\rm
Sp}_{2(n-1)},1}$ and is paired with ${\rm St}_{{\rm Sp}_{2(n-1)}}$, and this
Steinberg character is parametrized by the symbol
$\binom{n,n-1,\ldots,1}{n-1,n-2,\ldots,0}$.

  2. (2)

If $\epsilon=-$, then one of the Steinberg characters of ${\rm O}^{-}_{2n}$
first occurs in the correspondence $\Theta^{\psi}_{{\rm O}^{-}_{2n},{\rm
Sp}_{2(n-1)},1}$ and is paired with ${\rm St}_{{\rm Sp}_{2(n-1)}}$, and this
Steinberg character is parametrized by the symbol
$\binom{n-1,n-2,\ldots,1}{n,n-1,\ldots,0}$.

##  6\. Lusztig Correspondence and Finite Theta Correspondence

###  6.1. Lusztig correspondences

Let G be a classical group, and let $s$ be a semisimple element in the
connected component $(G^{*})^{0}$ of $G^{*}$. A rational maximal torus
$\text{\bf T}^{*}$ in $\text{\bf G}^{*}$ contains $s$ if and only if it is a
rational maximal torus in $C_{\text{\bf G}^{*}}(s)$. From [Car85] theorem
7.3.4, it is known that

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

for any rational maximal tori $\text{\bf T}^{*},\text{\bf T}^{\prime*}$ of
$\text{\bf G}^{*}$ containing $s$. Then the mapping

$\epsilon_{\text{\bf G}}R^{\text{\bf G}}_{\text{\bf
T}^{*},s}\mapsto\epsilon_{C_{\text{\bf G}^{*}}(s)}R_{\text{\bf
T}^{*},1}^{C_{\text{\bf G}^{*}}(s)}$

for any $\text{\bf T}^{*}$ containing $s$ can be extended uniquely to an
isometry from ${\mathcal{V}}(\text{\bf G},s)^{\sharp}$ onto
${\mathcal{V}}(C_{\text{\bf G}^{*}},1)^{\sharp}$. Now a Lusztig correspondence
${\mathfrak{L}}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)$, i.e., a bijective
mapping satisfying (1.3), can be extended linearly to be an isometry, still
denoted by ${\mathfrak{L}}_{s}$, of inner product spaces

(6.1) ${\mathfrak{L}}_{s}\colon{\mathcal{V}}(\text{\bf
G},s)\longrightarrow{\mathcal{V}}(C_{\text{\bf G}^{*}}(s),1)$

whose restriction to ${\mathcal{V}}(\text{\bf G},s)^{\sharp}$ is uniquely
determined.

Suppose that $\text{\bf G}=\prod_{i=1}^{k}\text{\bf G}_{k}$. Then $\text{\bf
G}^{*}=\prod_{i=1}^{k}\text{\bf G}^{*}_{i}$ where $\text{\bf G}_{i}^{*}$ is
the dual group of $\text{\bf G}_{i}$. If $s\in G^{*}$ semisimple, the we can
write $s=(s_{1},\ldots,s_{k})$ where each $s_{i}\in G^{*}_{i}$ is semisimple,
and then $C_{\text{\bf G}^{*}}(s)=\prod_{i=1}^{k}C_{\text{\bf
G}^{*}_{i}}(s_{i})$. Now a rational maximal torus $\text{\bf T}^{*}$
containing $s$ can be written as $\text{\bf T}^{*}=\prod_{i=1}^{k}\text{\bf
T}^{*}_{i}$ where $\text{\bf T}^{*}_{i}$ is a rational maximal torus in
$\text{\bf G}^{*}_{i}$. Therefore, we have $R^{\text{\bf G}}_{\text{\bf
T}^{*},s}=\bigotimes_{i=1}^{k}R^{\text{\bf G}_{i}}_{\text{\bf
T}^{*}_{i},s_{i}}$.

