¹¹institutetext: Dan Barbasch ²²institutetext: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853, USA ²²email: [email protected] ³³institutetext: Jia-Jun Ma ⁴⁴institutetext: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
Department of Mathematics, Xiamen University Malaysia campus, Sepang, Selangor Darul Ehsan, 43900, Malaysia ⁴⁴email: [email protected] ⁵⁵institutetext: Binyong Sun⁶⁶institutetext: Institute for Advanced Study in Mathematics & New Cornerstone Science Laboratory, Zhejiang University, Hangzhou, 310058, China ⁶⁶email: [email protected] ⁷⁷institutetext: Chen-Bo Zhu ⁸⁸institutetext: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 ⁸⁸email: [email protected]

Genuine special unipotent representations of spin groups

Dan Barbasch Jia-Jun Ma Binyong Sun and Chen-Bo Zhu

Abstract

We determine all genuine special unipotent representations of real spin groups and quaternionic spin groups, and show in particular that all of them are unitarizable. We also show that there are no genuine special unipotent representations of complex spin groups.

To Toshiyuki Kobayashi with friendship and admiration

1 Introduction and the main results

In two earlier works BMSZ1 ; BMSZ2 , the authors construct and classify special unipotent representations of real classical groups (in the sense of Arthur and Barbasch-Vogan; the terminology comes from Lu ). As a direct consequence of the construction and the classification, the authors show that all special unipotent representations of real classical groups are unitarizable, as predicted by the Arthur-Barbasch-Vogan conjecture ((ArUni, , Section 4), (ABV, , Introduction)). For quasi-split classical groups, the unitarity is independently established in AAM ; AM .

In this paper we first consider a real or quaternionic spin group $G$ and determine all genuine special unipotent representations of $G$ . (The analogous but simpler case of the complex spin group is briefly discussed at the end of this section.) In particular, we show that all of them are unitarizable. The paper may be considered as a companion paper of BMSZ1 ; BMSZ2 . The general strategy follows those of BMSZ1 ; BMSZ2 and it consists of two steps. First we use the coherent continuation representation to count the number of special unipotent representations (attached to a nilpotent $\check{G}$ -orbit $\check{\mathcal{O}}$ in $\check{\mathfrak{g}}$ ; notations to follow), as in BMSZ1 . Second we construct the exact number of special unipotent representations to match the counting and arrive at the classification as a result. This second task is made significantly easier in the current case of spin groups, due to the fact that there are only a very small number of genuine special unipotent representations. In the case of real classical groups, both tasks of the construction and distinguishing the constructed representations pose significant challenges, which we overcome in BMSZ2 by Howe’s method of theta lifting Howe79 ; Howe89 and Vogan’s theory of associated cycles Vo89 .

Let $G_{{\mathbb{C}}}=\mathrm{Spin}(m,{\mathbb{C}})$ ( $m\geq 2$ ) be the complex spin group. Its rank is $n:=\lfloor\frac{m}{2}\rfloor$ . Let $G$ be a real form of $G_{\mathbb{C}}$ , namely $G$ is the fixed point group of an involutive anti-holomorphic automorphism $\sigma$ of $G_{\mathbb{C}}$ . Up to conjugation by $G_{\mathbb{C}}$ , every real form of $G_{\mathbb{C}}$ is a real spin group or a quaternionic spin group. We may thus assume without loss of generality that $G$ is the real spin group $\mathrm{Spin}(p,q)$ ( $p+q=m$ ), or the quaternionic spin group $\mathrm{Spin}^{*}(2n)$ (when $m$ is even). We will be concerned with special unipotent representations of $G$ (in the sense of Arthur and Barbasch-Vogan; see BVUni ; ABV ).

It will be convenient to introduce a slightly larger group $\widetilde{G}$ . Consider the natural exact sequence

\{1\}\rightarrow\{\pm 1\}\rightarrow G_{\mathbb{C}}\rightarrow\mathrm{SO}(m,{\mathbb{C}})\rightarrow\{1\}.

The automorphism $\sigma$ descends to an involutive anti-holomorphic automorphism of $\mathrm{SO}(m,{\mathbb{C}})$ , to be denoted by $\sigma_{0}$ . Let $G_{0}$ denote the fixed point group of $\sigma_{0}$ , and write $\widetilde{G}$ for the preimage of $G_{0}$ under the covering homomorphism $G_{\mathbb{C}}\rightarrow\mathrm{SO}(m,{\mathbb{C}})$ . Then $G_{0}$ is a real special orthogonal group $\mathrm{SO}(p,q)$ ( $p+q=m$ ), or a quaternionic orthogonal group $\mathrm{SO}^{*}(2n)$ (when $m$ is even). We summarize with an exact sequence

\{1\}\rightarrow\{\pm 1\}\rightarrow\widetilde{G}\rightarrow G_{0}\rightarrow\{1\},

where $G_{0}=\mathrm{SO}(p,q)$ or $\mathrm{SO}^{*}(2n)$ .

If $G=\mathrm{Spin}(p,q)$ and $pq=0$ , or $G=\mathrm{Spin}^{*}(2n)$ , then $\widetilde{G}=G$ . Otherwise, $\widetilde{G}$ contains $G$ as a subgroup of index $2$ .

We first consider representations of $\widetilde{G}$ instead of $G$ , and we then relate the representation theory of $G$ to that of $\widetilde{G}$ by Clifford theory.

We will work in the category of Casselman-Wallach representations (Wa2, , Chapter 11). An irreducible Casselman-Wallach representation of $\widetilde{G}$ either descends to a representation of $G_{0}$ or is genuine in the sense that the central subgroup $\{\pm 1\}\subset\widetilde{G}$ (the kernel of the covering homomorphism $\widetilde{G}\rightarrow G_{0}$ ) acts via the non-trivial character. (The notion of “genuine” will be used in some similar situations without further explanation.) Since special unipotent representations of $G_{0}$ have been classified BMSZ1 ; BMSZ2 , we will focus on genuine special unipotent representations of $\widetilde{G}$ .

The Langlands dual of $G$ is

\check{G}:=\begin{cases}\mathrm{SO}(m,{\mathbb{C}})/\{\pm 1\},&\text{if $m$ is even};\\ \mathrm{Sp}(m-1,{\mathbb{C}})/\{\pm 1\},&\text{if $m$ is odd}.\end{cases}

(1)

Denote by $\check{\mathfrak{g}}$ the Lie algebra of $\check{G}$ .

Let $\check{\mathcal{O}}$ be a nilpotent $\check{G}$ -orbit in $\check{\mathfrak{g}}$ (see CM for basic facts about nilpotent orbits). Denote by $\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}(\widetilde{G})$ the set of isomorphism classes of genuine special unipotent representations of $\widetilde{G}$ attached to $\check{\mathcal{O}}$ . We refer the reader to (BMSZ1, , Section 2) for a comprehensive discussion of special unipotent representations in the case of real classical groups.

Denote by $\mathfrak{g}$ and $\mathfrak{g}_{\mathbb{C}}$ the Lie algebras of $G$ and $G_{\mathbb{C}}$ , respectively. Let ${\mathcal{O}}\subset\mathfrak{g}_{\mathbb{C}}^{*}$ denote the Barbasch-Vogan dual of $\check{\mathcal{O}}$ (see (BVUni, , Appendix) and (BMSZ0, , Section 1)).

Our first result will count the set ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ . As usual we will not distinguish between a nilpotent orbit $\check{\mathcal{O}}$ and its Young diagram (when there is no confusion). Recall that $\check{\mathcal{O}}$ is called very even if all its row lengths are even.

Theorem 1.1

(a) If some row length of $\check{\mathcal{O}}$ occurs with odd multiplicity, then the set ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ is empty.

(b) Suppose that all row lengths of $\check{\mathcal{O}}$ occur with even multiplicity. Then

			$\displaystyle\sharp({\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G}))$
		$\displaystyle=$	$\displaystyle\begin{cases}1,&\text{if $G=\mathrm{Spin}(p,q)$, $\left\|{p-q}\right\|=1$};\\ 1,&\text{if $G=\mathrm{Spin}(p,q)$, $p=q$ and $\check{\mathcal{O}}$ is very even};\\ 2,&\text{if $G=\mathrm{Spin}(p,q)$, $p=q$ and $\check{\mathcal{O}}$ is not very even};\\ \sharp(\widetilde{G}\backslash\sqrt{-1}\mathfrak{g}^{}\cap{\mathcal{O}}),&\text{if $G=\mathrm{Spin}^{}(2n)$ and $\check{\mathcal{O}}$ is very even};\\ 0,&\text{otherwise}.\end{cases}$

Here and henceforth, $\sharp$ indicates the cardinality of a finite set.

Note that when $m$ is even and $\check{\mathcal{O}}$ is very even, there are actually two nilpotent $\check{G}$ -orbits in $\check{\mathfrak{g}}$ that have the same Young diagram as $\check{\mathcal{O}}$ . The two orbits are given labels $I/II$ as $\check{\mathcal{O}}_{I}$ and $\check{\mathcal{O}}_{II}$ . Write ${\mathcal{O}}_{I}$ and ${\mathcal{O}}_{II}$ respectively for their Barbasch-Vogan duals. If $G=\mathrm{Spin}^{*}(2n)$ and $\check{\mathcal{O}}$ is very even, then among the sets $\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}_{I}$ and $\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}_{II}$ , one is empty and the other is not. We will use the convention for the labels so that

\sqrt{-1}\mathfrak{g}^{*}\cap\mathcal{O}_{I}\neq\emptyset

(2)

and we say ${\check{\mathcal{O}}}_{I}$ is $G$ -relevant. We refer the reader to (BMSZ1, , Section 2.6) for the notion of $G$ -relevance in a more general context.

Let $\widetilde{\mathrm{sgn}}$ denote the quadratic character on $\widetilde{G}$ whose kernel equals $G$ .

Theorem 1.2

Let $\pi\in{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ . If $G=\mathrm{Spin}(p,q)$ and $p=q$ is odd, then $\pi\otimes\widetilde{\mathrm{sgn}}$ is not isomorphic to $\pi$ . In all other cases, $\pi\otimes\widetilde{\mathrm{sgn}}$ is isomorphic to $\pi$ .

Similar to ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ , define the set ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)$ of isomorphism classes of genuine special unipotent representations of $G$ attached to $\check{\mathcal{O}}$ . By Clifford theory and Theorem 1.2, Theorem 1.1 implies the following counting result for $G$ .

Theorem 1.3

(a) If some row length of $\check{\mathcal{O}}$ occurs with odd multiplicity, then the set ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)$ is empty.

(b) Suppose that all row lengths of $\check{\mathcal{O}}$ occur with even multiplicity. Then

			$\displaystyle\sharp({\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G))$
		$\displaystyle=$	$\displaystyle\begin{cases}2,&\text{if $G=\mathrm{Spin}(p,q)$, $\left\|{p-q}\right\|=1$};\\ 2,&\text{if $G=\mathrm{Spin}(p,q)$, $p=q$ is even and $\check{\mathcal{O}}$ is very even};\\ 4,&\text{if $G=\mathrm{Spin}(p,q)$, $p=q$ is even and $\check{\mathcal{O}}$ is not very even};\\ 1,&\text{if $G=\mathrm{Spin}(p,q)$, $p=q$ is odd};\\ \sharp(G\backslash\sqrt{-1}\mathfrak{g}^{}\cap{\mathcal{O}}),&\text{if $G=\mathrm{Spin}^{}(2n)$ and $\check{\mathcal{O}}$ is very even};\\ 0,&\text{otherwise}.\end{cases}$

We aim to describe all representations in ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ and ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)$ . In the rest of this introduction, we assume that the set ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ is non-empty, or equivalently, the set ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)$ is non-empty. Theorem 1.1 in particular implies that all row lengths of $\check{\mathcal{O}}$ occur with even multiplicity. Write

r_{1}=r_{1}\geq r_{2}=r_{2}\geq\dots\geq r_{k}=r_{k}>0\quad(k\geq 1)

for the row lengths of $\check{\mathcal{O}}$ .

