On the Associated Variety of a Highest Weight Harish-Chandra Module

Abstract

1 Introduction

1.1 Highest weight Harish-Chandra modules

Let |$G_{\mathbb{R}}$| be a simple Lie group. It follows from the work of Harish-Chandra [14, 15] that there exist infinite-dimensional highest weight Harish-Chandra modules for |$G_{\mathbb{R}}$| if and only if |$G_{\mathbb{R}}$| is of Hermitian type. Our main result, Theorem 1.2, is a very simple formula for the associated varieties of highest weight Harish-Chandra modules. Therefore, we assume that |$G_{\mathbb{R}}$| is of Hermitian type. We also assume that |$G_{\mathbb{R}}$| has finite center.

Standard Lie theory tells us that |${{\mathfrak{g}}}_{\mathbb{R}}={\mathscr Lie}(G_{\mathbb{R}})$| has a Cartan decomposition |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{k}}}_{\mathbb{R}}\oplus{{\mathfrak{p}}}_{\mathbb{R}}$| and the subgroup |$K_{\mathbb{R}}$| of |$G_{\mathbb{R}}$| with Lie algebra |${{\mathfrak{k}}}_{\mathbb{R}}$| is compact. Therefore, |$K_{\mathbb{R}}$| has a complexification |$K$|⁠. Writing the complexified Cartan decomposition as |${{\mathfrak{g}}}={{\mathfrak{k}}}\oplus{{\mathfrak{p}}}$| (since |$G_{\mathbb{R}}$| is of Hermitian type) there is a decomposition |${{\mathfrak{p}}}={{\mathfrak{p}}}^{+}\oplus{{\mathfrak{p}}}^{-}$| into nonzero irreducible |$K$|-subrepresentations. This gives a triangular decomposition |${{\mathfrak{g}}}={{\mathfrak{p}}}^{-}\oplus{{\mathfrak{k}}}\oplus{{\mathfrak{p}}}^{+}$| and a maximal parabolic subalgebra |${{\mathfrak{q}}}={{\mathfrak{k}}}\oplus{{\mathfrak{p}}}^{+}$|⁠. It follows from Harish-Chandra that a highest weight Harish-Chandra module is the irreducible quotient of a Verma module and the highest weight |$\lambda $| is integral for the roots in |${{\mathfrak{k}}}$| (with respect to a Cartan subalgebra |${{\mathfrak{h}}}$| of |${{\mathfrak{g}}}$| contained in |${{\mathfrak{k}}}$|⁠). See Section 3.2 for precisely which Verma modules have irreducible quotients occurring as highest weight Harish-Chandra modules.

1.2 Associated varieties

The associated variety of a |$({{\mathfrak{g}}},K)$|-module, as defined in [26], is a union of (closures of) nilpotent |$K$|-orbits in |$({{\mathfrak{g}}}/{{\mathfrak{k}}})^{*}$|⁠. When |$L(\lambda )$| is a highest weight Harish-Chandra module for |$G_{\mathbb{R}}$|⁠, the associated variety is contained in |$({{\mathfrak{g}}}/{{\mathfrak{q}}})^{*}\simeq{{\mathfrak{p}}}^{+}$|⁠. It is well known that the closures of the |$K$|-orbits in |${{\mathfrak{p}}}^{+}$| form a chain

where |$r$| is the real rank of |${{\mathfrak{g}}}_{\mathbb{R}}$| (⁠|$=\operatorname{rank}(G_{\mathbb{R}}/K_{\mathbb{R}})$|⁠). It follows that

The purpose of this paper is to express |$k=k(\lambda )$| in terms of the highest weight |$\lambda $| in a simple way that is uniform across Cartan types and is valid for arbitrary infinitesimal character. In the special case when |$L(\lambda )$| is the Harish-Chandra module of a unitary representation of |$G_{\mathbb{R}}$| this was accomplished in [2] by using previous work of Joseph [20]. However, a uniform formula for arbitrary highest weight Harish-Chandra modules has remained elusive.

1.3 Statement of the main result

To state our main result precisely, we need some notation for sets of roots. Let |$\Delta $| and |$\Delta ({{\mathfrak{k}}})$| denote the root systems of |$({{\mathfrak{g}}},{{\mathfrak{h}}})$| and |$({{\mathfrak{k}}},{{\mathfrak{h}}})$|⁠, respectively. Let |$\Delta ^{+}$| be a positive system of |$\Delta $| containing |$\Delta ({{\mathfrak{p}}}^{+})$| and define |$\Delta ^{+}({{\mathfrak{k}}})=\Delta ({{\mathfrak{k}}})\cap \Delta ^{+}$|⁠. Let |$\Lambda ^{+}({{\mathfrak{k}}})$| denote the lattice of integral and dominant weights for |$\Delta ({{\mathfrak{k}}})$|⁠. We view all sets of roots as partially ordered sets via the usual partial ordering, where |$\alpha \leq \beta $| means that |$\beta - \alpha $| is nonnegative integer linear combination of positive roots.

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. Our main result states that the associated variety of a highest weight Harish-Chandra module |$L(\lambda )$| is completely determined by the width of |$Y_{\lambda }$|⁠.

At first glance, it may seem difficult to compute |$\operatorname{width}(Y_{\lambda })$|⁠. However, this is not the case.

Example 1.3.

Suppose |$G_{\mathbb{R}}$| is the metaplectic double cover of the symplectic group |$Sp(2n,{\mathbb{R}})$| and |$L(\lambda )$| is a highest weight Harish-Chandra module with half-integral highest weight |$\lambda $|⁠. Using the conventions for roots and weights as in [12], we then have |$\lambda +\rho =(t_{1},\ldots ,t_{n})$|⁠, where the entries |$t_{i}$| are half-integers such that |$t_{1}> \cdots >t_{n}$|⁠. Since |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{i} +\varepsilon _{j} \mid 1\leq i < j\leq n\}\cup \{2\varepsilon _{i} \mid 1\leq i\leq n\}$| and two short roots |$\varepsilon _{i_{1}}+\varepsilon _{j_{1}}, \varepsilon _{i_{2}}+\varepsilon _{j_{2}}\in \Delta ({{\mathfrak{p}}}^{+})$| are incomparable if and only if either |$i_{1}<i_{2}<j_{2}<j_{1}$| or |$i_{2}<i_{1}<j_{1}<j_{2}$|⁠, it follows that |$\operatorname{width}(Y_{\lambda })$| is the maximal nonnegative integer |$m$| for which there exists a sequence of indices |$1\leq i_{1}< i_{2} < \cdots < i_{m} < j_{m} < \cdots < j_{2}< j_{1}\leq n$| such that

$$ \begin{align*} & t_{i_1}+t_{j_1}\leq 0\,\ t_{i_2}+t_{j_2}\leq 0\,\ \ldots\, \ t_{i_m}+t_{j_m}\leq 0. \end{align*} $$

Since the sequence |$(t_{1},\ldots ,t_{n})$| is strictly decreasing, we may furthermore assume that |$j_{k}=n-k+1$| for |$k=1,\ldots , m$|⁠. This leads to the following simple algorithm to find |$\operatorname{width}(Y_{\lambda })$| with a running time that is linear in |$n$|⁠, which is optimal. Set |$j_{1}=n$| and let |$i_{1}$| be the smallest index such that |$i_{1}<j_{1}$| and |$t_{i_{1}}+t_{j_{1}}\leq 0$|⁠. If no such |$i_{1}$| exists, then |$m=0$|⁠. Next set |$j_{2}=n-1$| and let |$i_{2}$| be the smallest index such that |$i_{1}<i_{2}<j_{2}$| and |$t_{i_{2}}+t_{j_{2}}\leq 0$|⁠. If no such |$i_{2}$| exists, then |$m=1$|⁠. Continue doing this until it is no longer possible which happens after at most |$[n/2]$| steps. For a concrete example, suppose

Here brackets were drawn from the |$j_{1}$|-th entry to the |$i_{1}$|-th entry, from the |$j_{2}$|-th entry to the |$i_{2}$|-th entry, and from the |$j_{3}$|-th entry to the |$i_{3}$|-th entry. We see that |$\operatorname{width}(Y_{\lambda })=3$| and the theorem says that |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{7}}$|⁠.

1.4 Gelfand–Kirillov dimension

Our main result has another interpretation in terms of the Gelfand–Kirillov (GK) dimension of Harish-Chandra modules. The GK dimension of a Harish-Chandra module is equal to the dimension of its associated variety. Since the associated variety of a highest weight Harish-Chandra module |$L(\lambda )$| is one of the varieties in the chain (1.1), it follows that the associated variety of |$L(\lambda )$| is determined by the GK dimension of |$L(\lambda )$|⁠. The following theorem gives |$\operatorname{GKdim}(L(\lambda ))$| when |$\lambda $| is nonintegral. One sees that in this case |$\Delta _{\lambda }({{\mathfrak{p}}}^{+})$| is either empty or is isomorphic to a poset |$\Delta ({{\mathfrak{p}}}^{\prime +})$| corresponding to a noncompact Hermitian symmetric space |$G_{\mathbb{R}}^{\prime}/K_{\mathbb{R}}^{\prime}$|⁠, which in fact has simply laced root system. This follows from Cartan’s classification of Hermitian symmetric spaces and the work of Jakobsen [17] on the posets of noncompact roots (see the diagrams in [17, Appendix]).

Let |${{\mathcal{O}}}_{k}^{\prime}, k=0,1,\dots ,r^{\prime}$|⁠, be the |$K^{\prime}$|-orbits in |${{\mathfrak{p}}}^{\prime +}$| (analogous to the chain described in 1.1). Set |$\delta =\#(\Delta ^{+})-\#(\Delta _{\lambda }^{+})$|⁠, which equals |$\#(\Delta ({{\mathfrak{p}}}^{+}))-\#(\Delta _{\lambda }({{\mathfrak{p}}}^{+}))$| since |$\lambda $| is |$\Delta ({{\mathfrak{k}}})$|-integral.

It turns out that |$\Delta _{\lambda }({{\mathfrak{p}}}^{+})$| is nonempty (for nonintegral |$\lambda $|⁠) only when |$\Delta $| is non-simply laced and |$\lambda $| is half-integral; see [12, Lem. 3.17]. When |$\Delta _{\lambda }({{\mathfrak{p}}}^{+})=\emptyset $|⁠, then |${{\mathcal{O}}}_{m}^{\prime}=\{0\}$| and |$\delta =\#(\Delta ({{\mathfrak{p}}}^{+}))=\dim ({{\mathfrak{p}}}^{+})$|⁠, so |$\operatorname{AV}(L(\lambda ))=\overline{{{\mathcal{O}}}_{r}}={{\mathfrak{p}}}^{+}$|⁠.

