March 23, 2004 19:30 WSPC/Trim Size: 9in x 6in for Proceedings idha2003rev

ISOTROPY REPRESENTATION FOR HARISH-CHANDRA MODULE

HIROSHI YAMASHITA Department of Mathematics, Hokkaido University Sapporo 060-0810, JAPAN E-mail: yamasita@math.sci.hokudai.ac.jp

We study the isotropy representation attached to an irreducible Harish-Chandra module with irreducible associated variety. It is shown that, under some assumptions, the dual of the isotropy representation in question can be characterized by means of the principal symbol of a differential operators of gradient type. By using this, the case of Harish-Chandra module of discrete series is more closely examined.

1. Introduction

Let 𝔤 be a complex semisimple Lie algebra with a nontrivial involutive automorphism θ of 𝔤. We write 𝔤 = 𝔨 ⊕ 𝔭 for the symmetric decomposition of 𝔤 given by θ, where 𝔨 and 𝔭 denote the +1 and −1 eigenspaces for θ, respectively. Let K_{ℂ} be a connected complex algebraic group with Lie algebra 𝔨. We assume that the natural inclusion 𝔨 ↪ 𝔤 gives rise to a group homomorphism from K_{ℂ} to G_{ℂ}^{ad} through the exponential map. Here G_{ℂ}^{ad} denotes the adjoint group of 𝔤. Then, this homomorphism naturally induces the adjoint representation Ad of K_{ℂ} on 𝔤.

We say that a finitely generated 𝔤-module X is a (𝔤, K_{ℂ})-module, or a Harish-Chandra module, if the action on X of the Lie subalgebra 𝔨 is locally finite and if it lifts up to a representation of K_{ℂ} on X through the exponential map in such a way as (k ·X ·k^{−1})v = (Ad(k)X)v for X ∈ 𝔤, k ∈ K_{ℂ} and v ∈ X. It is a fundamental result of Harish-Chandra that the study of irreducible admissible representations of a real semisimple Lie group essentially reduces to that of irreducible (𝔤, K_{ℂ})-modules.

Let X be an irreducible (𝔤, K_{ℂ})-module. A K_{ℂ}-stable good filtration of X naturally gives rise to a graded, compatible (S(𝔤), K_{ℂ})-module gr X annihilated by 𝔨, where S(𝔤) denotes the symmetric algebra of 𝔤. By Borho-Brylinski [1] and Vogan [22], [23], the associated cycle C(X) of X is defined to be the support V(X) of gr X combined with the multiplicity at each

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irreducible component of V(X). The support V(X) is called the associated variety of X. It is a K_{ℂ}-stable affine algebraic cone contained in the set of nilpotent elements in 𝔭, and each irreducible component of V(X) is the closure \overline{O} of a nilpotent K_{ℂ}-orbit O in 𝔭. As we have shown in [6] and [27], the variety V(X) controls some fundamental properties for X.

The algebraic cycle C(X) describes a sort of asymptotic behavior of X (cf. [21]). Moreover, it is shown by Vogan [22, Theorem 2.13] that the multiplicity of X at an irreducible component \overline{O} of V(X) can be interpreted as the dimension of a certain finite-dimensional representation (ϖ_{O}, W) of the isotropy subgroup K_{ℂ}(X) of K_{ℂ} at an X ∈ O. We call ϖ_{O} an isotropy representation attached to X. In terms of ϖ_{O}, the associated cycle C(X) of X is expressed as

C(X) = ∑_{O} dim ϖ_{O} · [\overline{O}].  (1.1)

Now, we assume that the associated variety V(X) of X is the closure of a single nilpotent K_{ℂ}-orbit O in 𝔭. This assumption does not exclude important (𝔤, K_{ℂ})-modules related to elliptic orbits. In reality, it is wellknown that the Harish-Chandra modules of discrete series (more generally Zuckerman derived functor modules) and also the irreducible admissible highest weight modules of hermitian Lie algebras satisfy this hypothesis.

