A characterization of unitarity of some highest weight Harish-Chandra modules

Let $G_{\mathbb{R}}$ be a connected non-compact simple Lie group with finite center, and let $K_{\mathbb{R}}$ be a maximal compact subgroup. From the work of Harish-Chandra (see comments in [BHXZ, §3.2]), it follows that infinite-dimensional highest weight Harish-Chandra modules for $G_{\mathbb{R}}$ exist if and only if ($G_{\mathbb{R}}$, $K_{\mathbb{R}}$) is a Hermitian symmetric pair. The problem of determining when a highest weight Harish-Chandra module is unitarizable has been extensively studied by various authors (see, for example, the references in [EHW]). The full classification was independently completed in [EHW] and [Ja1], though the classification itself is rather intricate. In this paper, we provide a simple and uniform characterization of unitarity for Harish-Chandra modules with a given associated variety, expressed in terms of the highest weight (see Theorem 1.1).

From now, we assume that $(G_{\mathbb{R}},K_{\mathbb{R}})$ is a Hermitian symmetric pair. We denote by $K$ the complexification of the compact group $K_{\mathbb{R}}$ and by $(\mathfrak{g},\mathfrak{k})$ the complexified Lie algebras of $(G_{\mathbb{R}},K_{\mathbb{R}})$. Then we have the usual decompositition $\mathfrak{g}=\mathfrak{p}^{-}\oplus\mathfrak{k}\oplus\mathfrak{p}^{+}$ of $\mathfrak{g}$ as a $K$-representation. Let $\mathfrak{h}\subseteq\mathfrak{k}$ be a Cartan subalgebra. Then $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$. Let $\Delta$ and $\Delta(\mathfrak{k})$ denote the root systems of $(\mathfrak{g},\mathfrak{h})$ and $(\mathfrak{k},\mathfrak{h})$, respectively. Let $\Delta^{+}$ be the positive system of $\Delta$, and define $\Delta^{+}(\mathfrak{k})=\Delta(\mathfrak{k})\cap\Delta^{+}$ and $\Delta(\mathfrak{p}^{+})=\Delta^{+}\setminus\Delta^{+}(\mathfrak{k})$. Let $\beta$ denote the unique maximal noncompact root of $\Delta^{+}$. Now choose $\zeta\in\mathfrak{h}^{*}$ so that $\zeta$ is orthogonal to $\Delta(\mathfrak{k})$ and ($\zeta,\beta^{\vee}$)=1. Let $\lambda\in\mathfrak{h}^{*}$ be $\Delta^{+}(\mathfrak{k})$-dominant integral and $F(\lambda)$ be the irreducible $\mathfrak{k}$-module with highest weight $\lambda$. By letting the nilradical act by zero, we may consider $F(\lambda)$ as a module of the parabolic subalgebra $\mathfrak{q}=\mathfrak{k}+\mathfrak{p}^{+}$. Then we define:

Let $L(\lambda)$ denote the irreducible quotient of $N(\lambda)$, which is a highest weight module of $\mathfrak{g}$.

From [EHW], $L(\lambda)$ is a highest weight Harish-Chandra module if and only if $\lambda\in\Lambda^{+}(\mathfrak{k})$, where

Write $\rho$ for half the sum of positive roots in $\Delta^{+}$. Then we can write $\lambda=\lambda_{0}+z\zeta$, with $\lambda_{0}\in\mathfrak{h}^{*}$ such that ($\lambda_{0}+\rho,\beta$)=0, and $z=(\lambda+\rho,\beta^{\vee})\in\mathbb{R}$.

The associated variety of a highest weight Harish-Chandra module is known to be the closure of a single $K$-orbit in $\mathfrak{p}^{+}$ (see §3 for more details). Furthermore, the closures of the $K$-orbits in $\mathfrak{p}^{+}$ form a linear chain of varieties

where $r$ is the $\mathbb{R}$-rank of $G_{\mathbb{R}}$, i.e., the dimension of a Cartan subgroup of the group $G_{\mathbb{R}}$, which is also equal to the rank of the symmetric space $G_{\mathbb{R}}/K_{\mathbb{R}}$. Therefore, if $L(\lambda)$ is a highest weight Harish-Chandra module with highest weight $\lambda$, then there is an integer $0\leq k(\lambda)\leq r$ such that the associated variety of $L(\lambda)$ is $\overline{\mathcal{O}_{k(\lambda)}}$.

Denote

for $0\leq k\leq r$. Here $c$ is a real number associated with the Hermitian type Lie group $G_{\mathbb{R}}$, see Table 1.

In this paper, we will prove the following result.

In [BH], we have found a uniform formula for the Gelfand–Kirillov dimensions of unitary highest weight Harish-Chandra modules. Now we want to generalize our formula to all highest weight Harish-Chandra modules.

Here $\left\lceil c\right\rceil$ denotes the smallest integer $n$ such that $n\geq c$.