1.5 About the proofs

Theorems 1.2 and 1.4 are proved in Section 4. Theorem 1.2 is first proved when |$\lambda +\rho $| is regular integral; it suffices to assume the infinitesimal character is |$\rho $|⁠. The argument is to first observe that if |$\operatorname{width}(Y_{\lambda })=m$|⁠, then |$\operatorname{AV}(L(\lambda ))\supseteq \overline{{{\mathcal{O}}}_{m}}$|⁠. A counting argument is then used. The general theory of Harish-Chandra cells gives a lower bound on the size of a cell in terms of the Springer correspondence. This, along with the number of |$L(\lambda )$| (of infinitesimal character |$\rho $|⁠) for which |$\operatorname{width}(Y_{\lambda })=m$|⁠, completes the proof. The number with a given width is calculated in Section 2 using some combinatorics of the posets. The necessary background on Harish-Chandra cells and some specific data on the Springer correspondence is given in Section 3. The case of arbitrary integral infinitesimal character follows from the translation principle, illustrating that our definition of the diagram |$Y_{\lambda }$| is “correct”. This leaves the nonintegral case. When |$\Delta $| is non-simply laced and |$\lambda $| is half-integral, there is a group |$G_{\mathbb{R}}^{\prime}$|⁠, as described above. Lusztig’s formula for the GK dimension proves Theorem 1.4 and this reduces the proof of Theorem 1.2 to the integral case for |$G_{\mathbb{R}}^{\prime}$|⁠. In the other nonintegral cases |$L(\lambda )$| is precisely a (full) generalized Verma module for the parabolic |${{\mathfrak{q}}}$|⁠; the associated variety is then easily seen to be |$\overline{{{\mathcal{O}}}_{m}}={{\mathfrak{p}}}^{+}$|⁠.

Another proof of Theorem 1.2 is given in Section 5. In [3], Bai–Xiao–Xie used Robinson–Schensted insertion algorithm to compute the GK dimensions of highest weight modules for classical type Lie algebras and cell decompositions of maximal parabolic subgroups of Hermitian type for exceptional type Lie algebras. Then, we translate these results into the world of antichains of posets, which gives us another proof for Theorem 1.2.

1.6 Connections with other work

In 1992, Joseph [20] described the associated varieties of unitary highest weight modules in a case-by-case fashion. Nishiyama–Ochiai–Taniguchi [24] and Enright–Willenbring [13] independently gave some characterizations of the associated varieties of unitary highest weight modules appearing in the dual pair settings. More recently, Bai–Hunziker [2] found a uniform formula for the GK dimensions and associated varieties of all unitary highest weight modules. For |$G_{\mathbb{R}}=Sp(2n,{\mathbb{R}})$|⁠, Barchini–Zierau [8] computed the characteristic cycles of highest weight Harish-Chandra modules for regular integral infinitesimal character and associated varieties are given implicitly by an inductive procedure.

2 Posets of Positive Noncompact Roots

Let |${{\mathfrak{g}}}={{\mathfrak{p}}}^{-} \oplus{{\mathfrak{k}}}\oplus{{\mathfrak{p}}}^{+}$| be the triangular decomposition corresponding to an irreducible Hermitian symmetric space (of noncompact type) as in the introduction. Therefore, |${{\mathfrak{q}}}={{\mathfrak{k}}}\oplus{{\mathfrak{p}}}^{+}$| is a maximal parabolic subalgebra of |${{\mathfrak{g}}}$| with Levi subgroup |${{\mathfrak{k}}}$| and abelian nilradical |${{\mathfrak{p}}}^{+}\!$|⁠. We fix a Cartan subalgebra |${{\mathfrak{h}}}$| of |${{\mathfrak{g}}}$| that is contained in |${{\mathfrak{k}}}$| and a Borel subalgebra |${{\mathfrak{b}}}={{\mathfrak{h}}}\oplus{{\mathfrak{n}}}$| of |${{\mathfrak{g}}}$| such that |${{\mathfrak{b}}}\subset{{\mathfrak{q}}}$|⁠. Let |$\Delta $| and |$\Delta ({{\mathfrak{k}}})$| denote the root systems of |$({{\mathfrak{g}}},{{\mathfrak{h}}})$| and |$({{\mathfrak{k}}},{{\mathfrak{h}}})$|⁠, respectively, and set |$\Delta ^{+}=\{ \alpha \in \Delta \mid{{\mathfrak{g}}}_{\alpha } \subseteq{{\mathfrak{n}}}\}$|⁠, |$\Delta ^{+}({{\mathfrak{k}}})=\Delta ({{\mathfrak{k}}})\cap \Delta ^{+}$|⁠, and |$\Delta ({{\mathfrak{p}}}^{+})=\{\alpha \in \Delta \mid{{\mathfrak{g}}}_{\alpha }\subset{{\mathfrak{p}}}^{+}\}=\Delta ^{+}\setminus \Delta ^{+}({{\mathfrak{k}}})$|⁠. The roots in |$\Delta ^{+}({{\mathfrak{k}}})$| are called the positive compact roots and the roots in |$\Delta ({{\mathfrak{p}}}^{+})$| are called the positive noncompact roots. Let |$\Pi $| denote the set of simple roots for |$\Delta ^{+}$|⁠. Exactly one simple root is in |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. As usual, |$\rho $| is half the sum of the positive roots. Denote by |$W$| the Weyl group of |$\Delta $|⁠.

Define |$\Lambda ^{+}({{\mathfrak{k}}}):=\{\lambda \in{{\mathfrak{h}}}^{*}\mid \langle \lambda +\rho \,, \alpha ^{\vee }\rangle \in{\mathbb{Z}}_{\geq 0}\textrm{ for all}\ \alpha \in \Delta ^{+}({{\mathfrak{k}}})\}$|⁠. Note that |$\Lambda ^{+}({{\mathfrak{k}}})$| is the set of highest weights of irreducible finite-dimensional representations of |${{\mathfrak{k}}}$|⁠. We consider arbitrary |$\lambda \in \Lambda ^{+}({{\mathfrak{k}}})$|⁠. When |$\lambda $| is not integral we need to consider the integral root system |$\Delta _{\lambda }=\{\alpha \in \Delta \mid \langle \lambda +\rho \,, \alpha ^{\vee }\rangle \in{\mathbb{Z}}\}$| and its Weyl group |$W_{\lambda }$|⁠.

We view all sets of roots as partially ordered sets with the usual partial ordering, where |$\alpha \leq \beta $| means that |$\beta - \alpha $| is a sum of roots in |$\Delta ^{+}$|⁠. In what follows, the set of positive noncompact roots |$\Delta ({{\mathfrak{p}}}^{+})$|⁠, viewed as a poset, will play a fundamental role.

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. That is, we draw the Hasse diagram as a graph with vertices labelled by the positive noncompact roots and a directed edge from |$\beta $| to |$\beta ^{\prime}$| when |$\beta ^{\prime}-\beta $| is a simple (necessarily compact) root; the edge is sometimes labelled by the simple compact root. The unique minimal element of the poset is the simple noncompact root (drawn at the bottom) and the unique maximal element, which we denote by |$\theta $|⁠, is the highest root (drawn at the top). These diagrams appear in the appendix of Jakobsen’s paper [17]. We have included the Hasse diagrams for the simply-laced types in the appendix.

2.2 Lower-order ideals

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$|⁠.

For |$\lambda \in \Lambda ^{+}({{\mathfrak{k}}})$|⁠, define the diagram of |$\lambda $| is the set

viewed as a subposet of |$\Delta ({{\mathfrak{p}}}^{+})$|⁠.

It is easy to describe the lower order ideals in |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. Define

It follows that |${\mathcal{W}}=\{w\in W \mid{{\mathfrak{n}}} \cap \operatorname{Ad}(w) {{\mathfrak{n}}} \ \subseteq \ {{\mathfrak{p}}}^{+}\}$|⁠. Note that the cardinality of |${\mathcal{W}}$| is |$\#{\mathcal{W}}=\#(W/W({{\mathfrak{k}}}))$|⁠, where |$W({{\mathfrak{k}}})$| is the Weyl group of |${{\mathfrak{k}}}$|⁠.

Example 2.4.

Suppose |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{s}}}{{\mathfrak{u}}}(4,3)$| and |$\lambda +\rho =-w\rho = (2,1,-1,-2 \mid 3,0,-3)$|⁠. Then  

It is useful to know how to associate an element of |${\mathcal{W}}$| to a diagram |$Y_{\lambda }$|⁠. Given |$\lambda \in \Lambda ^{+}({{\mathfrak{k}}})$| that is integral (for |$\Delta $|⁠), we show how to determine |$w\in{\mathcal{W}}$| so that |$Y_{\lambda }=Y_{-w\rho -\rho }$|⁠. Let |$C^{-}:=\{y\in{{\mathfrak{h}}}^{*}\mid \langle y\,, \alpha \rangle <0,\textrm{ for all}\ \alpha \in \Delta ^{+}\}$|⁠, the negative Weyl chamber. When |$\lambda +\rho $| is regular there is a unique |$w\in W$| such that |$w^{-1}(\lambda +\rho )\in C^{-}$|⁠. This is the correct |$w$| (since |$w^{-1}(\lambda +\rho )$| and |$-\rho $| lie in the same Weyl chamber, which implies |$\lambda +\rho $| and |$-w\rho $| are also in the same chamber). However, when |$\lambda +\rho $| is singular the situation is more subtle. The following proposition shows how to determine the correct |$w$|⁠. The importance lies in Proposition 4.1.

2.3 Antichains and width of lower-order ideals

An antichain in any poset |$Y$| is a subset of |$Y$| consisting of pairwise noncomparable elements. The width of the poset, denoted by |$\operatorname{width}(Y)$|⁠, is the greatest cardinality of antichains in |$Y$|⁠.

Recall that two roots |$\gamma _{1},\gamma _{2}\in \Delta $| are called strongly orthogonal if neither |$\gamma _{1}\pm \gamma _{2}$| is a root. When |$\gamma _{1},\gamma _{2}\in \Delta ({{\mathfrak{p}}}^{+})$|⁠, |$\gamma _{1}+\gamma _{2}$| is never a root (as |${{\mathfrak{p}}}^{+}$| is abelian), therefore, they are strongly orthogonal if and only if |$\gamma _{1}-\gamma _{2}$| is not a root. Thus, |$\gamma _{1},\gamma _{2}$| are not comparable implies they are strongly orthogonal. (The converse does not hold; e.g., in |${\mathfrak{so}}(2,2n-1)$|⁠, |$(\varepsilon _{1}+\varepsilon _{n})-(\varepsilon _{1}-\varepsilon _{n})$| is a sum of roots, but is not a root.) The following lemma therefore holds.

2.4 The number of lower-order ideals of a given width

The number of lower order ideals of width |$m$| is determined in this section. This is important for our counting arguments in Section 4.

The proof of this proposition is a case-by-case argument.

2.4.1 Proof for |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{s}}}{{\mathfrak{u}}}(p,q)$|

Write |$n=p+q$|⁠. Take |$\Delta ^{+}$| to be the usual positive system of roots for Type |$A_{n-1}$|⁠, so |$\Delta ^{+}=\{\varepsilon _{i}-\varepsilon _{j}\mid 1\leq i<j\leq n\}$| and |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{i}-\varepsilon _{j}\mid 1\leq i\leq p<j\leq n\}$|⁠.