As usual, let $\mathbb{H}$ denote the division algebra of Hamilton’s quaternions. We omit the proof of the following elementary lemma.

Lemma 1

Up to conjugation by $G_{0}$ , there is a unique Levi subgroup $L_{0}$ of $G_{0}$ with the following properties:

(a)

L_{0}\cong\begin{cases}\mathrm{GL}_{r_{1}}(\mathbb{R})\times\mathrm{GL}_{r_{2}}(\mathbb{R})\times\dots\times\mathrm{GL}_{r_{k}}(\mathbb{R}),&\text{if $G_{0}=\mathrm{SO}(p,q)$};\\ \mathrm{GL}_{\frac{r_{1}}{2}}(\mathbb{H})\times\mathrm{GL}_{\frac{r_{2}}{2}}(\mathbb{H})\times\dots\times\mathrm{GL}_{\frac{r_{k}}{2}}(\mathbb{H}),&\text{if $G_{0}=\mathrm{SO}^{*}(2n)$};\end{cases}

(b)

the orbit ${\mathcal{O}}$ equals the induction of the zero orbit from $\mathfrak{l}_{\mathbb{C}}$ to $\mathfrak{g}_{\mathbb{C}}$ , where $\mathfrak{l}_{\mathbb{C}}$ denotes the complexified Lie algebra of $L_{0}$ .

We remark that the second condition in the above lemma is implied by the first one, unless $G_{0}=\mathrm{SO}(p,q)$ , $p=q$ , and $\check{\mathcal{O}}$ is very even.

Let $L_{0}$ be as in Lemma 1 and write $\widetilde{L}$ for the preimage of $L_{0}$ under the covering homomorphism $G_{\mathbb{C}}\rightarrow\mathrm{SO}(m,{\mathbb{C}})$ . Write $\mathrm{N}_{\widetilde{G}}(\widetilde{L})$ for the normalizer of $\widetilde{L}$ in $\widetilde{G}$ .

Lemma 2

(a) Suppose that $G_{0}=\mathrm{SO}(p,q)$ . If either $\left|{p-q}\right|=1$ , or $p=q$ and $\check{\mathcal{O}}$ is very even, then up to conjugation by $\mathrm{N}_{\widetilde{G}}(\widetilde{L})$ , there exists a unique genuine character on $\widetilde{L}$ of finite order.

(b) Suppose that $G_{0}=\mathrm{SO}(p,q)$ . If $p=q$ and $\check{\mathcal{O}}$ is not very even, then up to conjugation by $\mathrm{N}_{\widetilde{G}}(\widetilde{L})$ , there are precisely two genuine characters on $\widetilde{L}$ of finite order.

(c) Suppose that $G_{0}=\mathrm{SO}^{*}(2n)$ . Then the covering homomorphism $\widetilde{L}\rightarrow L_{0}$ uniquely splits, and $\widetilde{L}$ has a unique genuine character of finite order.

Proof

Suppose that $G_{0}=\mathrm{SO}(p,q)$ and $p=q$ . Let $Z$ denote the inverse image of $\{\pm 1\}$ under the covering homomorphism $\widetilde{G}\rightarrow\mathrm{SO}(p,p)$ . Then $Z$ is a central subgroup of $\widetilde{G}$ which is contained in $\widetilde{L}$ . Moreover,

Z\cong\begin{cases}\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z},&\text{if $p$ is even};\\ \mathbb{Z}/4\mathbb{Z},&\text{if $p$ is odd}.\end{cases}

When $\check{\mathcal{O}}$ is not very even, it is easy to see that there are two genuine characters on $\widetilde{L}$ of finite order that have distinct restrictions to $Z$ . Therefore up to conjugation by $\mathrm{N}_{\widetilde{G}}(\widetilde{L})$ , there are at least two genuine characters on $\widetilde{L}$ of finite order. The rest of the lemma is easy to prove and we omit the details.

Let $\widetilde{P}$ be a parabolic subgroup of $\widetilde{G}$ containing $\widetilde{L}$ as a Levi factor. Let $\chi$ be a genuine character on $\widetilde{L}$ of finite order, to be viewed as a character of $\widetilde{P}$ as usual. Put

I(\chi):=\mathrm{Ind}_{\widetilde{P}}^{\widetilde{G}}\chi\qquad(\textrm{normalized smooth parabolic induction}).

This is called a degenerate principal series representation.

Theorem 1.4

(a) Suppose that $G=\mathrm{Spin}(p,q)$ . Then $I(\chi)$ is irreducible and belongs to ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ . Moreover

{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})=\begin{cases}\{I(\chi)\},&\text{if either $\left|{p-q}\right|=1$, or $p=q$ and $\check{\mathcal{O}}$ is very even};\\ \{I(\chi_{1}),I(\chi_{2})\},&\text{if $p=q$ and $\check{\mathcal{O}}$ is not very even}.\end{cases}

Here $\chi_{1}$ and $\chi_{2}$ are two genuine characters on $\widetilde{L}$ of finite order that are not conjugate by $\mathrm{N}_{\widetilde{G}}(\widetilde{L})$ .

(b) Suppose that $G=\mathrm{Spin}^{*}(2n)$ . Then for every $\mathbf{o}\in\widetilde{G}\backslash\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}$ , there is a unique (up to isomorphism) irreducible Casselman-Wallach representation $\pi_{\mathbf{o}}$ of $\widetilde{G}$ that occurs in $I(\chi)$ whose wave front set equals the closure of $\mathbf{o}$ . Moreover,

{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})=\{\pi_{\mathbf{o}}\mid\mathbf{o}\in\widetilde{G}\backslash\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}\}

and

I(\chi)\cong\bigoplus_{\mathbf{o}\in\widetilde{G}\backslash\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}}\pi_{\mathbf{o}}.

Now we return to consider the group $G$ . Assume that $G=\mathrm{Spin}(p,q)$ . Put $P:=\widetilde{P}\cap G$ and $L=\widetilde{L}\cap G$ . Then we have the identification

I(\chi)|_{G}=\mathrm{Ind}_{P}^{G}\chi|_{L}

of representations of $G$ . By Clifford theory and Theorem 1.2, if $p=q$ is odd, then $I(\chi)|_{G}$ is irreducible. In all other cases, $I(\chi)|_{G}$ is the direct sum of two irreducible subrepresentations that are not isomorphic to each other. Write $I_{1}(\chi)$ and $I_{2}(\chi)$ for these two irreducible subrepresentations.

By Clifford theory, part (a) of Theorem 1.4 easily implies the following result. We omit the details.

Theorem 1.5

Assume that $G=\mathrm{Spin}(p,q)$ . Then

			$\displaystyle{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)$
		$\displaystyle=$	$\displaystyle\begin{cases}\{I_{1}(\chi),I_{2}(\chi)\},&\text{if $\left\|{p-q}\right\|=1$};\\ \{I_{1}(\chi),I_{2}(\chi)\},&\text{if $p=q$ is even},\\ &\quad\text{and $\check{\mathcal{O}}$ is very even};\\ \{I_{1}(\chi_{1}),I_{2}(\chi_{1}),I_{1}(\chi_{2}),I_{2}(\chi_{2})\},&\text{if $p=q$ is even},\\ &\quad\text{and $\check{\mathcal{O}}$ is not very even};\\ \{I(\chi)\|_{G}\},&\text{if $p=q$ is odd}.\end{cases}$

Here $\chi_{1}$ and $\chi_{2}$ are two genuine characters on $\widetilde{L}$ of finite order that are not conjugate by $\mathrm{N}_{\widetilde{G}}(\widetilde{L})$ .

The following unitarity result is an obvious consequence of Theorems 1.4 and 1.5.

Theorem 1.6

All genuine special unipotent representations of $\widetilde{G}$ or $G$ are unitarizable.

Remark 1

Theorem 1.6, together with the result of BMSZ2 on special unipotent representations of $G_{0}$ , implies that all special unipotent representations of $\widetilde{G}$ or $G$ are unitarizable.

We now examine the case of the complex spin group. Recall that $G_{\mathbb{C}}=\mathrm{Spin}(m,\mathbb{C})$ . View $G_{\mathbb{C}}$ as a real reductive group. Then its Langlands dual is $\check{G}\times\check{G}$ , where $\check{G}$ is given in (1). As before, denote by $\mathrm{Unip}^{\mathrm{gen}}_{{\check{\mathcal{O}}}_{\mathbb{C}}}(G_{\mathbb{C}})$ the set of isomorphism classes of genuine special unipotent representations of $G_{\mathbb{C}}$ attached to $\check{\mathcal{O}}_{\mathbb{C}}$ , where $\check{\mathcal{O}}_{\mathbb{C}}$ is a nilpotent $\check{G}\times\check{G}$ -orbit in $\check{\mathfrak{g}}\times\check{\mathfrak{g}}$ (in the obvious notation).

The following theorem should be known within the expert community. As the proof is along the same line as that of Theorem 1.1, we will state the result without proof.

Theorem 1.7

The set $\mathrm{Unip}^{\mathrm{gen}}_{{\check{\mathcal{O}}}_{\mathbb{C}}}(G_{\mathbb{C}})$ is empty, namely there exists no genuine special unipotent representation of $G_{\mathbb{C}}=\mathrm{Spin}(m,\mathbb{C})$ .

We remark that Losev, Mason-Brown and Matvieievskyi LMBM have recently proposed a notion of unipotent representations for a complex reductive group, extending the notion of special unipotent representations. See also (B17, , Section 2.3) for a different version of the notion proposed by Barbasch earlier. Genuine unipotent representations of $\mathrm{Spin}(m,{\mathbb{C}})$ do exist and they play a key role in the determination of the unitary dual of $\mathrm{Spin}(m,{\mathbb{C}})$ (see for example, Brega and WZ ).

2 Proof of Theorem 1.1

In this section, we apply the method developed in BMSZ1 to prove Theorem 1.1. We will also follow BMSZ1 in our choice of notations and conventions.

2.1 Coherent continuation representation of a spin group

We adopt the formulation of coherent continuation representation in (BMSZ1, , Sections 3 and 4). The original references include Sch ; Zu ; SpVo ; Vg .

We make the following identifications:

•

The dual $\mathfrak{h}^{*}$ of the abstract Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}_{\mathbb{C}}$ is identified with ${\mathbb{C}}^{n}$ .
•

The analytic weight lattice $Q\subset\mathfrak{h}^{*}$ of $G_{{\mathbb{C}}}$ is identified with ${\mathbb{Z}}^{n}\sqcup({\tfrac{1}{2}}\mathbf{1}+{\mathbb{Z}}^{n})$ . Here $\mathbf{1}:=(1,1,\cdots,1)\in{\mathbb{Z}}^{n}$ .
•

The analytic weight lattice $Q_{0}$ of $\mathrm{SO}(m,{\mathbb{C}})$ is identified with the sublattice ${\mathbb{Z}}^{n}$ of $Q$ via the natural quotient map $G_{\mathbb{C}}\rightarrow\mathrm{SO}(m,{\mathbb{C}})$ .