######  Corollary 6.2.

Then a bijection ${\mathfrak{L}}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)$ is a Lusztig
correspondence if and only if
${\mathfrak{L}}_{s}=\prod_{i=1}^{k}{\mathfrak{L}}_{s_{i}}$ where each
${\mathfrak{L}}_{s_{i}}\colon{\mathcal{E}}(\text{\bf
G}_{i},s_{i})\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}_{i}}(s_{i}),1)$ is a
Lusztig correspondence.

###### Proof.

This is obvious. ∎

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

(6.3) $\displaystyle\begin{split}\text{\bf G}^{(0)}=\text{\bf
G}^{(0)}(s)&=\prod_{\langle\lambda\rangle\subset\\{\lambda_{1},\ldots,\lambda_{n}\\},\
\lambda\neq\pm 1}\text{\bf G}_{[\lambda]}(s);\\\ \text{\bf G}^{(1)}=\text{\bf
G}^{(1)}(s)&=\text{\bf G}_{[-1]}(s);\\\ \text{\bf G}^{(2)}=\text{\bf
G}^{(2)}(s)&=\text{\bf G}_{[1]}(s)\end{split}$

where $\text{\bf G}_{[\lambda]}(s)$ is given in [AMR96] subsection 1.B (see
also [Pan19b] subsection 2.2). We know that

$C_{\text{\bf G}^{*}}(s)\simeq\text{\bf G}^{(0)}\times\text{\bf
G}^{(1)}\times\text{\bf G}^{(2)},$

and $\text{\bf G}^{(0)}$ is a product of general linear groups or unitary
group, and

(6.4) $(\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
SO}_{2n^{(2)}+1}),&\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 $n^{(1)}\geq 1$. The element $s$ can be
written as

(6.5) $s=s^{(0)}\times s^{(1)}\times s^{(2)}$

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 $-1$. In particular, $s^{(j)}$ is in the center of $\text{\bf
G}^{(j)}$. Then a Lusztig correspondence

(6.6) ${\mathfrak{L}}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(\text{\bf G}^{(0)}\times\text{\bf
G}^{(1)}\times\text{\bf G}^{(2)},1)$

can be written as

(6.7) ${\mathfrak{L}}_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$

where $\rho^{(j)}\in{\mathcal{E}}(\text{\bf G}^{(j)},1)$ for $j=0,1,2$. It is
known that $\rho$ is cuspidal if and only if $\text{\bf G}^{(0)}$ is a product
of unitary groups (i.e., no general linear groups) and each $\rho^{(j)}$ is
cuspidal.

Now ${\mathcal{S}}_{C_{\text{\bf G}^{*}}(s)}^{\sharp}={\mathcal{S}}_{\text{\bf
G}^{(0)}}^{\sharp}\times{\mathcal{S}}_{\text{\bf
G}^{(1)}}^{\sharp}\times{\mathcal{S}}_{\text{\bf G}^{(2)}}^{\sharp}$ and

$R_{\Sigma}^{C_{\text{\bf G}^{*}}(s)}=R^{\text{\bf G}^{(0)}}_{x}\otimes
R^{\text{\bf G}^{(1)}}_{\Sigma^{(1)}}\otimes R^{\text{\bf
G}^{(2)}}_{\Sigma^{(2)}}\in{\mathcal{V}}(C_{\text{\bf G}^{*}}(s),1)^{\sharp}$

for
$\Sigma=(x,\Sigma^{(1)},\Sigma^{(2)})\in{\mathcal{S}}^{\sharp}_{C_{\text{\bf
G}^{*}}(s)}$. We define

(6.8) $R_{\Sigma}^{\text{\bf
G}}={\mathfrak{L}}_{s}^{-1}(R_{\Sigma}^{C_{\text{\bf
G}^{*}}(s)})\in{\mathcal{V}}(\text{\bf G},s)^{\sharp}.$

Because $\\{\,R_{\Sigma}^{C_{\text{\bf
G}^{*}}(s)}\mid\Sigma\in{\mathcal{S}}^{\sharp}_{C_{\text{\bf G}^{*}}(s)}\,\\}$
is an orthonormal basis for ${\mathcal{V}}(C_{\text{\bf
G}^{*}}(s),1)^{\sharp}$ and