Suppose that |$\beta _{1}=\varepsilon _{i_{1}}-\varepsilon _{j_{1}}$| and |$\beta _{2}=\varepsilon _{i_{2}}-\varepsilon _{j_{2}}$| are noncomparable roots in |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. Then |$i_{1}\leq p<j_{1}$| and |$i_{2}\leq p< j_{2}$| and we may assume that |$i_{1}<i_{2}$|⁠. Then |$\beta _{1}-\beta _{2}=(\varepsilon _{i_{1}}-\varepsilon _{i_{2}})+(\varepsilon _{j_{2}}-\varepsilon _{j_{1}})$|⁠. The first is a positive root, so the second is necessarily a negative root (otherwise |$\beta _{1}\geq \beta _{2}$|⁠), so |$j_{1}<j_{2}$|⁠. We may conclude that antichains take the form

If |$Y$| has width |$m$|⁠, then |$Y$| contains a maximal antichain of the above form. Since |$Y$| is a lower order ideal we may subtract positive compact roots and stay in |$Y$|⁠, therefore |$Y$| contains the antichain

that is, |$i_{1}<i_{2}<\dots i_{m}= p<j_{1}<j_{2}<\dots <j_{m}$| are consecutive. (It is useful to view this antichain in the Hasse diagram for |$\Delta ({{\mathfrak{p}}}^{+})$|⁠: it is the horizontal row of nodes |$m-1$| rows up from the minimal root. See the Appendix for the Hasse diagram for |${{\mathfrak{s}}}{{\mathfrak{u}}}(4,3)$|⁠.) Note that |$m$| is necessarily less than |$r=\min \{p,q\}$|⁠. We conclude that all lower order ideals |$Y$| of width |$m$| have a maximal antichain in common, namely the one in (2.5).

Let |$\wp _{p,q}(m)=\#\{w\in{\mathcal{W}}\mid \operatorname{width}(Y_{-w\rho -\rho })=m\}$|⁠. (We interpret this to be |$0$| if |$p$| or |$q$| is |$0$|⁠.) Notice that |$\wp _{p,q}$| is symmetric in |$p,q$|⁠. We assume |$p\leq q$|⁠.

Claim: |$\wp _{p,q}(m)=$|

For our argument it is useful to keep in mind the Hasse diagram for |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. Note that |$\rho =(\frac{n-1}{2},\frac{n-3}{2},\dots ,-\frac{n-1}{2})$| and for |$w\in{\mathcal{W}}$| the entries of |$-w\rho $| are the same as those in |$\rho $| and the first |$p$| entries decrease as do the last |$q$| entries. Therefore, |$-w\rho $| takes the form

To prove the claim we consider the contributions to |$\wp _{p,q}(m)$| from the two forms of |$-w\rho $|⁠. The contribution from (a) is |$\wp _{p-1,q}(m)$|⁠, since no roots |$\varepsilon _{1}-\varepsilon _{j}$| appear in |$Y_{-w\rho -\rho }$|⁠. The contribution from (b) is

This is easily understood from the Hasse diagram of |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. The claim follows since |$\wp _{p-1,q}(m)=0$| when |$m=p$|⁠.

Now we show that for |$p\leq q$|

using induction on |$n=p+q$|⁠. When |$n=2$| this is immediate. There are several cases.

• (i)
  Case |$m<p$|⁠:

  $$ \begin{align*} \wp_{p,q}(m)&=\wp_{p-1,q}(m)+\wp_{p,q-1}(m-1), \text{ by the claim,} \\ &=\left(\binom{n-1}{m}-\binom{n-1}{m-1}\right)+\left(\binom{n-1}{m-1}-\binom{n-1}{m-2}\right), \text{ by induction,} \\ &=\binom{n-1}{m}-\binom{n-1}{m-2} \\ &=\binom{n}{m}-\binom{n}{m-1}, \textrm{ an easily checked identity}. \end{align*} $$
• (ii)
  Case |$m=p<q$|⁠:

  $$ \begin{align*} \wp_{p,q}(p)&=\wp_{p,q-1}(p)+\wp_{p,q-1}(p-1), \text{ by the claim,} \\ &=\left(\binom{n-1}{p}-\binom{n-1}{p-1}\right)+\left(\binom{n-1}{p-1}-\binom{n-1}{p-2}\right), \text{ by induction,} \\ &=\binom{n-1}{p}-\binom{n-1}{p-2} \\ &=\binom{n}{p}-\binom{n}{p-1}, \textrm{ same identity}. \end{align*} $$
• (iii)
  Case |$m=p=q$|⁠:

  $$ \begin{align*} \wp_{p,p}(p)&=\wp_{p,p-1}(p)+\wp_{p,p-1}(p-1), \text{ by the claim,} \\ &=\wp_{p,p-1}(p)+\wp_{p-1,p}(p-1), \text{ by symmetry,} \\ &=\binom{2p-1}{p-1}-\binom{2p-1}{p-2}, \text{ first term is {$0$},} \\ &=\binom{2p-1}{p}-\binom{2p-1}{p-2} \\ &=\binom{2p}{p}-\binom{2p}{p-1}, \textrm{ same identity}. \end{align*} $$

2.4.2 Proof for |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{s}}}{{\mathfrak{o}}}^{*}(2n)$|

This case is is quite similar to the |${\mathfrak{su}}(p,q)$| case above. The set of positive compact roots is |$\Delta ^{+}({{\mathfrak{k}}})=\{\varepsilon _{i}-\varepsilon _{j} \mid 1\leq i< j\leq n\}$| and the poset of positive noncompact roots is |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{i}+\varepsilon _{j} \mid 1\leq i< j\leq n\}$|⁠. Suppose |$\gamma _{1}=\varepsilon _{i_{1}}+\varepsilon _{j_{1}}$| and |$\gamma _{2}=\varepsilon _{i_{2}}+\varepsilon _{j_{2}}$| are two noncomparable roots in |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. We may assume that |$i_{1}<i_{2}, i_{1}<j_{1}$| and |$i_{2}<j_{2}$|⁠. Then in the expression

the first term is a positive root, therefore |$\gamma _{1},\gamma _{2}$| are noncomparable if and only if the second term is a negative root. This holds if and only if |$j_{2}<j_{1}$| (so |$i_{1}<i_{2}<j_{2}<j_{1}$|⁠). We may conclude from this that the antichains of length |$m$| in any lower order ideal are

For |$w\in{\mathcal{W}}$|⁠, |$-w\rho $| has decreasing entries, so if |$\operatorname{width}(Y_{-w\rho -\rho })=m$| then we may assume that |$i_{1}<\dots <i_{m}<j_{m}<\dots ,j_{1}$| are consecutive ending in |$j_{1}=n$|⁠: |$\{\varepsilon _{n-2m+s} + \varepsilon _{n+1-s}\mid s=1,\dots ,m\}.$| Therefore, whenever a lower order ideal has width |$m$| it contains this maximal antichain, which appears in the Hasse diagram as the horizontal row of nodes appearing |$2m-2$| rows up from the bottom. For |$n=6$|⁠, the Hasse diagram is shown in the appendix.

Let |$\wp _{n}(m)$| be the number of |$w\in{\mathcal{W}}$| such that |$\operatorname{width}(Y_{-w\rho -\rho })=m$|⁠. Since |$\operatorname{rank}_{\mathbb{R}}({\mathfrak{so}}^{*}(2n))=\lfloor \frac{n}{2}\rfloor $|⁠, |$\wp _{n}(m)=0$| when |$m>\lfloor \frac{n}{2}\rfloor $|⁠.

2.4.3 Proof for |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{s}}}{{\mathfrak{p}}}(2n,{\mathbb{R}})$|

The poset |$\Delta ({{\mathfrak{p}}}^{+})$| is isomorphic to |$\Delta ({{\mathfrak{p}}}^{\prime +})$|⁠, where |${{\mathfrak{g}}}^{\prime}_{\mathbb{R}}={{\mathfrak{s}}}{{\mathfrak{o}}}^{*}(2n+2)$|⁠. Therefore, this case is already done. Note that the real rank of |${{\mathfrak{s}}}{{\mathfrak{p}}}(2n,{\mathbb{R}})$| is |$n$| whereas the real rank of |${{\mathfrak{s}}}{{\mathfrak{o}}}^{*}(2n+2)$| is |$\lfloor \frac{n+1}{2}\rfloor $|⁠. This explains the entry |$0$| for |$m>\frac{n+1}{2}$| in the third column of the Table 1.

Using the isomorphism with the poset |$\Delta ({{\mathfrak{p}}}^{\prime +})$| and the description of antichains given in Section 2.4.2, we see that if |$\operatorname{width}(Y)=m$|⁠, then there is an maximal antichain having one of the forms

2.4.4 Proof for |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{s}}}{{\mathfrak{o}}}(2,2n-1)$|

The poset of |$\Delta ({{\mathfrak{p}}}^{+})$| is a linear graph. The result is immediate because |$\#\Delta ({{\mathfrak{p}}}^{+})=2n-1$|⁠.

2.4.5 Proof for |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{s}}}{{\mathfrak{o}}}(2,2n-2)$|

The poset |$\Delta ({{\mathfrak{p}}}^{+})$| in this case is quite simple and is illustrated below. Noting that the only antichain of width |$2$| is |$\{\varepsilon _{1}-\varepsilon _{2},\varepsilon _{1}+\varepsilon _{2}\}$|⁠, the lower order ideals of widths |$0,1$| and |$2$| are easily counted. We get |$\wp _{n}(0)=1, \wp _{n}(1)=n,$| and |$\wp _{n}(2)=n-1$|⁠. For |$n=6$|⁠, the Hasse diagram of |$\Delta ({{\mathfrak{p}}}^{+})$| is shown in the appendix.

2.4.6 Proof for |${{\mathfrak{g}}}_{\mathbb{R}}=\mathfrak{e}_{6(-14)} $|

The poset |$\Delta ({{\mathfrak{p}}}^{+})$| for |${{\mathfrak{g}}}_{\mathbb{R}}=\mathfrak{e}_{6(-14)} $| is shown in the Appendix. A lower-order ideal |$Y\subseteq \Delta ({{\mathfrak{p}}}^{+})$| has width |$\geq 1$| if and only if |$\alpha _{1}\in \Delta ({{\mathfrak{p}}}^{+})$| and width |$\geq 2$| if and only if |$\beta _{1},\beta _{2}\in \Delta ({{\mathfrak{p}}}^{+})$|⁠. A quick inspection shows that there are exactly |$6$| lower-order ideals of |$\Delta ({{\mathfrak{p}}}^{+})$| that contain |$\alpha _{1}$|⁠, but not both |$\beta _{1}$| and |$\beta _{2}$|⁠. Thus, there are exactly |$6$| posets of |$\Delta ({{\mathfrak{p}}}^{+})$| that have width |$1$|⁠. Since the total number of lower-order ideals of |$\Delta ({{\mathfrak{p}}}^{+})$| is |$\operatorname{card}({\mathcal{W}})=27$|⁠, the number of lower-order ideals of |$\Delta ({{\mathfrak{p}}}^{+})$| width |$2$| is |$27-1-6=20$|⁠.