We adopt the following notations:

•

Let $\mathcal{K}(\widetilde{G})$ be the Grothendieck group (with coefficients in ${\mathbb{C}}$ ) of the category of Casselman-Wallach representations of $\widetilde{G}$ .
•

Let $\mathcal{K}_{\lambda}(\widetilde{G})$ be the subgroup of $\mathcal{K}(\widetilde{G})$ generated by irreducible representations of $\widetilde{G}$ with infinitesimal character $\lambda\in\mathfrak{h}^{*}$ ,
•

Let $\mathcal{K}^{\mathrm{gen}}(\widetilde{G})$ be the subgroup of $\mathcal{K}(\widetilde{G})$ generated by irreducible genuine representations of $\widetilde{G}$ .
•

Let $\mathcal{K}^{\mathrm{cls}}(\widetilde{G})$ be the subgroup of $\mathcal{K}(\widetilde{G})$ generated by irreducible representations of $\widetilde{G}$ which factor through $G_{0}$ . We identify $\mathcal{K}^{\mathrm{cls}}(\widetilde{G})$ with the Grothendick group $\mathcal{K}(G_{0})$ of the category of Casselman-Wallach representations of $G_{0}$ .

Let $W$ be abstract Weyl group of $G_{\mathbb{C}}$ , which acts on $\mathfrak{h}^{*}$ in the standard way.

For the rest of this section, let $\Lambda\subset\mathfrak{h}^{*}$ be a $Q$ -coset. Denote by $W_{\Lambda}$ the stabilizer of $\Lambda$ in $W$ . Also we have the integral Weyl group

W(\Lambda):=\set{w\in W}{w\lambda-\lambda\textrm{ is in the root lattice for every }\lambda\in\Lambda},

(3)

which is a subgroup of $W_{\Lambda}$ .

Definition 1

The space $\mathrm{Coh}_{\Lambda}(\mathcal{K}(\widetilde{G}))$ of $\mathcal{K}(\widetilde{G})$ -valued coherent families based on $\Lambda$ is the vector space of all maps $\Psi\colon\Lambda\to\mathcal{K}(\widetilde{G})$ such that, for all $\lambda\in\Lambda$ ,

•

$\Psi(\lambda)\in\mathcal{K}_{\lambda}(\widetilde{G})$ , and
•

for any holomorphic finite-dimensional representation $F$ of $G_{\mathbb{C}}$ ,

$F\otimes\Psi(\lambda)=\sum_{\mu}\Psi(\lambda+\mu),$

where $\mu$ runs over the set of all weights (counting multiplicities) of $F$ .

The space $\mathrm{Coh}_{\Lambda}(\mathcal{K}(\widetilde{G}))$ is a $W_{\Lambda}$ -module under the action

(w\cdot\Psi)(\lambda)=\Psi(w^{-1}\lambda),\qquad w\in W_{\Lambda}.

This is called the coherent continuation representation.

Fix a decomposition

\Lambda=\Lambda_{1}\sqcup\Lambda_{2}

where $\Lambda_{1}$ and $\Lambda_{2}$ are $Q_{0}$ -cosets. Then $\Lambda_{1}={\tfrac{1}{2}}\mathbf{1}+\Lambda_{2}$ . Similar to (3), we define the integral Weyl groups $W(\Lambda_{1})$ and $W(\Lambda_{2})$ . Then

W(\Lambda)=W(\Lambda_{1})=W(\Lambda_{2}).

Let $\mathrm{Coh}_{\Lambda_{1}}(\mathcal{K}^{\mathrm{cls}}(\widetilde{G})):=\mathrm{Coh}_{\Lambda_{1}}(\mathcal{K}(G_{0}))$ be the coherent continuation representation defined for $G_{0}$ based on $\Lambda_{1}$ (replace $\mathcal{K}(\widetilde{G})$ by $\mathcal{K}(G_{0})$ , $\Lambda$ by $\Lambda_{1}$ and $G_{\mathbb{C}}$ by $\mathrm{SO}(m,{\mathbb{C}})$ in Definition 1). Similarly, let $\mathrm{Coh}_{\Lambda_{2}}(\mathcal{K}^{\mathrm{gen}}(\widetilde{G}))$ be the coherent continuation representation defined for $\widetilde{G}$ based on $\Lambda_{2}$ (replace $\mathcal{K}(\widetilde{G})$ by $\mathcal{K}^{\mathrm{gen}}(\widetilde{G})$ , $\Lambda$ by $\Lambda_{2}$ and $G_{\mathbb{C}}$ by $\mathrm{SO}(m,{\mathbb{C}})$ in Definition 1).

Define a subspace of $\mathrm{Coh}_{\Lambda}(\widetilde{G})$ by

\mathrm{Coh}_{\Lambda_{1},\Lambda_{2}}(\widetilde{G}):=\Set{\Psi\in\mathrm{Coh}_{\Lambda}(\widetilde{G})}{\begin{array}[]{l}\mathrm{Im}(\Psi|_{\Lambda_{1}})\subset\mathcal{K}^{\mathrm{cls}}(\widetilde{G}),\\ \mathrm{Im}(\Psi|_{\Lambda_{2}})\subset\mathcal{K}^{\mathrm{gen}}(\widetilde{G})\end{array}}.

The following lemma reduces the counting of genuine representations of $\widetilde{G}$ to that of $G_{0}$ -representations.

Lemma 3

Restrictions induce the following isomorphisms of $W(\Lambda)$ -modules:

\mathrm{Coh}_{\Lambda_{1}}(\mathcal{K}(G_{0}))\xleftarrow{\ \ \ \cong\ \ }\mathrm{Coh}_{\Lambda_{1},\Lambda_{2}}(\mathcal{K}(\widetilde{G}))\xrightarrow{\ \ \cong\ \ \ }\mathrm{Coh}_{\Lambda_{2}}(\mathcal{K}^{\mathrm{gen}}(\widetilde{G})).

Moreover,

\mathrm{Coh}_{\Lambda}(\mathcal{K}(\widetilde{G}))=\mathrm{Coh}_{\Lambda_{1},\Lambda_{2}}(\mathcal{K}(\widetilde{G}))\oplus\mathrm{Coh}_{\Lambda_{2},\Lambda_{1}}(\mathcal{K}(\widetilde{G})).

(4)

Proof

The $W(\Lambda)$ -equivariance of the maps is clear. Let $\Psi$ be in $\mathrm{Coh}_{\Lambda}(\mathcal{K}(\widetilde{G}))$ . Since $\Psi|_{\Lambda_{1}}$ and $\Psi|_{\Lambda_{2}}$ are related by tensoring with genuine finite dimensional representations, we see that the following conditions are equivalent:

•

$\Psi\in\mathrm{Coh}_{\Lambda_{1},\Lambda_{2}}(\mathcal{K}(\widetilde{G}))$ ,
•

$\Psi(\lambda_{1})\in\mathcal{K}^{\mathrm{cls}}(\widetilde{G})$ for some regular element $\lambda_{1}\in\Lambda_{1}$ ,
•

$\Psi(\lambda_{1})\in\mathcal{K}^{\mathrm{cls}}(\widetilde{G})$ for all elements $\lambda_{1}\in\Lambda_{1}$ ,
•

$\Psi(\lambda_{2})\in\mathcal{K}^{\mathrm{gen}}(\widetilde{G})$ for some regular element $\lambda_{2}\in\Lambda_{2}$ ,
•

$\Psi(\lambda_{2})\in\mathcal{K}^{\mathrm{gen}}(\widetilde{G})$ for all elements $\lambda_{2}\in\Lambda_{2}$ ,

The maps are isomorphisms since the evaluation $\Psi\mapsto\Psi(\lambda)$ induces an isomorphism for any regular element $\lambda$ in $\Lambda$ . See (Vg, , Section 7).

The following counting result on genuine special unipotent representations will follow from Lemma 3. Recall that attached to ${\check{\mathcal{O}}}$ , there is an infinitesimal character determined by ${\tfrac{1}{2}}(^{L}h)$ in the notation of (BVUni, , Section 5), which is represented by $\lambda_{\check{\mathcal{O}}}\in\mathfrak{h}^{*}$ .

Proposition 1

Suppose $\check{\mathcal{O}}$ is a nilpotent $\check{G}$ -orbit in $\check{\mathfrak{g}}$ . Then

\left|{{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})}\right|=\sum_{\sigma\in{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}}[\sigma:\mathrm{Coh}_{\Lambda_{1}}(\mathcal{K}(G_{0}))],

where

•

$\Lambda_{1}=\lambda_{\check{\mathcal{O}}}+{\tfrac{1}{2}}\mathbf{1}+Q_{0}$ , and
•

${}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}\subset\mathrm{Irr}(W(\Lambda_{1}))$ is the Lusztig left cell attached to $\lambda_{\check{\mathcal{O}}}$ (see (BMSZ1, , Definition 3.34)).

Here and henceforth, $\mathrm{Irr}(E)$ denotes the set of isomorphism classes of irreducible representations of a finite group $E$ ; $[\ :\ ]$ indicates the multiplicity of the first (irreducible) representation in the second representation, of a finite group. If it is necessary to specify the finite group, a subscript will be included.

Proof

Let $\Lambda_{2}:=\lambda_{\check{\mathcal{O}}}+Q_{0}$ and $\Lambda:=\lambda_{\check{\mathcal{O}}}+Q=\Lambda_{1}\sqcup\Lambda_{2}$ . We apply (BMSZ1, , Corollary 5.4) twice:

\begin{split}\left|{{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})}\right|&=\left|{\mathrm{Unip}_{\check{\mathcal{O}}}(\widetilde{G})}\right|-\left|{\mathrm{Unip}_{\check{\mathcal{O}}}(G_{0})}\right|\\ &=\sum_{\sigma\in{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}}[\sigma:\mathrm{Coh}_{\Lambda_{2}}(\mathcal{K}(\widetilde{G}))]-\sum_{\sigma\in{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}}[\sigma:\mathrm{Coh}_{\Lambda_{2}}(\mathcal{K}(G_{0}))]\\ &=\sum_{\sigma\in{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}}\left([\sigma:\mathrm{Coh}_{\Lambda}(\mathcal{K}(\widetilde{G}))]-[\sigma:\mathrm{Coh}_{\Lambda_{2},\Lambda_{1}}(\mathcal{K}(\widetilde{G}))]\right)\\ &=\sum_{\sigma\in{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}}[\sigma:\mathrm{Coh}_{\Lambda_{1},\Lambda_{2}}(\mathcal{K}(\widetilde{G}))]\quad\text{(by \eqref{eq:coheq})}\\ &=\sum_{\sigma\in{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}}[\sigma:\mathrm{Coh}_{\Lambda_{1}}(G_{0})].\hskip 144.54pt\Box\end{split}

Before we proceed to count explicitly genuine special unipotent representations case by case, we recall the following notations and definitions in BMSZ1 . The relevant labels (which specify the types of groups we consider) are $\star=B,D,D^{*}$ .

Denote by $\mathsf{S}_{n}\subset\mathrm{GL}_{n}(\mathbb{Z})$ the group of the permutation matrices. The Weyl group

W=\begin{cases}\mathsf{W}_{n},&\text{if $\star=B$};\\ \mathsf{W}_{n}^{\prime},&\text{if $\star\in\{D,D^{*}\}$},\\ \end{cases}

where $\mathsf{W}_{n}\subset\mathrm{GL}_{n}(\mathbb{Z})$ is the subgroup generated by $\mathsf{S}_{n}$ and all the diagonal matrices with diagonal entries $\pm 1$ , and $\mathsf{W}^{\prime}_{n}\subset\mathrm{GL}_{n}(\mathbb{Z})$ is the subgroup generated by $\mathsf{S}_{n}$ and all the diagonal matrices with diagonal entries $\pm 1$ and determinant $1$ .