In the special case when $\lambda_{0}=-(\rho,\beta^{\vee})\zeta$, we also write $z_{k}$ instead of $z_{k}(\lambda_{0})$ (This coincides with our definition of $z_{k}$ in (1.1)).

Our new formula for the Gelfand–Kirillov dimensions of all highest weight Harish-Chandra modules is as follows.

Note that for any integer $1\leq h\leq\operatorname{ht}(\beta)$, the set

is an antichain in $\Delta(\mathfrak{p}^{+})$.

Let $\Pi$ denote the set of simple roots in $\Delta^{+}$.

In light of this lemma, the Hasse diagram of $\Delta(\mathfrak{p^{+}})$ is an upward planar graph of order dimension two and hence can be drawn on a two-dimensional orthogonal lattice that has been rotated by a $45$-degree angle.

The antichains in $\Delta(\mathfrak{p}^{+})$ are given in Appendix.

A subset $Y\subseteq\Delta({\mathfrak{p}}^{+})$ is called a lower-order ideal if, for $\alpha\in\Delta({\mathfrak{p}}^{+})$ and $\beta\in Y$, $\alpha\leq\beta$ implies that $\alpha\in Y$.

By [BHXZ, Lem 2.2], the poset $Y_{\lambda}$ is a lower order ideal of $\Delta({\mathfrak{p}}^{+})$ when $\lambda$ is integral.

An antichain in a poset is a subset consisting of pairwise noncomparable elements. The width of a poset is the cardinality of maximal antichain in the poset. Suppose $\lambda\in\Lambda^{+}(\mathfrak{k})$, we use $m=m(\lambda)$ to denote the width of $Y_{\lambda}$.

By inspection of the Hasse diagram of $\Delta(\mathfrak{p}^{+})$, we have the following lemma.

Recall that $A_{k}=\Delta(\mathfrak{p}^{+})_{k\left\lceil c\right\rceil+1}$ is an antichain in $\Delta(\mathfrak{p}^{+})$.

In this section, we will recall some preliminaries on Gelfand–Kirillov dimensions and associated varieties of highest weight modules. See [Vo78, Vo91] for more details.

Let $M$ be a finite generated $U(\mathfrak{g})$-module. Fix a finite dimensional generating space $M_{0}$ of $M$. Let $U_{n}(\mathfrak{g})$ be the standard filtration of $U(\mathfrak{g})$. Set $M_{n}=U_{n}(\mathfrak{g})\cdot M_{0}$ and $\text{gr}(M)=\bigoplus\limits_{n=0}^{\infty}\text{gr}_{n}M,$ where $\text{gr}_{n}M=M_{n}/{M_{n-1}}$. Thus $\text{gr}(M)$ is a graded module of $\text{gr}(U(\mathfrak{g}))\simeq S(\mathfrak{g})$.

The Gelfand–Kirillov dimension of $M$ is defined by

The associated variety of $M$ is defined by

These two definitions are independent of the choice of $M_{0}$, and $\dim V(M)=\operatorname{GKdim}M$ (e.g., [NOT]). If $M_{0}$ is $\mathfrak{a}$-invariant for a subalgebra $\mathfrak{a}\subset\mathfrak{g}$, then

When $M=L(\lambda)$ is a highest weight Harish-Chandra module, we can choose $M_{0}$ to be the finite dimensional $U(\mathfrak{k})$-module generated by $\mathbb{C}_{\lambda}$. Then $M_{0}$ is $\mathfrak{k}\oplus\mathfrak{p}^{+}$-invariant. In view of (3.1),

where the last isomorphism is induced from the Killing form. As shown in [Vo91], the associated variety $\mathrm{AV}(M)$ is also $K$-invariant. In fact, Yamashita [Hir01] proved that $\mathrm{AV}(M)$ must be one of $\overline{\mathcal{O}_{k}}$.

We have the following table from [EHW]:

In [BH], we have found a uniform expression for the GK dimensions and associated varieties of unitary highest weight Harish-Chandra modules.

In this section, we assume that $\Delta$ is simply-laced.

For any integer $0\leq k\leq r-1$, $A_{k}$ is the antichain in $\Delta(\mathfrak{p}^{+})$ such that $|A_{k}|=k+1$ and $A_{k}=\Delta(\mathfrak{p}^{+})_{h}$ with $h$ minimal.

To prove our Theorem 1.1, we need the following useful lemma.

Now we can prove our Theorem 1.1. The idea is very simple. From Lemma 2.5, since $\mathrm{AV}(L(\lambda))=\overline{\mathcal{O}_{k}}$, we have $(\lambda+\rho,\alpha^{\vee})>0$ for some $\alpha\in A_{k}=\Delta(\mathfrak{p^{+}})_{k\left\lceil c\right\rceil+1}$. Then $\operatorname{ht}(\alpha)=kc+1$. So if $(\lambda+\rho,\beta^{\vee})=z=z_{k}=(\rho,\beta^{\vee})-kc$, we will have

This condition is very restrictive. In the following, we will give a case-by-case discussion for this condition, which will imply the unitarity of $L(\lambda)$.