2.4.7 Proof for |${{\mathfrak{g}}}_{\mathbb{R}}=\mathfrak{e}_{7(-25)} $|

The poset |$\Delta ({{\mathfrak{p}}}^{+})$| for |${{\mathfrak{g}}}_{\mathbb{R}}=\mathfrak{e}_{7(-25)} $| is shown in the Appendix. A lower-order ideal |$Y\subseteq \Delta ({{\mathfrak{p}}}^{+})$| has width |$\geq 1$| if and only if |$\alpha _{7}\in \Delta ({{\mathfrak{p}}}^{+})$|⁠, width |$\geq 2$| if and only if |$\beta _{1},\beta _{2}\in \Delta ({{\mathfrak{p}}}^{+})$|⁠, and width |$\geq 3$| if and only if |$\gamma _{1},\gamma _{2},\gamma _{3}\in \Delta ({{\mathfrak{p}}}^{+})$|⁠. Again, a quick inspection shows that there are exactly |$7$| lower-order ideals of |$\Delta ({{\mathfrak{p}}}^{+})$| that contain |$\alpha _{1}$|⁠, but not both |$\beta _{1}$| and |$\beta _{2}$|⁠. A more tedious inspection shows that there are exactly |$27$| lower-order ideals of |$\Delta ({{\mathfrak{p}}}^{+})$| that contain |$\beta _{1}$| and |$\beta _{2}$| but not all three roots |$\gamma _{1}$|⁠, |$\gamma _{2}$|⁠, and |$\gamma _{3}$|⁠. Thus, there are exactly |$7$| lower-order ideals of |$\Delta ({{\mathfrak{p}}}^{+})$| of width |$1$|⁠, exactly |$27$| lower-order ideals of width |$2$|⁠, and |$56-1-7-27=21$| lower-order ideals of width |$3$|⁠.

This concludes the proof of Proposition 2.9.

2.5 Half-integral case

We need to consider highest weight Harish-Chandra modules |$L(\lambda )$| when |$\lambda $| is half-integral. As we shall see, all the information needed is obtained by reducing to the integral case.

Let |$\lambda \in \Lambda ^{+}({{\mathfrak{k}}})$| be half-integral. This means that |$\langle \lambda +\rho \,, \beta ^{\vee }\rangle \in \frac 12+{\mathbb{Z}}$|⁠, for some |$\beta \in \Delta ({{\mathfrak{p}}}^{+})$|⁠. We consider the integral root system

Let |$W_{\lambda }$| denote the Weyl group of |$\Delta _{\lambda }$| and fix the positive system |$\Delta _{\lambda }^{+}:=\Delta ^{+}\cap \Delta _{\lambda }$|⁠. Let |${{\mathfrak{g}}}_{\lambda }$| be the complex semisimple Lie algebra with root system |$\Delta _{\lambda }$|⁠. Then there is a triangular decomposition

where |$\Delta ({{\mathfrak{p}}}_{\lambda }^{\pm }):=\Delta ({{\mathfrak{p}}}^{\pm })\cap \Delta _{\lambda }$|⁠. Note that |$\Delta ({{\mathfrak{k}}})\subset \Delta _{\lambda }$|⁠, since |$\lambda $| is |$\Delta ({{\mathfrak{k}}})$|-integral, so |${{\mathfrak{k}}}_{\lambda }\simeq{{\mathfrak{k}}}$|⁠. Therefore, we may apply Section 2.2 and Section 2.3 to |$\Delta ({{\mathfrak{p}}}_{\lambda }^{+})\subset \Delta _{\lambda }^{+}$| in place of |$\Delta ({{\mathfrak{p}}}^{+})\subset \Delta ^{+}$|⁠. In particular, |$Y_{\lambda }$| is a lower order ideal in |$\Delta _{\lambda }$| and defining |${\mathcal{W}}_{\lambda }$| in |$W_{\lambda }$| there is a bijection |${\mathcal{W}}_{\lambda }\leftrightarrow \{\textrm{lower order ideals in} \Delta ({{\mathfrak{p}}}_{\lambda }^{+})\}$|⁠. Note that for |$\alpha ,\beta \in \Delta ({{\mathfrak{p}}}_{\lambda }^{+})$|⁠, |$\alpha \leq \beta $| is the same in |$\Delta $| (with respect to |$\Delta ^{+}$|⁠) as in |$\Delta _{\lambda }$| (with respect to |$\Delta _{\lambda }^{+}$|⁠), since |$\beta -\alpha $| is a sum of compact roots and |$\Delta ({{\mathfrak{k}}})\subset \Delta _{\lambda }$|⁠. Therefore, |$Y_{\lambda }$| is the same whether defined in |$\Delta $| or in |$\Delta _{\lambda }$|⁠.

There are two examples we return to in Section 4.4.

• (a)
  |${{\mathfrak{g}}}_{\mathbb{R}}={\mathfrak{sp}}(2n,{\mathbb{R}}),n\geq 2$|⁠. If |$\lambda =(\lambda _{1},\dots ,\lambda _{n})$| is half-integral, then |$\lambda _{j}\in \frac 12+{\mathbb{Z}}$|⁠, all |$j$|⁠. Therefore, the integral root system is

  $$ \begin{align*}& \Delta_{\lambda}=\{\varepsilon_{j}\pm\varepsilon_{k}\mid 1\leq j<k\leq n\}, \end{align*} $$

  a root system of type |$D_{n}$|⁠. Observe that the complex Lie algebra |${{\mathfrak{g}}}_{\lambda }$| corresponding to |$\Delta _{\lambda }$| is of type |$D_{n}$| and has a real form |${{\mathfrak{g}}}_{\lambda ,{\mathbb{R}}}\simeq{\mathfrak{so}}^{*}(2n)$| of Hermitian type. The set of positive noncompact roots is |$\Delta ({{\mathfrak{p}}}_{\lambda }^{+})=\Delta ({{\mathfrak{p}}}^{+})\cap \Delta _{\lambda }=\{\varepsilon _{j}+\varepsilon _{k}:1\leq j<k\leq n\}$|⁠.
• (b)
  |${{\mathfrak{g}}}={\mathfrak{so}}(2,2n-1), n\geq 2$|⁠. Then

  $$ \begin{align*}& \Delta_{\lambda}=\{\pm\varepsilon_{1}\}\cup\Delta({{\mathfrak{k}}}), \end{align*} $$

  a root system of type |$A_{1}\times B_{n-1}$|⁠. The corresponding Lie Algebra |${{\mathfrak{g}}}_{\lambda }$| has a real form |${{\mathfrak{g}}}_{\lambda ,{\mathbb{R}}}\simeq{\mathfrak{sl}}(2,{\mathbb{R}})\times{\mathfrak{so}}(2n-1)$| of Hermitian type.

3 Associated Varieties and Highest Weight Harish-Chandra Modules

In this section we review some background information on associated varieties, cells and the Springer correspondence.

3.1 Associated varieties and cells for |$({{\mathfrak{g}}},K)$|-modules

Assume that |$G_{\mathbb{R}}$| is a connected semisimple Lie group with finite center. Let |${{\mathfrak{g}}}_{\mathbb{R}}={{\mathfrak{k}}}_{\mathbb{R}}\oplus{{\mathfrak{p}}}_{\mathbb{R}}$| be a Cartan decomposition; since |$G_{\mathbb{R}}$| has finite center, there is a maximal compact subgroup |$K_{\mathbb{R}}$| with Lie algebra |${{\mathfrak{k}}}_{\mathbb{R}}$|⁠. Let |${{\mathfrak{g}}}={{\mathfrak{k}}}\oplus{{\mathfrak{p}}}$| be the complexified Cartan decomposition and |$K$| the connected complex algebraic group containing |$K_{\mathbb{R}}$| with Lie algebra |${{\mathfrak{k}}}$|⁠. Any irreducible |$K_{\mathbb{R}}$| representation extends to an algebraic representation of |$K$|⁠. Therefore, |$({{\mathfrak{g}}},K)$| is a pair as in [28, §1] and the Harish-Chandra module of an admissible representation of |$G_{\mathbb{R}}$| is a |$({{\mathfrak{g}}},K)$|-module. The action of the algebraic group |$K$| is needed for definition of associated variety. We follow the definition of associated variety of a finitely generated |$({{\mathfrak{g}}},K)$|-module as given in [28]. Note that we are including finite covers of linear groups, such as the metaplectic cover of |$Sp(2n,{\mathbb{R}})$| which occurs for half integral highest weights. However, groups with infinite center, such as the universal cover of |$G_{\mathbb{R}}=Sp(2n,{\mathbb{R}})$|⁠, are not included in our discussion.

Suppose that |$({{\mathfrak{g}}},K)$| is a pair as above and |$X$| is a |$({{\mathfrak{g}}},K)$|-module of finite length. The associated variety of |$X$| is defined by starting with a |$K$|-invariant “good” filtration |$\{X_{n}\}$| of |$X$|⁠. Then |$\underline{\operatorname{Gr}}\, X$| is a module for |$\underline{\operatorname{Gr}}\, {\mathcal{U}}({{\mathfrak{g}}})\simeq S({{\mathfrak{g}}})\simeq P({{\mathfrak{g}}}^{*})$|⁠, and by |$K$|-invariance of the filtration, is a module for |$S({{\mathfrak{g}}}/{{\mathfrak{k}}})$|⁠. The associated variety of |$X$| is defined as the support of |$\underline{\operatorname{Gr}}\, X$|⁠:

Identifying |$({{\mathfrak{g}}}/{{\mathfrak{k}}})^{*}$| with |${{\mathfrak{p}}}$| by the Killing form, |$\operatorname{AV}(X)$| is a |$K$|-stable subvariety of |${{\mathfrak{p}}}$|⁠. In fact, |$\operatorname{AV}(X)\subseteq{{\mathfrak{p}}}\cap{{\mathcal{N}}}$|⁠, where |${{\mathcal{N}}}$| is the nilpotent cone in |${{\mathfrak{g}}}$|⁠. Therefore,

a union of closures of nilpotent |$K$|-orbits in |${{\mathfrak{p}}}$| and we write this union in a way that no |$\overline{{{\mathcal{O}}}}_{\alpha }$| is contained in any other.

When |$X$| is irreducible, the |${{\mathcal{O}}}_{\alpha }$| occurring in (3.1) are all equidimensional and all have the same |$G$|-saturation, that is, there is a complex nilpotent orbit |${{\mathcal{O}}}^{\mathbb{C}}$| such that

It is a fact that |$\overline{{{\mathcal{O}}}^{\mathbb{C}}}$| is the associated variety of the annihilator of |$X$|⁠. See [28] for details.

Suppose |$X$| and |$Y$| are irreducible Harish-Chandra modules having infinitesimal character of Harish-Chandra parameter |$\Lambda \in{{\mathfrak{h}}}^{*}$|⁠. We say that |$X\lesssim Y$| when |$X$| occurs as a subquotient of |$Y\otimes F$|⁠, with |$F$| some finite-dimensional |$G$|-subrepresentation of the tensor algebra of |${{\mathfrak{g}}}$|⁠. We say |$X\sim Y$| when |$X\lesssim Y$| and |$Y\lesssim X$|⁠; this generates an equivalence relation on the set of irreducible representations of infinitesimal character |$\Lambda $|⁠. The Harish-Chandra cells are defined to be the equivalence classes. Since |$X\lesssim Y$| implies |$\operatorname{AV}(X)\subseteq \operatorname{AV}(Y)$|⁠, we see that the associated variety is constant on any given cell.