As always, $\operatorname{sgn}$ denotes the sign character (of an appropriate Weyl group). By inflating the sign character of $\mathsf{S}_{n}$ (viewed as a quotient of $\mathsf{W}_{n}$ ), we obtain a quadratic character of $\mathsf{W}_{n}$ , to be denoted by $\overline{\operatorname{sgn}}$ . In addition let

\mathsf{H}_{t}:=\mathsf{W}_{t}\ltimes\set{\pm 1}^{t},\quad(t\in{\mathbb{N}}),

to be viewed as a subgroup in $\mathsf{W}_{2t}$ , and let $\eta$ be the quadratic character of $H_{t}$ , as in (BMSZ1, , Section 8.1).

We define

\mathcal{C}^{\mathrm{b}}_{\star,n}:=\begin{cases}\bigoplus_{\begin{subarray}{c}2t+c+d=n\end{subarray}}\mathrm{Ind}_{\mathsf{H}_{t}\times\mathsf{W}_{c}\times\mathsf{W}_{d}}^{\mathsf{W}_{n}}\eta\otimes 1\otimes 1,&\text{if $\star=B$};\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \bigoplus_{\begin{subarray}{c}2t+a=n\end{subarray}}\mathrm{Ind}_{\mathsf{H}_{t}\times\mathsf{S}_{a}}^{\mathsf{W}_{n}}\eta\otimes 1,&\text{if $\star=D$};\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \mathrm{Ind}_{\mathsf{H}_{\frac{n}{2}}}^{\mathsf{W}^{\prime}_{n}}\eta,&\text{if $\star=D^{*}$}.\end{cases}

(5)

We also define

\mathcal{C}^{\mathrm{g}}_{p,q}:=\begin{cases}\displaystyle\bigoplus_{\begin{subarray}{c}0\leq p-(2t+a+2r)\leq 1,\\ 0\leq q-(2t+a+2s)\leq 1\end{subarray}}\mathrm{Ind}_{\mathsf{H}_{t}\times\mathsf{S}_{a}\times\mathsf{W}_{s}\times\mathsf{W}_{r}}^{\mathsf{W}_{n}}\eta\otimes 1\otimes\operatorname{sgn}\otimes\operatorname{sgn},&\text{if $p+q=2n+1$};\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \displaystyle\bigoplus_{\begin{subarray}{c}2t+c+d+2r=p\\ 2t+c+d+2s=q\end{subarray}}\mathrm{Ind}_{\mathsf{H}_{t}\times\mathsf{W}_{s}\times\mathsf{W}_{r}\times\mathsf{W}^{\prime}_{c}\times\mathsf{W}_{d}}^{\mathsf{W}_{n}}\eta\otimes\overline{\operatorname{sgn}}\otimes\overline{\operatorname{sgn}}\otimes 1\otimes 1,&\text{if $p+q=2n$},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \end{cases}

(6)

and

\mathcal{C}^{\mathrm{g}}_{D^{*},n}:=\bigoplus_{\begin{subarray}{c}2t+a=n\end{subarray}}\mathrm{Ind}_{\mathsf{H}_{t}\times\mathsf{S}_{a}}^{\mathsf{W}^{\prime}_{n}}\eta\otimes\operatorname{sgn},\qquad\qquad\text{if $\star=D^{*}$}.

(7)

We introduce some notations relevant to irreducible representations and Young diagrams. As usual, we identify $\mathrm{Irr}(\mathsf{W}_{n})$ with the set of bipartions (or a pair of Young diagrams) of total size $n$ ((Carter, , Section 11.4)). In what follows, we let $\left[a_{1},a_{2},\cdots,a_{k}\right]_{\mathrm{col}}$ (resp. $\left[a_{1},a_{2},\cdots,a_{k}\right]_{\mathrm{row}}$ ) denote the Young diagram whose $i$ -th column (resp. row) has length $a_{i}$ if $1\leq i\leq k$ and has length $0$ otherwise. Let ${\mathbf{r}}_{i}({\check{\mathcal{O}}})$ denote the length of $i$ -th row of the Young diagram of a nilpotent orbit ${\check{\mathcal{O}}}$ of a classical group.

2.2 Real odd spin groups

In this case $\widetilde{G}=\mathrm{Spin}(p,q)$ and $G_{0}=\mathrm{SO}(p,q)$ , with $p+q=2n+1$ . Let $N$ be the smallest positive integer such that ${\mathbf{r}}_{2N}({\check{\mathcal{O}}})=0$ . Suppose the multiset $\set{{\mathbf{r}}_{i}({\check{\mathcal{O}}}):1\leq i\leq 2N}$ is

\set{2r^{\prime}_{1}+1,2r^{\prime}_{1}+1,2r^{\prime}_{2}+1,2r^{\prime}_{2}+1,\cdots,2r^{\prime}_{l}+1,2r^{\prime}_{l}+1,2r^{\prime\prime}_{1},2r^{\prime\prime}_{2},\cdots,2r^{\prime\prime}_{2k}}.

with $2l+2k=2N$ , $r^{\prime}_{i}\geq r^{\prime}_{i+1}$ (for $1\leq i<l$ ) and $r^{\prime\prime}_{i}\geq r^{\prime\prime}_{i+1}$ (for $1\leq i<2k$ ). Note that $r^{\prime\prime}_{2k}=0$ by definition.

Let

n_{\mathbf{b}}=l+\sum_{i=1}^{l}2r^{\prime}_{i}\qquad\text{ and }\qquad n_{\mathbf{g}}=\sum_{i=1}^{2k}r^{\prime\prime}_{i}.

Then

\Lambda_{1}=(\underbrace{{\tfrac{1}{2}},\cdots{\tfrac{1}{2}}}_{n_{\mathbf{b}}\text{-terms}},\underbrace{0,\cdots 0}_{n_{\mathbf{g}}\text{-terms}})+{\mathbb{Z}}^{n}\qquad\text{ and }\qquad W(\Lambda_{1})=\mathsf{W}_{n_{\mathbf{b}}}\times\mathsf{W}_{n_{\mathbf{g}}}.

As a $W(\Lambda_{1})$ -module, the coherent continuation representation is

\mathrm{Coh}_{\Lambda_{1}}(\mathcal{K}(G_{0}))=\begin{cases}\mathcal{C}^{\mathrm{g}}_{p-n_{\mathbf{g}},q-n_{\mathbf{g}}}\otimes\mathcal{C}^{\mathrm{b}}_{B,n_{\mathbf{g}}},&\text{if $p,q\geq n_{\mathbf{g}}$;}\\ 0,&\text{otherwise.}\end{cases}

See (BMSZ1, , Propositions 8.1 and 8.2).

As in (BMSZ1, , Section 2.8), let

\mathrm{PP}({\check{\mathcal{O}}}):=\set{(2i,2i+1)}{r^{\prime\prime}_{2i}>r^{\prime\prime}_{2i+1}\,\text{ and }\,1\leq i<k}

be the set of primitive pairs attached to ${\check{\mathcal{O}}}$ . The Lusztig left cell ${}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}$ in $\mathrm{Irr}(W(\Lambda_{1}))$ is given by

{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}=\set{\tau_{b}\otimes\tau_{\wp}}{\wp\subseteq\mathrm{PP}({\check{\mathcal{O}}})},

where

\tau_{b}=[r^{\prime}_{1}+1,r^{\prime}_{2}+1,\cdots,r^{\prime}_{l}+1]_{\text{col}}\times[r^{\prime}_{1},r^{\prime}_{2},\cdots,r^{\prime}_{l}]_{\text{col}},

and

\tau_{\wp}=[l_{\wp,1},l_{\wp,2},\cdots,l_{\wp,k+1}]_{\text{col}}\times[r_{\wp,1},r_{\wp,2},\cdots,r_{\wp,k+1}]_{\text{col}}.

Here $(l_{\wp,1},l_{\wp,k+1})=(r^{\prime\prime}_{1},0)$ and for $1<i\leq k$ ,

(l_{\wp,i},r_{\wp,i+1})=\begin{cases}(r^{\prime\prime}_{2i+1},r^{\prime\prime}_{2i}),&\text{$(2i,2i+1)\in\wp$};\\ (r^{\prime\prime}_{2i},r^{\prime\prime}_{2i+1}),&\text{otherwise}.\end{cases}

See (BMSZ1, , Proposition 8.3).

Lemma 4

We have

[\tau_{b}:\mathcal{C}^{\mathrm{g}}_{p-n_{\mathbf{g}},q-n_{\mathbf{g}}}]=\begin{cases}1,&\text{ if $|p-q|=1$;}\\ 0,&\text{otherwise.}\end{cases}

Proof

The multiplicity is computed by the Littlewood-Richardson rule which is reformulated in terms of counting painted bipartitions as in (BMSZ1, , Section 2.8).

Lemma 5

For $\wp\subseteq\mathrm{PP}({\check{\mathcal{O}}})$ , we have

[\tau_{\wp}:\mathcal{C}^{\mathrm{b}}_{B,n_{g}}]=\begin{cases}1,&\text{if $\wp=\emptyset$ and ${\mathbf{r}}_{2i+1}({\check{\mathcal{O}}})={\mathbf{r}}_{2i+2}({\check{\mathcal{O}}})$ for all $i\in{\mathbb{N}}$;}\\ 0,&\text{otherwise.}\end{cases}

Proof

Note that we always have $l_{\wp,i}\leq r_{\wp,i}$ for $1\leq i\leq k+1$ . Again the multiplicity is computed by the Littlewood-Richardson rule and the result can be formulated in terms of counting certain painted bipartitions. In a bit more detail, from the painting rules of type $B$ in (BMSZ1, , Definition 8.5), the symbols “ $c$ ” and “ $d$ ” are painted in the left diagram of $\tau_{\wp}$ . The only possible painting allowed on $\tau_{\wp}$ will be to put the symbol “ $\bullet$ ” in every box, in which case we will have $r^{\prime\prime}_{2i-1}=r_{\wp,i}=l_{\wp,i}=r^{\prime\prime}_{2i}$ , $\wp=\emptyset$ and the multiplicity is $1$ .

Together with Proposition 1, the above two lemmas imply Theorem 1.1 for real odd spin groups.

2.3 Real even spin groups and quaternionic spin groups

We now assume $G_{0}=\mathrm{SO}(p,q)$ with $p+q=2n$ , or $G_{0}=\mathrm{SO}^{*}(2n)$ . Let $N$ be the smallest positive integer such that ${\mathbf{r}}_{2N+1}({\check{\mathcal{O}}})=0$ . Suppose the multiset $\set{{\mathbf{r}}_{i}({\check{\mathcal{O}}}):1\leq i\leq 2N}$ is

\set{2r^{\prime}_{1},2r^{\prime}_{1},2r^{\prime}_{2},2r^{\prime}_{2},\cdots,2r^{\prime}_{l},2r^{\prime}_{l},2r^{\prime\prime}_{1}+1,2r^{\prime\prime}_{2}+1,\cdots,2r^{\prime\prime}_{2k}+1}.

with $2l+2k=2N$ , $r^{\prime}_{i}\geq r^{\prime}_{i+1}$ (for $1\leq i<l$ ) and $r^{\prime\prime}_{i}\geq r^{\prime\prime}_{i+1}$ (for $1\leq i<2k$ ).