${\mathfrak{L}}_{s}\colon{\mathcal{V}}(\text{\bf
G},s)\rightarrow{\mathcal{V}}(C_{\text{\bf G}^{*}}(s),1)$

is an isometry which maps ${\mathcal{V}}(\text{\bf G},s)^{\sharp}$ onto
${\mathcal{V}}(C_{\text{\bf G}^{*}}(s),1)^{\sharp}$, we see that
$\\{\,R_{\Sigma}^{\text{\bf
G}}\mid\Sigma\in{\mathcal{S}}^{\sharp}_{C_{\text{\bf G}^{*}}(s)}\,\\}$ forms
an orthonormal basis for the space ${\mathcal{V}}(\text{\bf G},s)^{\sharp}$.
For $\rho\in{\mathcal{E}}(\text{\bf G},s)$, we have

(6.9)
$\displaystyle\begin{split}\rho^{\sharp}&=\sum_{\Sigma\in{\mathcal{S}}_{C_{\text{\bf
G}^{*}}(s)}^{\sharp}}\langle\rho,R_{\Sigma}^{\text{\bf G}}\rangle_{\text{\bf
G}}R_{\Sigma}^{\text{\bf G}},\\\
{\mathfrak{L}}_{s}(\rho)^{\sharp}&=\sum_{\Sigma\in{\mathcal{S}}_{C_{\text{\bf
G}^{*}}(s)}^{\sharp}}\langle{\mathfrak{L}}_{s}(\rho),R_{\Sigma}^{C_{\text{\bf
G}^{*}}(s)}\rangle_{C_{\text{\bf G}^{*}}(s)}R_{\Sigma}^{C_{\text{\bf
G}^{*}}(s)}.\end{split}$

Therefore we have
${\mathfrak{L}}_{s}(\rho^{\sharp})={\mathfrak{L}}_{s}(\rho)^{\sharp}$ for any
$\rho\in{\mathcal{E}}(\text{\bf G},s)$.

######  Lemma 6.10.

Let
${\mathfrak{L}}_{s},{\mathfrak{L}}^{\prime}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)$ be two Lusztig
correspondences, and write
${\mathfrak{L}}_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$,
${\mathfrak{L}}^{\prime}_{s}(\rho)=\rho^{\prime(0)}\otimes\rho^{\prime(1)}\otimes\rho^{\prime(2)}$.
Then

$\rho^{(0)}=\rho^{\prime(0)},\qquad(\rho^{(1)})^{\sharp}=(\rho^{\prime(1)})^{\sharp},\qquad(\rho^{(2)})^{\sharp}=(\rho^{\prime(2)})^{\sharp}.$

###### Proof.

Because ${\mathfrak{L}}_{s}(\rho^{\sharp})={\mathfrak{L}}_{s}(\rho)^{\sharp}$,
we have

${\mathfrak{L}}_{s}(\rho^{\sharp})=(\rho^{(0)})^{\sharp}\otimes(\rho^{(1)})^{\sharp}\otimes(\rho^{(2)})^{\sharp},\qquad{\mathfrak{L}}^{\prime}_{s}(\rho^{\sharp})=(\rho^{\prime(0)})^{\sharp}\otimes(\rho^{\prime(1)})^{\sharp}\otimes(\rho^{\prime(2)})^{\sharp}.$

Because the restrictions of ${\mathfrak{L}}_{s}$ and
${\mathfrak{L}}^{\prime}_{s}$ to ${\mathcal{V}}(\text{\bf G},s)^{\sharp}$ are
the same, i.e.,
${\mathfrak{L}}_{s}(\rho^{\sharp})={\mathfrak{L}}_{s}^{\prime}(\rho^{\sharp})$,
we have $(\rho^{(0)})^{\sharp}=(\rho^{\prime(0)})^{\sharp}$,
$(\rho^{(1)})^{\sharp}=(\rho^{\prime(1)})^{\sharp}$,
$(\rho^{(2)})^{\sharp}=(\rho^{\prime(2)})^{\sharp}$. Now $\text{\bf G}^{(0)}$
is a product of general linear groups or unitary groups, so we have
$\rho^{(0)}=(\rho^{(0)})^{\sharp}=(\rho^{\prime(0)})^{\sharp}=\rho^{\prime(0)}$.
∎