Now assume that |$\Lambda $| is regular and integral and assume also that |$G_{\mathbb{R}}$| is linear. Then the coherent continuation representation of the Weyl group |$W$| of |${{\mathfrak{g}}}$| is defined on the Grothendieck group of Harish-Chandra modules of infinitesimal character |$\Lambda $|⁠. This gives a representation of |$W$| for each cell as follows. Fix an irreducible |$X_{0}$| (infinitesimal character |$\Lambda $|⁠) and let |${\mathscr C}$| be the cell of |$X_{0}$|⁠, that is, the cell containing |$X_{0}$|⁠. Set

Then both |$W_{\mathscr C}$| and |$U_{\mathscr C}$| are |$W$|-stable under coherent continuation. The cell representation for |${\mathscr C}$| is the quotient |$V_{\mathscr C}:=W_{\mathscr C}/U_{\mathscr C}$|⁠. We may view

(but keeping in mind that it is really a quotient). See [6] for cells and [26, Ch. 7] for coherent continuation.

Much is known about cells and the cell representations. We have already mentioned that the associated variety is constant on cells, as is the associated variety of annihilator. An important property of the cell representations is that if |$\overline{{{\mathcal{O}}}^{\mathbb{C}}}$| is the associated variety of annihilator, as in (3.2), then the |$W$| representation |$\pi ({{\mathcal{O}}}^{\mathbb{C}},1)$| corresponding to |${{\mathcal{O}}}^{\mathbb{C}}$| under the Springer correspondence occurs in |$V_{\mathscr C}$| [27, Cor. 14.11] and is a special representation (in the sense of Lusztig).

What makes this useful for us is that there are easy combinatorial procedures known (for the classical groups) and tables (for the exceptional groups) that (1) give the Springer correspondence and (2) determine which |$W$|-representations are special. See [10, §13.1-13.3]. The Atlas of Lie Groups software [1] also gives this information for Lie algebras of rank up to at least 8.

Most of the relevant data for us is contained in Table 2.

3.2 Highest weight Harish-Chandra modules

A highest weight Harish-Chandra module for our group |$G_{\mathbb{R}}$| (having finite center) is a |$({{\mathfrak{g}}},K)$|-module (for the pair described in Section 3.1) having a vector |$x_{\lambda }^{+}\in X$| satisfying

• (i)

  |$x_{\lambda }^{+}$| is a weight vector for |${{\mathfrak{h}}}$|⁠, a compact Cartan subalgebra of |${{\mathfrak{g}}}$|⁠, of weight |$\lambda \in{{\mathfrak{h}}}^{*}$|⁠,

• (ii)

  |${{\mathfrak{n}}}\cdot x_{\lambda }^{+}=0$|⁠, where |${{\mathfrak{n}}}$| is the nilradical of some Borel subalgebra |${{\mathfrak{b}}}={{\mathfrak{h}}}\oplus{{\mathfrak{n}}}\subset{{\mathfrak{g}}}$|⁠, and

• (iii)

  |$X={\mathcal{U}}({{\mathfrak{g}}})\cdot x_{\lambda }^{+}$|⁠.

Suppose |$X$| is an infinite-dimensional irreducible highest weight Harish-Chandra module for |$G_{\mathbb{R}}$|⁠. Then, by [14, § 1-2], |$G_{\mathbb{R}}$| is of Hermitian type. In this case, there is a triangular decomposition |${{\mathfrak{g}}}={{\mathfrak{p}}}^{-}\oplus{{\mathfrak{k}}}\oplus{{\mathfrak{p}}}^{+}$| as in Section 2.1. It is also shown in [14] that |${{\mathfrak{h}}}$| and |${{\mathfrak{b}}}$| in (i) and (ii) are given by our choices in Section 2.1. Therefore, |$\lambda \in \Lambda ^{+}({{\mathfrak{k}}})$| and |$X$| contains the |$K$|-type |$F(\lambda )$| having highest weight |$\lambda $|⁠. Furthermore,

We conclude that |$X$| is the unique irreducible quotient of the generalized Verma module

Therefore, |$X$| is also the irreducible quotient |$L(\lambda )$| of the full Verma module

The infinitesimal character is |$\lambda +\rho $|⁠. Observe that since |$X$| is a |$({{\mathfrak{g}}},K)$|-module we have |$\langle \lambda +\rho \,, \beta ^{\vee }\rangle \in{\mathbb{Q}}$|⁠, for |$\beta \in \Delta $|⁠. This follows from the fact that |$K_{\mathbb{R}}$| is compact, by our standing assumption that |$G_{\mathbb{R}}$| has finite center, so the highest weight |$\lambda $| of |$F(\lambda )$| is a character of the compact Cartan subgroup. In fact, for any |$\lambda \in{{\mathfrak{h}}}^{*}$|⁠, the irreducible highest weight module |$L(\lambda )$| (with respect to |${{\mathfrak{b}}}$| above) is a Harish-Chandra module for some |$G_{\mathbb{R}}$| of finite center if and only if |$\lambda \in \Lambda ^{+}({{\mathfrak{k}}})$| and |$\langle \lambda +\rho \,, \beta ^{\vee }\rangle \in{\mathbb{Q}}$|⁠, for all |$\beta \in \Delta $|⁠.

Our condition that |$G_{\mathbb{R}}$| has finite center implies, by [21], that our Harish-Chandra module |$X=L(\lambda )$| is the Harish-Chandra module of a continuous representation of |$G_{\mathbb{R}}$|⁠.

3.3 Associated varieties of highest weight Harish-Chandra modules

Suppose |$X$| is any irreducible highest weight Harish-Chandra module for |$G_{\mathbb{R}}$| of highest weight |$\lambda $|⁠. Then (as in Section 3.2) |$X={\mathcal{U}}({{\mathfrak{p}}}^{-})F(\lambda )$|⁠. Using the usual filtration of the enveloping algebra we obtain a filtration

One easily checks that it is a good filtration and is |$K$|-invariant. It is also |${{\mathfrak{k}}}+{{\mathfrak{p}}}_{+}$|-invariant (so also |${{\mathfrak{b}}}$|-invariant). We conclude that |$\operatorname{AV}(X)\subseteq \left ({{\mathfrak{g}}}/{{\mathfrak{k}}}+{{\mathfrak{p}}}^{+}\right )^{*}\simeq{{\mathfrak{p}}}^{+}$|⁠.

It is well known that the |$K$|-orbits in |${{\mathfrak{p}}}^{+}$| have a particularly nice form. Letting |$r=\operatorname{rank}_{\mathbb{R}}({{\mathfrak{g}}}_{\mathbb{R}})$|⁠, there are |$r+1$| such orbits. They may be written as follows. Let |$\{\gamma _{1},\dots ,\gamma _{r}\}$| be a maximal set of mutually strongly orthogonal long roots in |${{\mathfrak{p}}}^{+}$| (necessarily of cardinality |$r$|⁠). Any two sets of mutually strongly orthogonal long roots having the same cardinality are conjugate under the Weyl group |$W({{\mathfrak{k}}})$|⁠. The |$K$|-orbits in |${{\mathfrak{p}}}^{+}$| are

where |$X_{\gamma }$| is a root vector for |$\gamma $|⁠. Note that the choice (and numbering) of strongly orthogonal roots doesn’t matter, by the |$W({{\mathfrak{k}}})$|-conjugacy mentioned above. These orbits satisfy the inclusions

We conclude that the associated variety of any highest weight Harish-Chandra module is the closure of exactly one |${{\mathcal{O}}}_{k}$|⁠.

The following table collects relevant data on the |$K$|-orbits in |${{\mathfrak{p}}}^{+}$|⁠.

See [24] for more on associated varieties of highest weight Harish-Chandra modules.

Now consider any highest weight module |$L(\lambda )$|⁠. We may apply the the theory of associated varieties of highest weight modules (as in [9] and [20]). This tells us that the associated varieties are unions of orbital varieties, which we write as

As in [9] we write |$L_{w}=L(-w\rho -\rho ), w\in W$|⁠. Then the support of |$L_{w}$| is the Schubert variety |$Z_{w}=\overline{B\cdot w{{\mathfrak{b}}}}$| in the flag variety |${{\mathfrak{B}}}$| of |$G$|⁠. By [9, Prop. 6.11 ] the associated variety of |$L_{w}$| contains the moment map image |$\mu (\overline{T_{{\mathfrak{B}}}^{*}Z_{w}})=\nu (w)$|⁠, so

(However, the associated variety is sometimes larger and of greater dimension.)

Our first goal is a formula for associated varieties when |$L_{w}$| is a Harish-Chandra module, that is, when |$w\in{\mathcal{W}}$|⁠. Our description of |$\operatorname{AV}(L_{w})$| may be given in terms of some |${{\mathcal{O}}}_{k}$|  or in terms of orbital varieties. Theorem 1.2 expresses the associated variety in terms of |$K$|-orbits in |${{\mathfrak{p}}}^{+}$|⁠, however the orbital variety picture will allow us to use the combinatorics of posets discussed in Section 2. The following lemma puts this into perspective.

The proof of Theorem 1.2 will be largely a counting argument using Tables 1 and 2 along with the following important fact:

This fact follows from (3.7), (2.7) and (3.6).

4 Proof of Theorems 1.2 and 1.4

Theorem 1.2 is proved first for integral infinitesimal character. We begin by showing that it suffices to prove the theorem for infinitesimal character |$\rho $|⁠. The proof for |$X=L_{w} ~(w\in{\mathcal{W}})$| is then carried out separately for the simply laced and non-simply laced cases. Then we consider the half-integral case. We first prove Theorem 1.4, then reduce the half-integral case to the integral case.

4.1 The translation principle

The reduction of the proof of Theorem 1.2 from integral infinitesimal character to infinitesimal character |$\rho $| is accomplished using the translation principle. This principle is well-known; we use [16, Ch. 7] as a convenient reference; see also [18]. A translation functor |$T$| applied to |$X$| is tensoring by a finite-dimensional |${{\mathfrak{g}}}$|-representation followed by projection to the constituents of a generalized infinitesimal character. Therefore, one has |$\operatorname{AV}(T(X))\subseteq \operatorname{AV}(X)$|⁠. The following proposition is essentially contained in [4, Cor. 3.3]; we give the proof as it plays a key role for us.

4.2 Simply laced case

Assume |$\Delta $| is simply laced. We assume the infinitesimal character is integral, otherwise the highest weight module is irreducible and the associated variety is |$\overline{{{\mathcal{O}}}_{r}}={{\mathfrak{p}}}^{+}$|⁠.

The following lemma has been around and known for a long time. Its proof is scattered about in the literature. Here we give a short proof.

This lemma implies that there exists a cell |${\mathscr C}_{k}$| of associated variety |$\overline{{{\mathcal{O}}}_{k}}$|⁠, for each |$k=0,1,\dots ,r$|⁠. We know by Section 3.1 that |$\#{\mathscr C}_{k}\geq \dim (\pi ({{\mathcal{O}}}_{k}^{\mathbb{C}},1))$|⁠. Let |$M_{k}:=\{w\in{\mathcal{W}}\mid \operatorname{width}(\Delta ({{\mathfrak{n}}}\cap \operatorname{Ad}(w){{\mathfrak{n}}}))=k\}$| and let |$m_{k}$| be the cardinality of |$M_{k}$|⁠. Our Tables 1 and 2 show that

Theorem 1.2 follows from the following Proposition.

The following example illustrates the argument of our proof of the proposition.