Let

n_{\mathbf{b}}=\sum_{i=1}^{l}2r^{\prime}_{i}\qquad\text{ and }\qquad n_{\mathbf{g}}=k+\sum_{i=1}^{2k}r^{\prime\prime}_{i}.

Then

\Lambda_{1}=(\underbrace{0,\cdots 0}_{n_{\mathbf{b}}\text{-terms}},\underbrace{{\tfrac{1}{2}},\cdots{\tfrac{1}{2}}}_{n_{\mathbf{g}}\text{-terms}})\qquad\text{ and }\qquad W(\Lambda_{1})=\mathsf{W}^{\prime}_{n_{\mathbf{b}}}\times\mathsf{W}^{\prime}_{n_{\mathbf{g}}}.

Recall that irreducible representations of $\mathsf{W}_{n}^{\prime}$ are parameterized by unordered pairs of bipartitions with a label $I$ or $II$ attached when the pairs are equal to each other. For the labeling convention, see (BMSZ1, , Section 8.3).

As in (BMSZ1, , Section 2.8), let

\mathrm{PP}({\check{\mathcal{O}}}):=\set{(2i,2i+1)}{r^{\prime\prime}_{2i}>r^{\prime\prime}_{2i+1}>0\text{ and }1\leq i<k}

be the set of primitive pairs attached to ${\check{\mathcal{O}}}$ . The Lusztig left cell ${}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}$ in $\mathrm{Irr}(W(\Lambda_{1}))$ is given by

{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}=\set{\tau_{b}\otimes\tau_{\wp}}{\wp\subseteq\mathrm{PP}({\check{\mathcal{O}}})},

where

\tau_{b}=\set{[r^{\prime}_{1},r^{\prime}_{2},\cdots,r^{\prime}_{l}]_{\text{col}},[r^{\prime}_{1},r^{\prime}_{2},\cdots,r^{\prime}_{l}]_{\text{col}}}_{I/II}

(the label $I/II$ of $\tau_{b}$ is determined by ${\check{\mathcal{O}}}$ ) and

\tau_{\wp}=\set{[l_{\wp,1},l_{\wp,2},\cdots,l_{\wp,k}]_{\text{col}},[r_{\wp,1},r_{\wp,2},\cdots,r_{\wp,k}]_{\text{col}}}.

Here $(l_{\wp,1},r_{\wp,k})=(r^{\prime\prime}_{1}+1,r^{\prime\prime}_{2k})$ and for $1\leq i<k$ ,

(l_{\wp,i+1},r_{\wp,i})=\begin{cases}(r^{\prime\prime}_{2i}+1,r^{\prime\prime}_{2i+1}),&\text{$(2i,2i+1)\in\wp$};\\ (r^{\prime\prime}_{2i+1}+1,r^{\prime\prime}_{2i}),&\text{otherwise}.\end{cases}

See (BMSZ1, , Proposition 8.3).

The following lemma is clear.

Lemma 6

For $1\leq i\leq k$ and $\wp\subseteq\mathrm{PP}({\check{\mathcal{O}}})$ , we have

l_{\wp,i}>r_{\wp,i}.

The case when $G_{0}=\mathrm{SO}(p,q)$ , $p+q=2n$

In this case, the coherent continuation representation as a $W(\Lambda_{1})$ -module is

\mathrm{Coh}_{\Lambda_{1}}(\mathcal{K}(G_{0}))=\begin{cases}\mathcal{C}^{\mathrm{g}}_{p-n_{\mathbf{g}},q-n_{\mathbf{g}}}\otimes\mathcal{C}^{\mathrm{b}}_{n_{\mathbf{g}}},&\text{if $p,q\geq n_{\mathbf{g}}$};\\ 0,&\text{otherwise.}\end{cases}

Here the right-hand side is understood as its restriction to $W(\Lambda_{1})=\mathsf{W}^{\prime}_{n_{\mathbf{b}}}\times\mathsf{W}^{\prime}_{n_{\mathbf{g}}}$ . See (BMSZ1, , Propositions 8.1 and 8.2).

Lemma 7

We have

[\tau_{b}:\mathcal{C}^{\mathrm{g}}_{p-n_{\mathbf{g}},q-n_{\mathbf{g}}}]=\begin{cases}1,&\text{if $p=q$};\\ 0,&\text{otherwise.}\end{cases}

Proof

Since the pair of Young diagrams in $\tau_{b}$ have the same shape, $\tau_{b}$ does not occur in any of the terms of (6) with any of $s,r,c,d$ non-zero. On the other hand, $\tau_{b}$ occurs with multiplicity one in $\mathrm{Ind}_{\mathsf{H}_{\frac{n_{\mathbf{b}}}{2}}}^{\mathsf{W}_{n_{\mathbf{b}}}}\eta$ (see (BMSZ1, , (8.15)), for example).

Lemma 8

Suppose $n_{\mathbf{g}}>0$ and $\wp\subseteq\mathrm{PP}({\check{\mathcal{O}}})$ . Then

[\tau_{\wp}:\mathcal{C}^{\mathrm{b}}_{D,n_{\mathbf{g}}}]=\begin{cases}2,&\text{if $\wp=\emptyset$ and $r^{\prime\prime}_{2i+1}=r^{\prime\prime}_{2i+2}$ for all $i\in{\mathbb{N}}$};\\ 0,&\text{otherwise.}\end{cases}

Proof

By Lemma 6, $\tau_{\wp}=\tau_{\wp,L}\times\tau_{\wp,R}$ , with $\tau_{\wp,L}\neq\tau_{\wp,R}$ . Therefore,

\mathrm{Ind}_{\mathsf{W}^{\prime}_{n_{\mathbf{g}}}}^{\mathsf{W}_{n_{\mathbf{g}}}}=\tau_{\wp,L}\times\tau_{\wp,R}\oplus(\tau_{\wp,L}\times\tau_{\wp,R}\otimes\varepsilon),

where $\varepsilon$ is the quadratic character of $\mathsf{W}_{n_{\mathbf{g}}}$ whose kernel is $\mathsf{W}^{\prime}_{n_{\mathbf{g}}}$ . Now

\begin{split}[\tau_{\wp}:\mathcal{C}^{\mathrm{b}}_{D,n_{\mathbf{g}}}]_{\mathsf{W}^{\prime}_{n_{\mathbf{g}}}}&=[\tau_{\wp,L}\times\tau_{\wp,R}:\mathcal{C}^{\mathrm{b}}_{D,n_{\mathbf{g}}}]_{\mathsf{W}_{n_{\mathbf{g}}}}+[\tau_{\wp,L}\times\tau_{\wp,R}\otimes\varepsilon:\mathcal{C}^{\mathrm{b}}_{D,n_{\mathbf{g}}}]_{\mathsf{W}_{n_{\mathbf{g}}}}\\ &=2[\tau_{\wp,L}\times\tau_{\wp,R}:\mathcal{C}^{\mathrm{b}}_{D,n_{\mathbf{g}}}]_{\mathsf{W}_{n_{\mathbf{g}}}}\end{split}

since $\mathcal{C}^{\mathrm{b}}_{D,n_{\mathbf{g}}}\otimes\varepsilon=\mathcal{C}^{\mathrm{b}}_{D,n_{\mathbf{g}}}$ . It is easy to see that (by the painting rules of type $D$ in (BMSZ1, , Section 2.8), for example), $\tau_{\wp,L}\times\tau_{\wp,R}$ can only occur in the term of (5) with $a=k$ . When this is the case, it occurs with multiplicity one and

l_{\wp,i}=r_{\wp,i}+1\quad\text{for $1\leq i\leq k$.}

(8)

The equation (8) forces $r^{\prime\prime}_{2i-1}=r^{\prime\prime}_{2i}$ for $1\leq i\leq k$ and $\wp=\emptyset$ . This completes proof of the lemma.

Together with Proposition 1, the above two lemmas imply Theorem 1.1 for real even spin groups.

2.4 The case when $G_{0}=\mathrm{SO}^{*}(2n)$

In this case, the coherent continuation representation as a $W_{\Lambda_{1}}$ -module is

\mathrm{Coh}_{\Lambda_{1}}(G_{0})=\mathrm{Ind}_{\mathsf{W}^{\prime}_{n_{\mathbf{g}}}\times\mathsf{W}^{\prime}_{n_{\mathbf{b}}}}^{W_{\Lambda_{1}}}\mathcal{C}^{\mathrm{g}}_{D^{*},n_{\mathbf{b}}}\otimes\mathcal{C}^{\mathrm{b}}_{D^{*},n_{\mathbf{g}}},

where

W_{\Lambda_{1}}=(\mathsf{W}_{n_{\mathbf{b}}}\times\mathsf{W}_{n_{\mathbf{g}}})\cap\mathsf{W}^{\prime}_{n}.

See (BMSZ1, , Propositions 8.1 and 8.2).

Lemma 9

If $n_{\mathbf{g}}\neq 0$ , then $[\tau_{b}\otimes\tau_{\wp}:\mathrm{Coh}_{\Lambda_{1}}(G_{0})]=0$ for each $\wp\in\mathrm{PP}({\check{\mathcal{O}}})$ .

Proof

Applying the Littlewood-Richardson rule (see (BMSZ1, , (8.15))) to $\mathcal{C}^{\mathrm{b}}_{D^{*},n_{\mathbf{g}}}$ (see (5)), we see that if $\sigma_{1}\otimes\sigma_{2}\in\mathrm{Irr}(\mathsf{W}_{n_{\mathbf{b}}}\times\mathsf{W}_{n_{\mathbf{g}}})$ occurs in $\mathrm{Coh}_{\Lambda_{1}}(G)$ , then the pair of partitions in $\sigma_{2}$ must have the same shape. On the other hand, $\tau_{\wp}$ is represented by a pair of partitions with different shapes by Lemma 6. This proves the lemma.

We are now in the case when $n_{\mathbf{g}}=0$ (and so $n_{b}=n$ ). We have

{}^{L}\mathscr{C}_{{\check{\mathcal{O}}}}=\set{\tau_{b}}\qquad\text{and}\qquad\mathrm{Coh}_{\Lambda_{1}}(G_{0})=\mathcal{C}^{\mathrm{g}}_{D^{*},n}.

(9)

Recall the notion of $G$ -relevant orbit in (2).

Lemma 10

Suppose $n_{\mathbf{g}}=0$ . Then

[\tau_{b}:\mathcal{C}^{\mathrm{g}}_{D^{*},n}]=\begin{cases}\prod_{i=1}^{l}(r^{\prime}_{i}-r^{\prime}_{i+1}+1),&\text{if ${\check{\mathcal{O}}}$ is $G$-relevant;}\\ 0,&\text{otherwise.}\end{cases}

In all cases, it is equal to $\sharp(G\backslash\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}})$ . (Here $r^{\prime}_{l+1}:=0$ by convention. )

Proof

By the Littlewood-Richardson rule ((BMSZ1, , (8.15))), all representations occurring in $\mathcal{C}^{\mathrm{g}}_{D^{*},n}$ (see (7)) are labeled by $I$ . So if $\tau_{b}$ is to occur in it, $\tau_{b}$ must be labeled by $I$ , which is equivalent to ${\check{\mathcal{O}}}$ being $G$ -relevant. The rest of the claim about the multiplicity of $\tau_{b}$ in $\mathcal{C}^{\mathrm{g}}_{D^{*},n}$ also follows from the Littlewood-Richardson rule. Concretely, the multiplicity is counted by the number of paintings on the bipartition of shape $[r^{\prime}_{1},\cdots,r^{\prime}_{l}]_{\text{col}}\times[r^{\prime}_{1},\cdots,r^{\prime}_{l}]_{\text{col}}$ , using the painting rules of type $D^{*}$ in (BMSZ1, , Section 2.8). It is routine to check that there are $\prod_{i=1}^{l}(r^{\prime}_{i}-r^{\prime}_{i+1}+1)$ such paintings in total.