###  6.2. Lusztig correspondence and parabolic induction

Let $\text{\bf G}_{n}={\rm SO}_{2n+1}$, ${\rm Sp}_{2n}$ or ${\rm
O}^{\epsilon}_{2n}$ where $\epsilon=+$ or $-1$. The group $\text{\bf
G}_{n}\times{\rm GL}_{l}$ is the Levi factor of a parabolic subgroup of
$\text{\bf G}_{n+l}$. Let $\sigma$ be an irreducible cuspidal character of
${\rm GL}_{l}$, and so $\sigma\in{\mathcal{E}}({\rm GL}_{l},t)$ for some
semisimple element $t\in{\rm GL}_{l}(q)$. For $\rho\in{\mathcal{E}}(\text{\bf
G}_{n},s)$, an irreducible constituent of $R^{\text{\bf G}_{n+l}}_{\text{\bf
G}_{n}\times{\rm GL}_{l}}(\rho\otimes\sigma)$ is in ${\mathcal{E}}(\text{\bf
G}_{n+l},s^{\prime})$ where $s^{\prime}=(s,t)$ is regarded as an element in
$(G_{n+l}^{*})^{0}$. Then we define a relation
$\Omega_{t}\colon{\mathcal{E}}(\text{\bf
G}_{n},s)\rightarrow{\mathcal{E}}(\text{\bf G}_{n+l},s^{\prime})$ by

$\Omega_{t}(\rho)=\left\\{\,\rho^{\prime}\in{\mathcal{E}}(\text{\bf
G}_{n+l},s^{\prime})\mid\left\langle\rho^{\prime},R^{\text{\bf
G}_{n+l}}_{\text{\bf G}_{n}\times{\rm
GL}_{l}}(\rho\otimes\sigma)\right\rangle_{\\!\text{\bf G}_{n+l}}\neq
0\,\right\\}.$

Because we assume that $\sigma$ is cuspidal, one can see that $C_{\text{\bf
G}_{n}^{*}}(s)\times{\rm GL}_{1}^{\dagger}$ is a Levi subgroup of
$C_{\text{\bf G}_{n+l}^{*}}(s^{\prime})$ where ${\rm GL}_{1}^{\dagger}$
denotes the restriction to ${\mathbf{F}}_{q}$ of ${\rm GL}_{1}$ defined over a
finite extension (depending on $t$) of ${\mathbf{F}}_{q}$. Suppose that
$\rho^{\prime}\in\Omega_{t}(\rho)$, it is clear that $\rho^{\prime
c}\in\Omega_{t}(\rho^{c})$ for $\text{\bf G}_{n}={\rm Sp}_{2n},{\rm
O}^{\epsilon}_{2n}$ (cf. [Wal04] §4.4), and $\rho^{\prime}\cdot{\rm
sgn}_{\text{\bf G}_{n+l}}\in\Omega_{t}(\rho\cdot{\rm sgn}_{\text{\bf G}_{n}})$
for $\text{\bf G}_{n}={\rm O}^{\epsilon}_{2n}$.

We define the relation $\Omega\colon{\mathcal{E}}(C_{\text{\bf
G}_{n}^{*}}(s),1)\rightarrow{\mathcal{E}}(C_{\text{\bf
G}_{n+l}^{*}}(s^{\prime}),1)$ as in (3.1), i.e.,