4.3 Non-simply laced cases: integral infinitesimal character

The combinatorics in the simply vs. non-simply laced cases has a difference that is worth pointing out. As an example, consider |${{\mathfrak{g}}}$| of type |$B_{3}$|⁠. The Hasse diagram for |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{1}\}\cup \{\varepsilon _{1}\pm \varepsilon _{2},\varepsilon _{1}\pm \varepsilon _{3}\}$| a linear chain. There are five lower order ideals of width |$1$|⁠. Two contain a pair of strongly orthogonal long roots and three contain just one; the width is not the number of strongly orthogonal roots. So (3) of Proposition 4.3 fails. It turns out that for all five |$\operatorname{AV}(L_{w})=K\cdot (X_{\varepsilon _{1}-\varepsilon _{2}}+X_{\varepsilon _{1}+\varepsilon _{2}})=\overline{{{\mathcal{O}}}_{2}}$|⁠. So part (2) fails in type |$B_{3}$| for some |$w\in{\mathcal{W}}$|⁠. We shall see that (1) also fails. Similarly, these statements fail for types |$B_{n}$| and |$C_{n}$|⁠, |$n\geq 2$|⁠.

We prove the following for |${{\mathfrak{g}}}$| of type |$B_{n}$| and |$C_{n}$|⁠, |$n\geq 2$|⁠. Theorem 1.2 will follow (for all integral infinitesimal characters).

We need several known facts contained in [8] about the cell structure for type |$C_{n}$|⁠, which we collect here.

4.4 Half-integral case

Theorem 1.2 will now be proved for highest weight Harish-Chandra modules having half-integral highest weight. As mentioned in the introduction, when |$\lambda $| is half-integral then either |$L(\lambda )$| is irreducible (in which case the associated variety is |$\overline{{{\mathcal{O}}}_{r}}={{\mathfrak{p}}}^{+}$|⁠) or  |$\Delta =\Delta ({{\mathfrak{g}}},{{\mathfrak{h}}})$| is non-simply laced. We assume for the remainder of this section that |$\Delta $| is non-simply laced and |$\lambda $| is half-integral.

The two cases to consider are |${\mathfrak{sp}}(2n,{\mathbb{R}})$| and |${\mathfrak{so}}(2,2n-1)$|⁠. The integral root systems |$\Delta _{\lambda }$| and corresponding Lie algebras |${{\mathfrak{g}}}_{\lambda }$| are described in Section 2.5. Recall that |$W_{\lambda }$| denotes the Weyl group of |$\Delta _{\lambda }$| and we have fixed the positive system |$\Delta _{\lambda }^{+}=\Delta ^{+}\cap \Delta _{\lambda }$|⁠. Let |$\Pi _{\lambda }$| be the simple roots for this positive system. The Lie algebra |${{\mathfrak{g}}}_{\lambda }$| has a real form of Hermitian type and we my choose corresponding group |$G_{\lambda ,{\mathbb{R}}}$| to be either |$SO^{*}(2n)$| or |$SL(2,{\mathbb{R}})$|⁠. Then, since |$\lambda $| is integral for |$\Delta _{\lambda }$|⁠, |$L(\lambda )$| is an Harish-Chandra module for |$G_{\lambda ,{\mathbb{R}}}$|⁠. Furthermore, letting |$K_{\lambda }$| be the complexification of a maximal compact subgroup, the associated varieties of highest weight Harish-Chandra modules are the closures of the |$K_{\lambda }$|-orbits in |${{\mathfrak{p}}}_{\lambda }^{+}$|⁠, which we denote by |${{\mathcal{O}}}_{\lambda ,0},{{\mathcal{O}}}_{\lambda ,1},\dots $|⁠.

In our proof of Theorem 1.2 we first determine the GK dimension of |$L(\lambda )$|⁠. This is done by applying Lusztig’s formula for the GK dimension given in terms of his |$a$|-function. Since the possible associated varieties are the |$\overline{{{\mathcal{O}}}_{k}}$|⁠, |$k=0,1,\dots ,r$|⁠, which form an increasing sequence, the GK dimension determines the associated variety. The key fact is that the |$a$|-function depends only on the Coxeter system |$(W,\Pi )$|⁠, therefore we are able to relate the GK dimension of |$L(\lambda )$| to the GK dimension of a highest weight Harish-Chandra module for |${{\mathfrak{g}}}_{\lambda ,{\mathbb{R}}}$|⁠, then we may apply the integral case of Theorem 1.2 proved in the preceding sections.

For Lusztig’s |$a$|-function see [22], [23], [5], and [4]. However, details about the |$a$|-function are not needed here. Lusztig’s formula for the GK dimension may be stated as follows. If |$\lambda +\rho $| is regular and |$w\in W_{\lambda }$| is chosen so that |$w^{-1}(\lambda +\rho )$| is antidominant, then

where |$a_{\lambda }$| is Lusztig’s |$a$|-function for |$(W_{\lambda },\Pi _{\lambda })$|⁠. In particular,

where |$a$| is the |$a$|-function for |$(W,\Pi )$|⁠.

When |$\lambda +\rho $| is singular, then choose |$w$| to be the minimal length element of |$W_{\lambda }$| for which |$w^{-1}(\lambda +\rho )$| is antidominant, then

See [4, Prop. 1.1] for this formula.

Theorem 1.2 now follows easily for the half-integral case by comparing dimensions of orbits using Table 2.

Assume |$\operatorname{width}(Y_{\lambda })=m$|⁠.

(a) |${{\mathfrak{g}}}_{\mathbb{R}}={\mathfrak{sp}}(2n,{\mathbb{R}}), n\geq 2$|⁠. The real rank is |$r=n$|⁠.

(b) |${{\mathfrak{g}}}_{\mathbb{R}}={\mathfrak{so}}(2,2n-1), n\geq 2$|⁠. The real rank is |$r=2$|⁠.

Therefore,

in both cases.

5 Another Approach

In this section, we will give another proof for Theorem 1.2. In [4] and [3], Bai–Xie and Bai–Xiao–Xie have found an algorithm to compute the GK dimensions of highest weight modules of classical Lie algebras. We recall their results and will give another proof for our Theorem 1.2.

For a totally ordered set |$ \Gamma $|⁠, we denote by |$ \mathrm{Seq}_{n} (\Gamma )$| the set of sequences |$ x=(x_{1},x_{2},\cdots , x_{n}) $| of length |$ n $| with |$ x_{i}\in \Gamma $|⁠. We say |$q=(q_{1}, \cdots , q_{N})$| is the dual partition of a partition |$p=(p_{1}, \cdots , p_{N})$| and write |$q=p^{t}$| if |$q_{i}$| is the length of |$i$|-th column of the Young diagram |$p$|⁠. Let |$p(x)$| be the shape of the Young tableau |$Y(x)$| obtained by applying Robinson–Schensted algorithm to |$x\in \mathrm{Seq}_{n} (\Gamma )$|⁠. For convenience, we set |$q(x)=p(x)^{t}$|⁠.

For a Young diagram |$p$|⁠, use |$ (k,l) $| to denote the box in the |$ k $|-th row and the |$ l $|-th column. We say the box |$ (k,l) $| is even (resp. odd) if |$ k+l $| is even (resp. odd). Let |$ p_{i}^{\mathrm{ev}}$| (resp. |$ p_{i}^{\mathrm{odd}} $|⁠) be the number of even (resp. odd) boxes in the |$ i $|-th row of the Young diagram |$ p $|⁠. One can easily check that

Here for |$ a\in \mathbb{R} $|⁠, |$ \lfloor a \rfloor $| is the largest integer |$ n $| such that |$ n\leq a $|⁠, and |$ \lceil a \rceil $| is the smallest integer |$n$| such that |$ n\geq a $|⁠. For convenience, we set

For |$ x=(x_{1},x_{2},\cdots ,x_{n})\in \mathrm{Seq}_{n} (\Gamma ) $|⁠, set

Let |$\Delta $| be the root system of |$E_{6}$| or |$E_{7}$|⁠. As usual |$\Delta $| can be realized as a subset of |$\mathbb{R}^{8}$|⁠. The simple roots are

where |$n=6, 7$|⁠. Here we use notations in [12]. Let |$\Pi $| denote the set of simple roots in |$\Delta ^{+}$|⁠. Exactly one root in |$\Pi $| is in |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. In particular, for |$\mathfrak{e}_{6(-14)}$|⁠, this simple root is |$\alpha _{1}$|⁠; for |$\mathfrak{e}_{7(-25)}$|⁠, this simple root is |$\alpha _{7}$|⁠.

For an integral weight |$ \lambda \in \mathfrak{h}^{*}$|⁠, recall that |$ \Delta _{\lambda }^{+}=\{\alpha \in \Delta ^{+}\mid \langle \lambda +\rho \,, \alpha ^{\vee }\rangle>0\} $|⁠.

Now we will give our proof for Theorem 1.2 by the algorithms in [3, 4]. In the following we denote |$m(\lambda ):=\operatorname{width}(Y_{\lambda })$|⁠. We will show that the values of |$k^{\prime}(\lambda )$| in Proposition 5.1, Proposition 5.2, and Proposition 5.3 will equal to the values of |$k(\lambda )$| given in Theorem 1.2.

5.1 |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{su}(p,q)$|

We use the notations from [12]. Suppose |$L(\lambda )$| is a highest weight Harish-Chandra (⁠|$\mathfrak{g}$|⁠,|$K$|⁠)-module with highest weight |$\lambda $|⁠. We denote |$\lambda +\rho =(t_{1},...,t_{n})$|⁠. Suppose |$\lambda $| is integral. Then from [12], we have |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$1\leq i\leq p$|⁠, |$p+1\leq j\leq n$|⁠. From [4], we say that |$(t_{1},...,t_{n})$| is |$(p,q)$|-dominant.

By using Schensted insertion algorithm for |$(t_{1},...,t_{n})$|⁠, we can get a Young tableau |$Y_{\lambda }$| which consists of at most two columns. Now we suppose the number of entries in the second column of |$Y_{\lambda }$| is |$c_{2}(Y_{\lambda })=q_{2}$|⁠, then |$c_{1}(Y_{\lambda })=n-q_{2}$|⁠. From Proposition 5.1, we know |$ \operatorname{AV}(L(\lambda ))= \overline{\mathcal{O}_{q_{2}}}. $|

Then from Lemma 5.4, we know |$q_{2}$| is equal to the maximum integer |$t$| for which there exists a sequence of indices

such that

For |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{su}(p,q)$|⁠, we know |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{i}-\varepsilon _{j}| 1\leq i\leq p, ~p+1\leq j\leq n\}$|⁠. So |$m=q_{2}$| is just our |$m(\lambda )$| in Theorem 1.2.

Thus, |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{q_{2}}}=\overline{\mathcal{O}_{m}}=\overline{\mathcal{O}_{m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

When |$\lambda $| is non-integral, we will have |$N(\lambda )=L(\lambda )$| by [12, Lem. 3.17]. So |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{r}}$|⁠.

5.2 |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{sp}(2n,\mathbb{R})$|

Suppose |$L(\lambda )$| is a highest weight Harish-Chandra module with highest weight |$\lambda $|⁠. We denote |$\lambda +\rho =(t_{1},...,t_{n})$|⁠. Suppose |$\lambda $| is integral. Then from [12], we have |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$1\leq i<j\leq n$|⁠. From [4], we say that |$(t_{1},...,t_{n},-t_{n},-t_{n-1},...,-t_{2},-t_{1})$| is |$(n,n)$|-dominant.