When ${\check{\mathcal{O}}}$ is $G$ -relevant, $\prod_{i=1}^{l}(r^{\prime}_{i}-r^{\prime}_{i+1}+1)$ also counts the number of signed Young diagrams having shape $\mathcal{O}$ , which is the number of real nilpotent orbits in $\mathcal{O}$ . When ${\check{\mathcal{O}}}$ is not $G$ -relevant, $\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}$ is empty. The lemma thus follows.

Together with Proposition 1, the above two lemmas imply Theorem 1.1 for $G=\mathrm{Spin}^{*}(2n)$ .

3 Genuine special unipotent representations of real spin groups

Based on Theorem 1.1, we will prove Theorems 1.2 to 1.4 in this section and the next section. We retain the notation and assumptions of Section 1.

For the time being, $G$ can be either $\mathrm{Spin}(p,q)$ or $\mathrm{Spin}^{*}(2n)$ . Recall the $\widetilde{G}$ -representation $I(\chi)=\mathrm{Ind}_{\widetilde{P}}^{\widetilde{G}}\chi$ , where $\chi$ is a genuine character on $\widetilde{L}$ of finite order.

The following lemma follows from a general result of Barbasch (B00, , Corollary 5.0.10). See also (MT, , Section 3).

Lemma 11

The wavefront cycle of $I(\chi)$ equals

\sum_{\mathbf{o}\in\widetilde{G}\backslash\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}}\mathbf{o}.

Proof

Let ${\mathfrak{n}}$ be the nilpotent radical of $\mathrm{Lie}(\widetilde{P})$ . By (B00, , Corollary 5.0.10), the wavefront cycle of $I(\chi)$ is given by

WF(I(\chi))=\sum_{\widetilde{G}\cdot X}\frac{\sharp(C(\widetilde{G}_{X}))}{\sharp(C(\widetilde{P}_{X}))}\widetilde{G}\cdot X,

where the summation runs over the set of orbits of the form $\widetilde{G}\cdot X$ in $\sqrt{-1}\mathfrak{g}^{*}\cap\mathcal{O}$ with $X\in\sqrt{-1}\mathfrak{n}^{*}$ . Here $\widetilde{G}_{X}$ (resp. $\widetilde{P}_{X}$ ) denotes the centralizer of $X$ in $\widetilde{G}$ (resp. $\widetilde{P}$ ), and the symbol $C$ indicates the component group.

It is routine to check that every orbit $\mathbf{o}\in\widetilde{G}\backslash\sqrt{-1}\mathfrak{g}^{*}\cap\mathcal{O}$ has a representative in $\sqrt{-1}{\mathfrak{n}}^{*}$ . Thus

WF(I(\chi))=\sum_{\mathbf{o}\in\widetilde{G}\backslash\sqrt{-1}\mathfrak{g}^{*}\cap{\mathcal{O}}}m_{\mathbf{o}}\mathbf{o},

where $m_{\mathbf{o}}=\frac{|C(\widetilde{G}_{X})|}{|C(\widetilde{P}_{X})|}$ , with $X$ an element of $\sqrt{-1}{\mathfrak{n}}^{*}\cap\mathbf{o}$ .

By a result of Kostant and Barbasch-Vogan (CM, , Lemma 3.7.3), $\widetilde{G}_{X}$ has a semidirect product decomposition:

\widetilde{G}_{X}=R\ltimes U_{X},

where the reductive part $R$ is the centralizer in $\widetilde{G}$ of the $\mathfrak{s}\mathfrak{l}_{2}$ -triple containing $X$ , and $U_{X}$ is the unipotant radical of $\widetilde{G}_{X}$ .

In view of condition (b) of Lemma 1 and by replacing $\widetilde{P}$ by a $\widetilde{G}$ -conjugate if necessary, the group $R$ will be contained in the Levi component $\widetilde{L}$ of $\widetilde{P}$ and therefore in $\widetilde{P}$ in particular. Since $\widetilde{P}_{X}=\widetilde{P}\cap\widetilde{G}_{X}$ , we conclude that $\widetilde{P}_{X}=R\ltimes(\widetilde{P}\cap U_{X})$ . The reductive part of $\widetilde{P}_{X}$ and $\widetilde{G}_{X}$ are therefore isomorphic and consequently,

\frac{\sharp(C(\widetilde{G}_{X}))}{\sharp(C(\widetilde{P}_{X}))}=1.

The lemma follows.

Lemma 12

The representation $I(\chi)$ is completely reducible and multiplicity free. Moreover, every irreducible subrepresentation of $I(\chi)$ belongs to ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ .

Proof

With the easy calculation of the infinitesimal character of $I(\chi)$ , Lemma 11 implies that every irreducible subquotient of $I(\chi)$ belongs to ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ . Since $I(\chi)$ is unitarizable, it is completely reducible. Lemma 11 implies that it is also multiplicity free.

In the rest of this section, we focus on the real spin groups.

Lemma 13

Suppose that $G=\mathrm{Spin}(p,q)$ and $\sharp({\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G}))=1$ . Then $I(\chi)$ is irreducible and ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})=\{I(\chi)\}$ .

Proof

This is a direct consequence of Lemma 12.

Lemma 14

Suppose that $G=\mathrm{Spin}(p,q)$ and $\sharp({\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G}))=2$ . Then $I(\chi)$ is irreducible and ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})=\{I(\chi),I(\chi^{\prime})\}$ . Here $\chi^{\prime}$ is a genuine character on $\widetilde{L}$ of finite order that is not conjugate to $\chi$ by $\mathrm{N}_{\widetilde{G}}(\widetilde{L})$ .

Proof

The conditions of the lemma imply that $p=q$ . As in the proof of Lemma 2, let $Z$ denote the inverse image of $\{\pm 1\}$ under the covering homomorphism $\widetilde{G}\rightarrow\mathrm{SO}(p,q)$ , which is a central subgroup of $\widetilde{G}$ and it is contained in $\widetilde{L}$ . Then $\chi|_{Z}\neq\chi^{\prime}|_{Z}$ . Note that $Z$ acts on $I(\chi)$ and $I(\chi^{\prime})$ through the characters $\chi|_{Z}$ and $\chi^{\prime}|_{Z}$ , respectively. Thus $I(\chi)$ and $I(\chi^{\prime})$ have no irreducible subrepresentation in common. In view of Lemma 12, the lemma easily follows by the condition that $\sharp({\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G}))=2$ .

In view of the counting result in Theorem 1.1, it is clear that Lemmas 13 and 14 imply part (a) of Theorem 1.4. Theorem 1.2 is an easy consequence of part (a) of Theorem 1.4. By Clifford theory, Theorem 1.3 is a direct consequence of Theorems 1.1 and 1.2.

4 Genuine special unipotent representations of quaternionic spin groups

In this section, we assume that $G=\mathrm{Spin}^{*}(2n)$ . We will prove part (b) of Theorem 1.4. Recall that $\widetilde{G}=G$ and the set ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(\widetilde{G})$ is assumed to be nonempty. This implies that $n$ is even and $\check{\mathcal{O}}$ is very even. Note that the Young diagram of ${\mathcal{O}}$ is the transpose of that of $\check{\mathcal{O}}$ . Thus ${\mathcal{O}}$ is also very even.

Fix a Cartan involution $\theta$ on $G$ such that its fixed point group, to be denoted by $K$ , is identified with $\widetilde{\operatorname{U}}(n)$ . Here $\widetilde{\operatorname{U}}(n)$ is the double cover of $\operatorname{U}(n)$ given by the square root of the determinant character on $\operatorname{U}(n)$ . The complexification of $K$ , to be denoted by $K_{\mathbb{C}}$ , is identified with $\widetilde{\mathrm{GL}}_{n}(\mathbb{C})$ , the double cover of $\mathrm{GL}_{n}(\mathbb{C})$ given by the square root of the determinant character on $\mathrm{GL}_{n}(\mathbb{C})$ . The category of Casselman-Wallach representations of $G$ is equivalent to the category of $(\mathfrak{g}_{\mathbb{C}},K_{\mathbb{C}})$ -modules of finite length (see (Wa2, , Chapter 11)). By using the trace form, we identify $\mathfrak{g}_{\mathbb{C}}^{*}$ with $\mathfrak{g}_{\mathbb{C}}$ . Denote the Lie algebra of $K_{\mathbb{C}}$ by $\mathfrak{k}_{\mathbb{C}}$ , and write $\mathfrak{s}_{\mathbb{C}}$ for its orthogonal complement in $\mathfrak{g}_{\mathbb{C}}$ under the trace form.

Let $\mathbf{o}^{\prime}$ be a $K_{\mathbb{C}}$ -orbit in $\mathfrak{s}_{\mathbb{C}}\cap{\mathcal{O}}$ . In what follows, we will first construct some auxiliary representations, to be precise irreducible $(\mathfrak{g}_{\mathbb{C}},K_{\mathbb{C}})$ -modules $V_{{\mathbf{o}}^{\prime},N}$ ( $N\in{\mathbb{N}}$ ) with the associated variety $\overline{{\mathbf{o}}^{\prime}}$ .

Let $\set{{\mathrm{e}},{\mathrm{h}},{\mathrm{f}}}$ be an ${\mathfrak{sl}}_{2}$ -triple in $\mathfrak{g}_{\mathbb{C}}$ such that ${\mathrm{h}}\in{\mathfrak{k}}_{\mathbb{C}}$ , ${\mathrm{e}},{\mathrm{f}}\in\mathfrak{s}_{\mathbb{C}}$ , and ${\mathrm{f}}\in{\mathbf{o}}^{\prime}$ . Then we have a decomposition

\mathfrak{g}_{\mathbb{C}}=\bigoplus_{i\in\mathbb{Z}}\mathfrak{g}_{{\mathbb{C}},2i}=\bar{\mathfrak{u}}\oplus\mathfrak{l}^{\prime}_{\mathbb{C}}\oplus\mathfrak{u},

where $\mathfrak{g}_{{\mathbb{C}},2i}$ is the eigenspace of the operator

\mathfrak{g}_{\mathbb{C}}\rightarrow\mathfrak{g}_{\mathbb{C}},\quad x\mapsto[\mathrm{h},x]

with eigenvalue $2i$ , and

\mathfrak{l}^{\prime}_{\mathbb{C}}:=\mathfrak{g}_{{\mathbb{C}},0},\qquad\mathfrak{u}:=\bigoplus_{i=1}^{\infty}\mathfrak{g}_{{\mathbb{C}},2i},\qquad\textrm{and}\qquad\bar{\mathfrak{u}}:=\bigoplus_{i=1}^{\infty}\mathfrak{g}_{{\mathbb{C}},-2i}.