$\Omega(\rho)=\Biggl{\\{}\,\rho^{\prime}\in{\mathcal{E}}(C_{\text{\bf
G}_{n+l}^{*}}(s^{\prime}),1)\mid\left\langle\rho^{\prime},R^{C_{\text{\bf
G}_{n+l}^{*}}(s^{\prime})}_{C_{\text{\bf G}_{n}^{*}}(s)\times{\rm
GL}_{1}^{\dagger}}(\rho\otimes{\bf 1})\right\rangle_{\\!C_{\text{\bf
G}_{n+l}^{*}}(s^{\prime})}\neq 0\,\Biggr{\\}}$

for $\rho\in{\mathcal{E}}(C_{\text{\bf G}_{n}^{*}}(s),1)$. Then we say that a
Lusztig correspondence

${\mathfrak{L}}_{s}\colon{\mathcal{E}}(\text{\bf
G},s)\rightarrow{\mathcal{E}}(C_{\text{\bf G}^{*}}(s),1)$

is _compatible with the parabolic induction_ if the following diagram

(6.11) $\begin{CD}{\mathcal{E}}(\text{\bf
G}_{n},s)@>{\Omega_{t}}>{}>{\mathcal{E}}(\text{\bf G}_{n+l},s^{\prime})\\\
@V{{\mathfrak{L}}_{s}}V{}V@V{}V{{\mathfrak{L}}_{s^{\prime}}}V\\\
{\mathcal{E}}(C_{\text{\bf
G}^{*}_{n}}(s),1)@>{\Omega}>{}>{\mathcal{E}}(C_{\text{\bf
G}_{n+l}^{*}}(s^{\prime}),1)\end{CD}$

commutes for any $s$ and $s^{\prime}=(s,t)$ given as above. Now write

$C_{\text{\bf G}^{*}_{n}}(s)=\text{\bf G}^{(0)}(s)\times\text{\bf
G}^{(1)}(s)\times\text{\bf G}^{(2)}(s),\qquad C_{\text{\bf
G}_{n+l}^{*}}(s^{\prime})=\text{\bf G}^{(0)}(s^{\prime})\times\text{\bf
G}^{(1)}(s^{\prime})\times\text{\bf G}^{(2)}(s^{\prime})$

and ${\mathfrak{L}}_{s}(\rho)=\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)}$ as
in (6.7). Then the diagram (6.11) can be described more precisely according to
the following three cases:

  1. (1)

If $t=1$ and so $l=1$, then $\text{\bf G}^{(0)}(s)=\text{\bf
G}^{(0)}(s^{\prime})$ and $\text{\bf G}^{(1)}(s)=\text{\bf
G}^{(1)}(s^{\prime})$, and then the relation $\Omega$ is given by

$\Omega(\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)})=\rho^{(0)}\otimes\rho^{(1)}\otimes\Omega(\rho^{(2)})$

where $\Omega(\rho^{(2)})$ is defined as in (3.1).

  2. (2)

If $t=-1$ and so $l=1$, then $\text{\bf G}^{(0)}(s)=\text{\bf
G}^{(0)}(s^{\prime})$ and $\text{\bf G}^{(2)}(s)=\text{\bf
G}^{(2)}(s^{\prime})$, and then

$\Omega(\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)})=\rho^{(0)}\otimes\Omega(\rho^{(1)})\otimes\rho^{(2)}$

where $\Omega(\rho^{(1)})$ is defined as in (3.1).

  3. (3)

If $t\neq\pm 1$, then $\text{\bf G}^{(1)}(s)=\text{\bf G}^{(1)}(s^{\prime})$
and $\text{\bf G}^{(2)}(s)=\text{\bf G}^{(2)}(s^{\prime})$, and then

$\Omega(\rho^{(0)}\otimes\rho^{(1)}\otimes\rho^{(2)})=\Omega(\rho^{(0)})\otimes\rho^{(1)}\otimes\rho^{(2)}$

where $\Omega(\rho^{(0)})$ is defined as in (3.1).

###  6.3. Modified Lusztig correspondence

For a semisimple element $s\in G^{*}$, we define

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

Combining ${\mathcal{L}}_{1}$ in Proposition 3.12 (for $\text{\bf
G}^{(0)}\times\text{\bf G}^{(-)}\times\text{\bf G}^{(+)}$) and the inverse of
${\mathfrak{L}}_{s}$ in (6.6), we obtain a bijection

(6.13)