By using Schensted insertion algorithm for |$(t_{1},...,t_{n},-t_{n},-t_{n-1},...,-t_{2},-t_{1})$|⁠, we can get a Young tableau |$Y(\lambda ^{-})$| which consists of at most two columns. Now we suppose the number of entries in the second column of |$Y(\lambda ^{-})$| is |$c_{2}(Y(\lambda ^{-}))=q_{2}$|⁠, then |$c_{1}(Y(\lambda ^{-}))=2n-q_{2}$|⁠. From Proposition 5.2, we know

Then from Lemma 5.4, we know |$q_{2}$| is equal to the maximum integer |$t$| for which there exists a sequence of indices

such that

Since |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$1\leq i<j\leq n$|⁠, we can make a new choice for these indices such that

Thus, we have

When |$q_{2}=t=2t^{\prime}+1$| is an odd number, we will have

From from Lemma 5.4, we know |$m=t^{\prime}$| is the maximum integer for which there exists a sequence of indices

such that

For |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{sp}(2n,\mathbb{R})$|⁠, we know |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{i}+\varepsilon _{j}| 1\leq i< j\leq n\}\cup \{2\varepsilon _{l}| 1\leq l \leq n\}$|⁠. So |$m+1$| is just our |$m(\lambda )$| in Theorem 1.2.

Thus, |$q_{2}+1=t+1=2(m+1)$| and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{q_{2}+1}}=\overline{\mathcal{O}_{2(m+1)}}=\overline{\mathcal{O}_{2m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

When |$q_{2}=t=2t^{\prime}$| is an even number, we will have

From the definition of |$t$|⁠, we know |$m=t^{\prime}$| is the maximum integer for which there exists a sequence of indices

such that

So |$m$| is just our |$m(\lambda )$| in Theorem 1.2.

Thus, |$q_{2}=t=2m$| and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{q_{2}}}=\overline{\mathcal{O}_{2m}}=\overline{\mathcal{O}_{2m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

Now we suppose |$\lambda $| is half-integral. Then from [12], we have |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$1\leq i<j\leq n$| and |$t_{k}-\frac{1}{2} \in \mathbb{Z}$| for |$1\leq k\leq n$|⁠. By using Schensted insertion algorithm for |$(t_{1},...,t_{n},-t_{n},-t_{n-1},...,-t_{2},-t_{1})$|⁠, we can get a Young tableau |$Y(\lambda ^{-})$| which consists of at most two columns. Now we suppose the number of entries in the second column of |$Y(\lambda ^{-})$| is |$c_{2}(Y(\lambda ^{-}))=q_{2}$|⁠, then |$c_{1}(Y(\lambda ^{-}))=2n-q_{2}$|⁠. From Proposition 5.2, we know

When |$q_{2}=t=2t^{\prime}+1$| is an odd number, similarly to the integral case, we will have

and |$m=t^{\prime}$| is the maximum integer for which there exists a sequence of indices

such that

For |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{sp}(2n,\mathbb{R})$|⁠, we know |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{i}+\varepsilon _{j}| 1\leq i< j\leq n\}\cup \{2\varepsilon _{l}| 1\leq l \leq n\}$|⁠. Since |$t_{l}\notin \mathbb{Z}$|⁠, |$m$| is just our |$m(\lambda )$| in Theorem 1.2.

Thus, |$q_{2}=t=2m+1$| and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{q_{2}}}=\overline{\mathcal{O}_{2m+1}}=\overline{\mathcal{O}_{2m(\lambda )+1}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

When |$q_{2}=t=2t^{\prime}$| is an even number, similarly to above case, we will have |$m=m(\lambda )=t^{\prime}$| and |$q_{2}+1=2m+1$|⁠. Thus, |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{q_{2}+1}}=\overline{\mathcal{O}_{2m+1}}=\overline{\mathcal{O}_{2m(\lambda )+1}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

5.3 |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{so}^{*}(2n)$|

Suppose |$L(\lambda )$| is a highest weight Harish-Chandra module with highest weight |$\lambda $|⁠. We denote |$\lambda +\rho =(t_{1},...,t_{n})$|⁠. Then from [12, Lem. 3.17], we may suppose |$\lambda $| is integral (otherwise |$N(\lambda )$| will be irreducible). Also we have |$t_{1}+t_{2}\in \mathbb{Z}$| and |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$1\leq i<j\leq n$|⁠. Then we have

From [4], we say that |$(t_{1},...,t_{n},-t_{n},-t_{n-1},...,-t_{2},-t_{1})$| is |$(n,n)$|-dominant. When |$t_{n}\geq 0$|⁠, |$L(\lambda )$| will be finite-dimensional. We only consider infinite-dimensional modules. Thus, we suppose |$t_{n}<0$|⁠.

By using Schensted insertion algorithm for |$(t_{1},\dots ,t_{n},-t_{n},-t_{n-1},\ldots ,-t_{2},-t_{1})$|⁠, we can get a Young tableau |$Y(\lambda ^{-})$| which consists of at most two columns. Now we suppose the number of entries in the second column of |$Y(\lambda ^{-})$| is |$c_{2}(Y(\lambda ^{-}))=q_{2}$|⁠, then |$c_{1}(Y(\lambda ^{-}))=2n-q_{2}$|⁠. From Proposition 5.2, we know

Then similar to the case of |${{\mathfrak{s}}}{{\mathfrak{p}}}(2n,{\mathbb{R}})$|⁠, we know |$q_{2}$| is equal to the maximum integer |$t$| for which there exists a sequence of indices

such that

When |$q_{2}=t=2t^{\prime}+1$| is an odd number, we will have

From the definition of |$t$|⁠, we know |$m=t^{\prime}$| is the maximum integer for which there exists a sequence of indices

such that

For |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{so}^{*}(2n)$|⁠, we know |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{i}+\varepsilon _{j}| 1\leq i< j\leq n\}$|⁠. So |$m=t^{\prime}=\frac{q_{2}-1}{2}$| is just our |$m(\lambda )$| in Theorem 1.2. Thus, |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{\frac{q_{2}-1}{2}}}=\overline{\mathcal{O}_{m}}=\overline{\mathcal{O}_{m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

When |$q_{2}=t=2t^{\prime}$| is an even number, similarly we will have |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{\frac{q_{2}}{2}}}=\overline{\mathcal{O}_{m}}=\overline{\mathcal{O}_{m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

5.4 |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{so}(2,2n-1)$|

Suppose |$L(\lambda )$| is a highest weight Harish-Chandra module with highest weight |$\lambda $|⁠. We denote |$\lambda +\rho =(t_{1},...,t_{n})$|⁠. Suppose |$\lambda $| is integral. Then from [12], we have |$t_{n}> 0$|⁠, |$t_{1}-t_{2}\in \mathbb{Z}$|⁠, |$2t_{k}\in \mathbb{Z}$| for |$1\leq k\leq n $| and |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$2\leq i<j\leq n$|⁠. Then we have |$t_{2}>t_{3}>...>t_{n}>-t_{n}>...>-t_{2} $|⁠. When |$t_{1}>t_{2}$|⁠, |$L(\lambda )$| will be finite-dimensional and |$V(L(\lambda ))=\overline{\mathcal{O}}_{0}$|⁠. We only consider infinite-dimensional modules. Thus, we suppose |$t_{1}\leq t_{2}$|⁠.

By using Schensted insertion algorithm for |$(t_{1},...,t_{n},-t_{n},-t_{n-1},...,-t_{2},-t_{1})$|⁠, we can get a Young tableau |$Y(\lambda ^{-})$| which consists of at most three columns. From the construction process, we can see that |$Y(\lambda ^{-})$| will be a Young tableau consisting of two columns with |$c_{1}(Y(\lambda ^{-}))=2n-2$| and |$c_{2}(Y(\lambda ^{-}))=2$| when |$t_{2}\geq t_{1}>-t_{n}$|⁠, and it will be a Young tableau consisting of three columns with |$c_{1}(Y(\lambda ^{-}))=2n-2$|⁠, |$c_{2}(Y(\lambda ^{-}))=1$| and |$c_{3}(Y(\lambda ^{-}))=1$| when |$t_{1}\leq -t_{n}$|⁠.

From Proposition 5.2, we have |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{2}}$| for both cases.

For |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{so}(2,2n-1)$|⁠, we know |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{1}\pm \varepsilon _{j}| 2\leq i< j\leq n\}\cup \{\varepsilon _{l}| 1\leq l \leq n\}$|⁠. When |$t_{2}\geq t_{1}>-t_{n}$|⁠, we have |$t_{1}-t_{2}\leq 0$|⁠. So |$m=1$| is just our |$m(\lambda )$| in Theorem 1.2 and |$AV(L(\lambda ))=\overline{\mathcal{O}_{2}}=\overline{\mathcal{O}_{2m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

When |$t_{1}\leq -t_{n}$|⁠, we have |$t_{1}+t_{n}\leq 0$|⁠. So |$m=1$| is just our |$m(\lambda )$| in Theorem 1.2 and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{2}}=\overline{\mathcal{O}_{2m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

Now we suppose |$\lambda $| is half-integral. Then from Enright–Howe–Wallach [12], we have |$t_{n}> 0$|⁠, |$t_{1}-t_{2}\in \frac{1}{2}+\mathbb{Z}$|⁠, |$2t_{k}\in \mathbb{Z}$| for |$1\leq k\leq n $| and |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$2\leq i<j\leq n$|⁠. Then we have |$t_{2}>t_{3}>...>t_{n}>-t_{n}>...>-t_{2} $|⁠.

From Proposition 5.2, we have

When |$t_{1}>0$|⁠, we find that |$m(\lambda )=0$|⁠. So |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{1}}=\overline{\mathcal{O}_{2m(\lambda )+1}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

When |$t_{1}\leq 0$|⁠, we have |$m(\lambda )=1$| and |$2m(\lambda )+1=3>2$|⁠. We still have |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{2}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

5.5 |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{so}(2,2n-2)$|

Suppose |$L(\lambda )$| is a highest weight Harish-Chandra module with highest weight |$\lambda $|⁠. We denote |$\lambda +\rho =(t_{1},...,t_{n})$|⁠. Then from [12, Lem. 3.17], we may suppose |$\lambda $| is integral (otherwise |$N(\lambda )$| will be irreducible). Also, we have |$t_{1}- t_{2}\in \mathbb{Z}$|⁠, |$t_{n-1}-|t_{n}|>0$| and |$t_{i}-t_{j} \in \mathbb{Z}_{> 0}$| for |$2\leq i<j\leq n$|⁠. Then we have

When |$t_{1}>t_{2}$|⁠, |$L(\lambda )$| will be finite-dimensional and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{0}}$|⁠. We only consider infinite-dimensional modules. Thus, we suppose |$t_{1}\leq t_{2}$|⁠.

By using Schensted insertion algorithm for |$(t_{1},...,t_{n},-t_{n},-t_{n-1},...,-t_{2},-t_{1})$|⁠, we can get a Young tableau |$Y(\lambda ^{-})$| which consists of at most four columns.

From Proposition 5.2, when |$-|t_{n}|< t_{1}\leq t_{2}$|⁠, then |$ q(\lambda ^{-})=(2n-2, 2)$| or |$(2n-3,3)$|⁠, we get |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{1}}$|⁠.