Write ${\mathfrak{q}}=\mathfrak{l}^{\prime}_{\mathbb{C}}\oplus\mathfrak{u}$ . Then ${\mathfrak{q}}$ is a $\theta$ -stable parabolic subalgebra of $\mathfrak{g}_{\mathbb{C}}$ . Note that

\mathfrak{l}^{\prime}_{\mathbb{C}}\cong\mathfrak{g}\mathfrak{l}_{r_{1}}({\mathbb{C}})\times\mathfrak{g}\mathfrak{l}_{r_{2}}({\mathbb{C}})\times\dots\times\mathfrak{g}\mathfrak{l}_{r_{k}}({\mathbb{C}}).

Write $C_{\mathbb{C}}$ for the connected closed subgroup of $K_{\mathbb{C}}$ whose Lie algebra equals $\mathfrak{l}^{\prime}_{\mathbb{C}}\cap\mathfrak{k}_{\mathbb{C}}$ .

An easy calculation shows the following lemma.

Lemma 15

Up to isomorphism, there is a unique one-dimensional $(\mathfrak{l}^{\prime}_{\mathbb{C}},C_{\mathbb{C}})$ -module whose tensor square is isomorphic to $\wedge^{\dim\mathfrak{u}}\mathfrak{u}$ . Moreover, this module is genuine.

Let $\rho_{\mathfrak{u}}$ be a one-dimensional $(\mathfrak{l}^{\prime}_{\mathbb{C}},C_{\mathbb{C}})$ -module as in Lemma 15. View it as a $(\mathfrak{q},C_{\mathbb{C}})$ -module via the trivial action of $\mathfrak{u}$ . Let $\mathcal{R}_{\mathfrak{q}}$ denote the functor from the category of $(\mathfrak{q},C_{\mathbb{C}})$ -modules to the category of $(\mathfrak{g},K_{\mathbb{C}})$ -modules given by

V_{0}\mapsto\mathrm{Hom}_{\mathcal{H}(\mathfrak{q},C_{\mathbb{C}})}(\mathcal{H}(\mathfrak{g},K_{\mathbb{C}}),V_{0})_{K_{\mathbb{C}}-\textrm{fin}}.

Here $\mathcal{H}$ indicates the Hecke algebra, $\mathrm{Hom}_{\mathcal{H}(\mathfrak{q},C_{\mathbb{C}})}(\mathcal{H}(\mathfrak{g},K_{\mathbb{C}}),V_{0})$ is viewed as an (left) $\mathcal{H}(\mathfrak{g},K_{\mathbb{C}})$ -module via its right multiplication on itself, and the subscript ${K_{\mathbb{C}}-\textrm{fin}}$ indicates the $K_{\mathbb{C}}$ -finite parts. See (KV, , Page 105) for more details. The functor $\mathcal{R}_{\mathfrak{q}}$ is left exact, and write $\mathcal{R}_{\mathfrak{q}}^{i}$ ( $i\in\mathbb{Z}$ ) for its $i$ -th derived functor.

For each integer $N$ , put

V_{{\mathbf{o}}^{\prime},N}:=\mathcal{R}_{\mathfrak{q}}^{S}((\rho_{\mathfrak{u}})^{\otimes(2N+1)}),\qquad\textrm{where }S:=\dim(\mathfrak{k}_{\mathbb{C}}\cap\mathfrak{u}).

Write $\pi_{{\mathbf{o}}^{\prime},N}$ for the Casselman-Wallach representation of $G$ corresponding to $V_{{\mathbf{o}}^{\prime},N}$ .

Fix a Cartan subalgebra ${\mathfrak{t}}_{\mathbb{C}}\subset{\mathfrak{k}}_{\mathbb{C}}$ , which is also a Cartan subalgebra of $\mathfrak{g}_{\mathbb{C}}$ . Fix an arbitrary positive system of the root system $\Delta({\mathfrak{t}}_{\mathbb{C}},\mathfrak{l}_{\mathbb{C}}^{\prime})$ , and write $\rho_{\mathfrak{l}_{\mathbb{C}}^{\prime}}\in{\mathfrak{t}}_{\mathbb{C}}^{*}$ for the half sum of the positive roots. By abuse of notation, still write $\rho_{\mathfrak{u}}\in{\mathfrak{t}}^{*}_{\mathbb{C}}$ for the weight of the module $\rho_{\mathfrak{u}}$ .

Lemma 16

When $N\geq 1$ , $\pi_{{\mathbf{o}}^{\prime},N}$ is an irreducible genuine representation of $G$ with infinitesimal character $\rho_{\mathfrak{l}_{\mathbb{C}}^{\prime}}+2N\rho_{\mathfrak{u}}$ and associated variety $\overline{{\mathbf{o}}^{\prime}}$ .

Proof

Write $\rho:=\rho_{\mathfrak{l}^{\prime}_{\mathbb{C}}}+\rho_{\mathfrak{u}}$ . By (Knapp, , Corollary 5.100), for every weight $\alpha\in\mathfrak{t}_{\mathbb{C}}^{*}$ of ${\mathfrak{u}}$ ,

\left\langle{\rho},{\check{\alpha}}\right\rangle>0\qquad\textrm{and}\qquad\left\langle{\rho_{\mathfrak{u}}},{\check{\alpha}}\right\rangle>0.

Here $\check{\alpha}\in\mathfrak{t}_{\mathbb{C}}$ is the coroot corresponding to $\alpha$ . Then

\langle\rho_{\mathfrak{l}_{\mathbb{C}}^{\prime}}+2N\rho_{\mathfrak{u}},\check{\alpha}\rangle>0,

and hence the $({\mathfrak{l}}^{\prime},C_{\mathbb{C}})$ -module $(\rho_{\mathfrak{u}})^{\otimes(2N-1)}$ is in the good range (see (KV, , Definition 0.4.9) for the definition of good range). So $\pi_{{\mathbf{o}}^{\prime},N}$ is irreducible since cohomological induction in the good range preserves irreducibility (see (KV, , Theorem 8.2)).

By a result of Barbasch-Vogan (BV.W, , Proposition 3.4), the associated variety of $\pi_{{\mathbf{o}}^{\prime},N}$ equals $\overline{K_{\mathbb{C}}\cdot({\mathfrak{s}}_{\mathbb{C}}\cap\bar{\mathfrak{u}})}$ (the Zariski closure). See also (Ko, , Lemma 2.7) and (Tr, , Proposition 5.4). Note that $\overline{\mathcal{O}}\subset\overline{G_{\mathbb{C}}\cdot\bar{\mathfrak{u}}}$ since $f\in\bar{\mathfrak{u}}$ . Then by (CM, , Theorem 7.3.3), we conclude that

\overline{\mathcal{O}}=\overline{G_{\mathbb{C}}\cdot\bar{\mathfrak{u}}}.

Similarly, $\overline{{\mathbf{o}}^{\prime}}\subset\overline{K_{\mathbb{C}}\cdot({\mathfrak{s}}_{\mathbb{C}}\cap\bar{\mathfrak{u}})}$ since $f\in{\mathfrak{s}}_{\mathbb{C}}\cap\bar{\mathfrak{u}}$ . Then by (Vo89, , Corollary 5.20), we conclude that

\overline{K_{\mathbb{C}}\cdot({\mathfrak{s}}_{\mathbb{C}}\cap\bar{\mathfrak{u}})}=\overline{{\mathbf{o}}^{\prime}}.

The computation of the infinitesimal character of $\pi_{{\mathbf{o}}^{\prime},N}$ is straightforward (see (KV, , Proposition 0.48)). This finishes the proof.

Lemma 17

There exists a representation in ${\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)$ whose associated variety equals $\overline{{\mathbf{o}}^{\prime}}$ .

Proof

We follow an idea of Barbasch-Vogan in BV.W . Retain the notation in Section 2. Recall that ${\check{\mathcal{O}}}$ is very even. Let $\Lambda_{2}=\lambda_{\check{\mathcal{O}}}+Q_{0}\subset\mathfrak{h}^{*}$ . It is easy to see that $W(\Lambda_{2})=\mathsf{W}^{\prime}_{n}$ equals the abstract Weyl group $W$ of $G_{\mathbb{C}}$ and $\Set{\tau_{b}}$ is the double cell of $\mathrm{Irr}(W)$ associated to the orbit $\mathcal{O}$ via the Springer correspondence. See (9).

For each $K_{\mathbb{C}}$ -stable closed subset $S$ in ${\mathfrak{s}}_{\mathbb{C}}$ , consider the $\mathsf{W}^{\prime}_{n}$ -submodule

\mathrm{Coh}_{\Lambda_{2},S}(\mathcal{K}^{\mathrm{gen}}(G)):=\Set{\Psi\in\mathrm{Coh}_{\Lambda_{2}}(\mathcal{K}^{\mathrm{gen}}(G))}{\mathrm{AV}(\Psi(\lambda))\subset S\text{ for all $\lambda\in\Lambda_{2}$}}

of the coherent continuation representation $\mathrm{Coh}_{\Lambda_{2}}(\mathcal{K}^{\mathrm{gen}}(G))$ . Here $\mathrm{AV}(\Psi(\lambda))$ denotes the union of associated varieties of irreducible genuine representations of $G$ occurring in $\Psi(\lambda)$ with non-zero coefficient.

Let $\Delta^{+}(\mathfrak{t}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})\subset\mathfrak{t}_{\mathbb{C}}^{*}$ denote the union of $\Delta^{+}(\mathfrak{t}_{\mathbb{C}},\mathfrak{l}^{\prime}_{\mathbb{C}})$ and the weights of ${\mathfrak{u}}$ , which is a positive system of the root system $\Delta(\mathfrak{t}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$ . Using this positive system, we identify $\mathfrak{t}_{\mathbb{C}}$ with the abstract Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}_{\mathbb{C}}$ . Note that $\rho_{{\mathfrak{l}}_{\mathbb{C}}^{\prime}}+2\rho_{{\mathfrak{u}}}$ belongs to $\Lambda_{2}$ and is regular in ${\mathfrak{t}}_{\mathbb{C}}^{*}$ .

Let $\Psi_{1}$ be the element in $\mathrm{Coh}_{\Lambda_{2}}(\mathcal{K}^{\mathrm{gen}}(G))$ such that $\Psi_{1}(\rho_{{\mathfrak{l}}_{\mathbb{C}}^{\prime}}+2\rho_{\mathfrak{u}})=\pi_{{\mathbf{o}}^{\prime},1}$ . Then $\Psi_{1}\in\mathrm{Coh}_{\Lambda_{2},\overline{{\mathbf{o}}^{\prime}}}(\mathcal{K}^{\mathrm{gen}}(G))$ by Lemma 16.

Let ${\mathbb{C}}[\mathfrak{h}^{*}]$ denote the space of polynomial functions on $\mathfrak{h}^{*}$ . By a result of Barbasch-Vogan and King (King, , Theorem 1.2), there is a well-defined $W$ -equivariant linear map

\mathrm{Coh}_{\Lambda_{2},\overline{{\mathbf{o}}^{\prime}}}(\mathcal{K}^{\mathrm{gen}}(G))\rightarrow{\mathbb{C}}[\mathfrak{h}^{*}]

sending $\Psi_{1}$ to the Goldie rank polynomial of the annihilator ideal of $V_{\overline{{\mathbf{o}}^{\prime}},1}$ . By a result of Joseph (Jo85, , 2.10), the Goldie rank polynomial generates an irreducible $W$ -module which is isomorphic to $\tau_{b}$ . Therefore $\tau_{b}$ occurs in the $W$ -module $\mathrm{Coh}_{\Lambda_{2},\overline{{\mathbf{o}}^{\prime}}}(\mathcal{K}^{\mathrm{gen}}(G))$ .