For |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{so}(2,2n-2)$|⁠, we know |$\Delta ({{\mathfrak{p}}}^{+})=\{\varepsilon _{1}\pm \varepsilon _{j}| 2\leq i< j\leq n\}$|⁠. For |$t_{n}> 0$| and |$t_{2}\geq t_{1}>-t_{n}$|⁠, we have |$t_{1}-t_{2}\leq 0$| and |$t_{1}+t_{n}>0$|⁠. For |$t_{n}\leq 0$| and |$t_{2}\geq t_{1}> t_{n}$|⁠, we have |$t_{1}-t_{2}\leq 0$| and |$t_{1}-t_{n}>0$|⁠. So for both cases, |$m=1$| is just our |$m(\lambda )$| in Theorem 1.2 and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{1}}=\overline{\mathcal{O}_{m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

From Proposition 5.2, when |$ t_{1}\leq -|t_{n}| $|⁠, then |$ q(\lambda ^{-})=(2n-2, 1^{2})$| or |$(2n-3, 1^{3})$|⁠, one has |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{2}}$|⁠.

When |$t_{1}\leq -t_{n}<0$|⁠, we have |$t_{1}+t_{n}\leq 0$| and |$t_{1}-t_{n}\leq 0$|⁠. When |$t_{1}\leq t_{n}\leq 0$|⁠, we have |$t_{1}-t_{n}\leq 0$| and |$t_{1}+t_{n}\leq 0$|⁠. So for both cases, |$m=2$| is just our |$m(\lambda )$| in Theorem 1.2 and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{2}}=\overline{\mathcal{O}_{m(\lambda )}}=\overline{\mathcal{O}_{k(\lambda )}}$|⁠.

5.6 |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{e}_{6(-14)}$|

Suppose |$L(\lambda )$| is a highest weight Harish-Chandra module with highest weight |$\lambda $|⁠. We denote |$\lambda +\rho =(t_{1},...,t_{8})$|⁠. Then from [12, Lem. 3.17], we may suppose |$\lambda $| is integral (otherwise |$N(\lambda )$| will be irreducible). Also, we have |$|t_{1}|< t_{2}<...< t_{5}$|⁠, |$t_{i}-t_{j}\in \mathbb{Z}$| and |$2t_{i}\in \mathbb{Z}$| for all |$1\leq i, j\leq 5$|⁠.

The Hasse diagram of |$\Delta ({{\mathfrak{p}}}^{+})$| for |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{e}_{6(-14)}$| is shown in the Appendix.

The highest root is |$\theta = \alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+2\alpha _{5}+\alpha _{6}=\frac{1}{2}(1,1,1,1,1,-1,-1,1)$|⁠.

We denote

and

From Proposition 5.3, we have|$A_{1}=\{\alpha _{1}\}$| and |$A_{2}=\{\beta _{1},\beta _{2}\}$|⁠.

If |$ \Delta _{\lambda }^{+}\cap A_{1}\neq \emptyset $|⁠, we will have |$\langle \lambda +\rho \,, \alpha _{1}^{\vee }\rangle>0$|⁠. Since |$\alpha _{1}$| is the minimal noncompact root in |$\Delta ({{\mathfrak{p}}}^{+})$|⁠, we will have |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle>0$| for all |$\alpha \in \Delta ({{\mathfrak{p}}}^{+})$| and |$m(\lambda )=0$|⁠. Thus, |$L(\lambda )$| will be finite-dimensional and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{0}}$|⁠. So |$k(\lambda )=m(\lambda )=0$|⁠.

If |$\Delta _{\lambda }^{+}\cap A_{1}=\emptyset $| and |$\Delta _{\lambda }^{+}\cap A_{2}\neq \emptyset $|⁠, we will have |$\langle \lambda +\rho \,, \alpha _{1}^{\vee }\rangle \leq 0$| and there is at least one root of |$A_{2}$| satisfying |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle> 0$|⁠. So we have |$m(\lambda )\geq 1$|⁠. From the diagram of |$\Delta ({{\mathfrak{p}}}^{+})$| for |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{e}_{6(-14)}$|⁠, we find that if an antichain |$A\subseteq \Delta ({{\mathfrak{p}}}^{+}) $| has |$m(\lambda )=2$|⁠, it must contain |$\beta _{1}$| and |$\beta _{2}$| since they are the smallest incompatible pair in |$\Delta ({{\mathfrak{p}}}^{+}) $|⁠. So we must have |$m(\lambda )=1$| since |$\Delta _{\lambda }^{+}\cap A_{2}\neq \emptyset $|⁠. Thus, |$k(\lambda )=m(\lambda )=1$|⁠.

If |$\Delta _{\lambda }^{+}\cap A_{2}=\emptyset $|⁠, |$\beta _{1}$| and |$\beta _{2}$| will satisfy that |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle \leq 0$|⁠. Thus, we have |$m(\lambda )\geq 2$|⁠. The equality must hold since |$m(\lambda )\leq 2$| by the diagram of |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. So |$k(\lambda )=m(\lambda )=2$|⁠.

5.7 |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{e}_{7(-25)}$|

Suppose |$L(\lambda )$| is a highest weight Harish-Chandra module with highest weight |$\lambda $|⁠. We denote |$\lambda +\rho =(t_{1},...,t_{8})$|⁠. Then from [12, Lem. 3.17], we may suppose |$\lambda $| is integral (otherwise |$N(\lambda )$| will be irreducible). Also, we have |$|t_{1}|< t_{2}<...< t_{5}$|⁠, |$t_{i}-t_{j}\in \mathbb{Z}$| and |$2t_{i}\in \mathbb{Z}$| for all |$1\leq i, j\leq 5$|⁠.

The Hasse diagram of |$\Delta ({{\mathfrak{p}}}^{+})$| for |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{e}_{7(-25)}$| is shown in the Appendix.

The highest root is |$\theta = 2\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}$|⁠.

We denote

and

From Proposition 5.3, we have |$A_{1}=\{\alpha _{7}\}$|⁠, |$A_{2}=\{\beta _{1},\beta _{2}\}$|⁠, and |$A_{3}=\{\gamma _{1},\gamma _{2}, \gamma _{3}\}$|⁠.

If |$ \Delta _{\lambda }^{+}\cap A_{1}\neq \emptyset $|⁠, we will have |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle>0$|⁠. Since |$\alpha _{7}$| is the minimal noncompact root in |$\Delta ({{\mathfrak{p}}}^{+})$|⁠, we will have |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle>0$| for all |$\alpha \in \Delta ({{\mathfrak{p}}}^{+})$| and |$m(\lambda )=0$|⁠. Thus, |$L(\lambda )$| will be finite-dimensional and |$\operatorname{AV}(L(\lambda ))=\overline{\mathcal{O}_{0}}$|⁠. So |$k(\lambda )=m(\lambda )=0$|⁠.

If |$\Delta _{\lambda }^{+}\cap A_{1}=\emptyset $| and |$\Delta _{\lambda }^{+}\cap A_{2}\neq \emptyset $|⁠, we will have |$\langle \lambda +\rho \,, \alpha _{7}^{\vee }\rangle \leq 0$| and there is at least one root of |$A_{2}$| satisfying |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle> 0$|⁠. So we have |$m(\lambda )\geq 1$|⁠. From the diagram of |$\Delta ({{\mathfrak{p}}}^{+})$| for |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{e}_{7(-25)}$|⁠, we find that if an antichain |$A\subseteq \Delta ({{\mathfrak{p}}}^{+}) $| has |$m(\lambda )=2$|⁠, it must contain |$\beta _{1}$| and |$\beta _{2}$| since they are the smallest incompatible pair in |$\Delta ({{\mathfrak{p}}}^{+}) $|⁠. So we must have |$m(\lambda )=1$| since |$\Delta _{\lambda }^{+}\cap A_{2}\neq \emptyset $|⁠. Thus, |$k(\lambda )=m(\lambda )=1$|⁠.

If |$\Delta _{\lambda }^{+}\cap A_{2}=\emptyset $| and |$\Delta _{\lambda }^{+}\cap A_{3}\neq \emptyset $|⁠, |$\beta _{1}$| and |$\beta _{2}$| will satisfy that |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle \leq 0$|⁠. Thus, we have |$m(\lambda )\geq 2$|⁠. From the diagram of |$\Delta ({{\mathfrak{p}}}^{+})$| for |$\mathfrak{g}_{\mathbb{R}}=\mathfrak{e}_{7(-25)}$|⁠, we find that if an antichain |$A\subseteq \Delta ({{\mathfrak{p}}}^{+}) $| has |$m(\lambda )=3$|⁠, it must contain |$\gamma _{1}$|⁠, |$\gamma _{2}$| and |$\gamma _{3}$| since they are the smallest incompatible triples in |$\Delta ({{\mathfrak{p}}}^{+}) $|⁠. So we must have |$m(\lambda )=2$| since |$\Delta _{\lambda }^{+}\cap A_{3}\neq \emptyset $|⁠. Thus, |$k(\lambda )=m(\lambda )=2$|⁠.

If |$\Delta _{\lambda }^{+}\cap A_{3}=\emptyset $|⁠, then |$\gamma _{1}$|⁠, |$\gamma _{2}$| and |$\gamma _{3}$| will satisfy that |$\langle \lambda +\rho \,, \alpha ^{\vee }\rangle \leq 0$|⁠. Thus, we have |$m(\lambda )\geq 3$|⁠. The equality must hold since |$m(\lambda )\leq 3$| by the diagram of |$\Delta ({{\mathfrak{p}}}^{+})$|⁠. So |$k(\lambda )=m(\lambda )=3$|⁠.

This finishes our second proof of Theorem 1.2.

Acknowledgments

Z.B. was supported by the National Natural Science Foundation of China (no. 12171344) and the National Key |$\mathrm{R}\,\&\,\mathrm{D}$| Program of China (nos. 2018YFA0701700 and 2018YFA0701701). X.X. was supported by the National Natural Science Foundation of China (nos. 12171030 and 12431002).

A Hasse Diagrams and Distinguished Antichains

The figures below show the Hasse diagrams of |$\Delta ({{\mathfrak{p}}}^{+})$| for |${{\mathfrak{g}}}_{\mathbb{R}}$| in the simply-laced types. Let |$c$| be the constant given in [12, Table on p. 115] and shown below. Then the roots in the distinguished antichains |$A_{k}=\{\alpha \in \Delta ({{\mathfrak{p}}}^{+}): \operatorname{ht}(\alpha )=(k-1)c+1\}$|⁠, |$k=1,2,\ldots , r$|⁠, are labelled so that |$A_{1}=\{\alpha _{*}\}$|⁠, |$A_{2}=\{\beta _{1},\beta _{2}\}$|⁠, and |$A_{3}=\{\gamma _{1},\gamma _{2},\gamma _{3}\}$|⁠.

|${{\mathfrak{s}}}{{\mathfrak{u}}}(p,q)$|⁠, here shown for |$p=4$|⁠, |$q=3$|⁠:

|${{\mathfrak{s}}}{{\mathfrak{o}}}(2,2n-2)$|⁠, here shown for |$n=6$|⁠:

|${{\mathfrak{s}}}{{\mathfrak{o}}}^{*}(2n)$|⁠, here shown for |$n=6$|⁠:

|$\mathfrak{e}_{6(-14)}$|⁠:

|$\mathfrak{e}_{7(-25)}$|⁠:

Communicated by Prof. Weiqiang Wang

References