Write $W_{\lambda_{\check{\mathcal{O}}}}$ for the stabilizer of $\lambda_{\check{\mathcal{O}}}$ in $W$ , and $1_{W_{\lambda_{\check{\mathcal{O}}}}}$ for the trivial representation of $W_{\lambda_{\check{\mathcal{O}}}}$ . By (BVUni, , Corollary 5.30), $1_{W_{\lambda_{\check{\mathcal{O}}}}}$ occurs in $\tau_{b}=(j_{W_{\lambda_{\check{\mathcal{O}}}}}^{W}\operatorname{sgn})\otimes\operatorname{sgn}$ with multiplicity-one. Here $j$ stands for the $j$ -operation, as in (Carter, , Section 11.2). We therefore conclude that

\begin{split}&\sharp\Set{\pi\in{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)}{\mathrm{AV}(\pi)=\overline{{\mathbf{o}}^{\prime}}}\\ =&\,\sharp\Set{\pi\in{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)}{\mathrm{AV}(\pi)\subset\overline{{\mathbf{o}}^{\prime}}}\qquad\text{(by \cite[cite]{(\@@bibref{AuthorsPhrase1Year}{Vo89}{\@@citephrase{, }}{}, Theorem~{}8.4)})}\\ =&\sum_{\sigma\in\mathrm{Irr}(W)}[1_{W_{\lambda_{\check{\mathcal{O}}}}}:\sigma][\sigma:\mathrm{Coh}_{\Lambda_{2},\overline{{\mathbf{o}}^{\prime}}}(\mathcal{K}^{\mathrm{gen}}(G))]\quad\quad\text{(by \cite[cite]{(\@@bibref{AuthorsPhrase1Year}{BMSZ1}{\@@citephrase{, }}{}, Proposition~{}5.1)})}\\ \geq&[\tau_{b}:\mathrm{Coh}_{\Lambda_{2},\overline{{\mathbf{o}}^{\prime}}}(\mathcal{K}^{\mathrm{gen}}(G))]\geq 1.\end{split}

Here $\mathrm{AV}$ stands for the associated variety. This completes the proof.

Together with the counting result in Theorem 1.1, the above proposition implies that there is a unique representation $\pi_{\mathbf{o}^{\prime}}\in{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)$ whose associated variety equals $\overline{{\mathbf{o}}^{\prime}}$ , and

{\mathrm{Unip}^{\mathrm{gen}}_{\check{\mathcal{O}}}}(G)=\{\pi_{\mathbf{o}^{\prime}}\mid\mathbf{o}^{\prime}\in K_{\mathbb{C}}\backslash(\mathfrak{s}_{\mathbb{C}}\cap{\mathcal{O}})\}.

Recall that the wavefront cycle agrees with the weak associated cycle (SV ). By using Lemma 11 and considering the wavefront cycle, we conclude that

I(\chi)\cong\bigoplus_{\mathbf{o}^{\prime}\in K_{\mathbb{C}}\backslash(\mathfrak{s}_{\mathbb{C}}\cap{\mathcal{O}})}\pi_{\mathbf{o}^{\prime}}.

This easily implies part (b) of Theorem 1.4.

Remark 2

In MT , Matumoto and Trapa consider degenerate principal series representations of the linear group $\mathrm{Sp}(p,q),\mathrm{SO}^{*}(2n),$ or ${\mathrm{U}}(m,n)$ with integral infinitesimal character. In particular, they prove that each irreducible constituent of maximal Gelfand–Kirillov dimension is a derived functor module.

Acknowledgements.

D. Barbasch is supported by NSF grant, Award Number 2000254. J.-J. Ma is supported by the National Natural Science Foundation of China (Grant No. 11701364 and Grant No. 11971305) and Xiamen University Malaysia Research Fund (Grant No. XMUMRF/2022-C9/IMAT/0019). B. Sun is supported by National Key R & D Program of China (No. 2022YFA1005300 and 2020YFA0712600) and the New Cornerstone Science Foundation. C.-B. Zhu is supported by MOE AcRF Tier 1 grant R-146-000-314-114, and Provost’s Chair grant E-146-000-052-001 in NUS. C.-B. Zhu is grateful to Max Planck Institute for Mathematics in Bonn, for its warm hospitality and conducive work environment, where he spent the academic year 2022/2023 as a visiting scientist. Some statements in the paper are (additionally) verified for low rank groups with the atlas software. J.-J. Ma thanks J. Adams for patiently answering questions on the atlas software.

References

(1) Adams, J., Barbasch, D. and Vogan, D. A.: The Langlands classification and irreducible characters for real reductive groups, Progr. Math., vol. 104, Birkhauser, 1991.
(2) Adams, J., Arancibia Robert, N. and Mezo, P.: Equivalent definitions of Arthur packets for real classical groups, arXiv:2108.05788.
(3) Arancibia Robert, N. and Mezo, P.: Equivalent definitions of Arthur packets for real unitary groups, arXiv:2204.19715.
(4) Arthur, J.: Unipotent automorphic representations: conjectures, Orbites unipotentes et représentations, II, Astérisque (171-172), 13–71, 1989.
(5) Arthur, J.: The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Amer. Math. Soc. Colloq. Publ., vol. 61, Amer. Math. Soc., Providence, RI, 2013.
(6) Barbasch, D.: The unitary dual for complex classical Lie groups, Invent. Math. (96), no. 1, 103–176, 1989.
(7) Barbasch, D.: Orbital integrals of nilpotent orbits, The mathematical legacy of Harish-Chandra, Proc. Sympos. Pure Math. no. 68, 97–110, Amer. Math. Soc., Providence, RI, 2000.
(8) Barbasch, D.: Unipotent representations and the dual pair correspondence, J. Cogdell et al. (eds.), Representation Theory, Number Theory, and Invariant Theory, In Honor of Roger Howe. Progr. Math., vol. 323, 47–85, 2017.
(9) Barbasch, D., Ma, J.-J., Sun, B. and Zhu, C.-B.: On the notion of metaplectic Barbasch-Vogan duality, arXiv:2010.16089. To appear in Int. Math. Res. Not. IMRN.
(10) Barbasch, D., Ma, J.-J., Sun, B. and Zhu, C.-B.: Special unipotent representations of real classical groups: counting and reduction, arXiv:2205.05266.
(11) Barbasch, D., Ma, J.-J., Sun, B. and Zhu, C.-B.: Special unipotent representations of real classical groups: construction and unitarity, arXiv:1712.05552.
(12) Barbasch, D. and Vogan, D. A.: Weyl Group Representations and Nilpotent Orbits, Representation Theory of Reductive Groups: Proceedings of the University of Utah Conference (1982), 21–33, Birkhäuser Boston, 1983.
(13) Barbasch, D. and Vogan, D. A.: Unipotent representations of complex semisimple groups, Annals of Math. (121), no. 1, 41–110, 1985.
(14) Brega, A.: On the unitary dual of $\mathrm{Spin}(2n,{\mathbb{C}})$ , Trans. AMS, vol. 351, no. 1, 403-415, 1999.
(15) Carter, R. W.: Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993.
(16) Collingwood, D. H. and McGovern, W. M.: Nilpotent orbits in semisimple Lie algebra: an introduction, Van Nostrand Reinhold Co., 1993.
(17) Howe, R.: $\theta$ -series and invariant theory, Automorphic Forms, Representations and $L$ -functions, Proc. Sympos. Pure Math., vol. 33, 275–285, 1979.
(18) Howe, R.: Transcending classical invariant theory, J. Amer. Math. Soc. (2), 535–552, 1989.
(19) Joseph, A.: On the associated variety of a primitive ideal, J. Algebra 93 (1985), no. 2, 509–523
(20) King, D.: The character polynomial of the annihilator of an irreducible Harish-Chandra module, Amer. J. Math. (103), 1195–-1240, 1981.
(21) Knapp, A. W.: Lie groups beyond an introduction. Second edition. Progr. Math., vol. 140, Birkhäuser Boston, Inc., Boston, Mass., 2002. xviii+812 pp.
(22) Knapp, A. W. and Vogan, D. A.: Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995.
(23) Kobayashi, T: Discrete decomposability of the restriction of $A_{\mathfrak{q}}(\lambda)$ with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties. Invent. Math. (131), no. 2, 229–-256, 1998.
(24) Losev, I., Mason-Brown, L. and Matvieievskyi, D.: Unipotent ideals and Harish-Chandra bimodules, arXiv:2108.03453.
(25) Lusztig, G.: Characters of reductive groups over a finite field, Ann. of Math. Stud., vol. 107, Princeton University Press, 1984.
(26) Matumoto, H. and Trapa, P. E.: Derived functor modules arising as large irreducible constituents of degenerate principal series, Compos. Math. (143), no. 1, 222–-256, 2007.
(27) Schmid, W.: Two character identities for semisimple Lie groups, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Lecture Notes in Math., Vol. 587, 196–225, Springer, Berlin, 1977.
(28) Schmid, W. and Vilonen, K.: Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math. (151), no. 3, 1071–1118, 2000.
(29) Speh, B. and Vogan, D. A.: Reducibility of generalized principal series representations, Acta Math. (145), no. 3–4, 227–299, 1980.
(30) Trapa, P. E.: Annihilators and associated varieties of $A_{\mathfrak{q}}(\lambda)$ modules for $\operatorname{U}(p,q)$ , Compos. Math. (129), no. 1, 1–-45, 2001.
(31) Vogan, D. A.: Representations of real reductive Lie groups, Progr. Math., vol. 15, Birkhäuser, Boston, Mass., 1981.
(32) Vogan, D. A.: Associated varieties and unipotent representations, Harmonic analysis on reductive groups, edited by W. Barker and P. Sally, Progr. Math., vol. 101, 315–388, Birkhäuser, Boston-Basel-Berlin, 1991.
(33) Wallach, N. R.: Real reductive groups II, Academic Press Inc., 1992.
(34) Wong, D. and Zhang, H.: The genuine unitary dual of $\mathrm{Spin}(2n,\mathbb{C})$ , arXiv: 2302.10803, 2023.
(35) Zuckerman, G.: Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math. (106), 295–309, 1977.

Genuine special unipotent representations of spin groups

Abstract

1 Introduction and the main results

Theorem 1.1

Theorem 1.2

Theorem 1.3

Lemma 1

Lemma 2

Proof

Theorem 1.4

Theorem 1.5

Theorem 1.6

Remark 1

Theorem 1.7

2 Proof of Theorem 1.1

2.1 Coherent continuation representation of a spin group

Definition 1

Lemma 3

Proof

Proposition 1

Proof

2.2 Real odd spin groups

Lemma 4

Proof

Lemma 5

Proof

2.3 Real even spin groups and quaternionic spin groups

Lemma 6

The case when G0=SO​(p,q)G_{0}=\mathrm{SO}(p,q), p+q=2​np+q=2n

Lemma 7

Proof

Lemma 8

Proof

2.4 The case when G0=SO∗​(2​n)G_{0}=\mathrm{SO}^{*}(2n)

Lemma 9

Proof

Lemma 10

Proof

3 Genuine special unipotent representations of real spin groups

Lemma 11

Proof

Lemma 12

Proof

Lemma 13

Proof

Lemma 14

Proof

4 Genuine special unipotent representations of quaternionic spin groups

Lemma 15

Lemma 16

Proof

Lemma 17

Proof

Remark 2

Acknowledgements.

References

The case when $G_{0}=\mathrm{SO}(p,q)$ , $p+q=2n$

2.4 The case when $G_{0}=\mathrm{SO}^{*}(2n)$