The purpose of this paper is to study the associated cycle C(X) and in particular the isotropy representation ϖ_{O} attached to a (𝔤, K_{ℂ})-module X with irreducible associated variety, by developing our arguments in [30] for unitary highest weight modules and also those in [29] for discrete series.

To do this, we first look at in Section 2 a relationship between the (S(𝔤), K_{ℂ})-module gr X and the induced representation Γ(W) = Ind_{K_{ℂ}(X)}^{K_{ℂ}}(ϖ_{O}, W) of K_{ℂ} equipped with a natural S(𝔤)-action. This amounts to a survey of some aspects of Vogan’s work ([22, Sections 2–4] and [23, Lectures 6 and 7]) in a slightly modified and simplified form (but for limited X’s). A reciprocity law of Frobenius type for such an induced module (Proposition 2.2) plays an important role. In fact, it is effectively used to prove an irreducibility criterion for ϖ_{O} (Theorem 2.1). Also, we include a remarkable result (Theoren 2.2) on the isotropy representations for singular unitary highest weight modules, given in [30], [33] and [24].

In order to identify the isotropy representation ϖ_{O}, it is useful to consider not only gr X but also its K_{ℂ}-finite dual realized as a space of certain (vector valued) polynomial functions on 𝔭. We present this idea in Section 3. A sufficient condition is given in Proposition 3.1 for gr X being annihi-

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lated by the whole prime ideal I of S(𝔤) defining \overline{O}. In such a case, the isotropy representation, more precisely, its dual ϖ_{O}^{∗}, can be described by means of the principal symbol of a differential operator on 𝔭 of gradient type (see Propositions 3.2 and 3.3).

In the last part of this paper, Section 4, we focus our attention on the irreducible Harish-Chandra modules X of discrete series. As is well known, the associated variety of such an X is irreducible (cf. [28], [29]). The multiplicities in the associated cycles for discrete series have been intensively studied by Chang [2], [3], by means of the localization theory of Harish-Chandra modules. He succeeds to describe C(X) explicitly for the real rank one case. Taniguchi applies in [19] and [20] the results of Chang in order to specify Whittaker functions associated with discrete series for SU(n, 1), Spin(n, 1) and SO_{0}(2n, 2). Here in this paper, we exploit another way to identify C(X), by using a realization of X as the kernel of an invariant differential operator of gradient type on the Riemannian symmetric space (cf. [9], [18]; see also [26], [34]). Based on our results in Section 3 and also on the discussion in [29], we can construct a certain K_{ℂ}(X)-submodule U_{λ}(Q_{c}) of the representation (ϖ_{O}^{∗}, W^{∗}) contragredient to ϖ_{O}. Moreover some evidences are given for this subrepresentation being large enough in the whole ϖ_{O}^{∗}. The gained results are summarized as Theorem 4.2 and Corollary 4.1.

This article is an updated and enlarged version of the informal reports [31] and [32] appeared in RIMS Kōkyūroku.

2. Graded module gr X and induced representation Γ(W)

As in Section 1, let X be an irreducible (𝔤, K_{ℂ})-module with irreducible associated variety V(X) = \overline{O}, where O is a nilpotent K_{ℂ}-orbit in 𝔭. For later use, this section introduces some elementary aspects of Vogan’s theory on the associated cycle and the isotropy representation attached to X. The results in this section may be read off from [22] and [23] with some effort. Nevertheless, we include the proofs for these important results in order to make this paper more accessible.

2.1. Associated cycle and isotropy representation

First, we introduce our key notion precisely. Take an irreducible K_{ℂ}- submodule (τ, V_{τ}) of X, which yields a K_{ℂ}-stable good filtration of X

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in the following way:

\begin{matrix}X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ ⋯, with \\ X_{n} := U_{n}(𝔤)V_{τ} (n = 0, 1, 2, …). \end{matrix}  (2.1)

Here U(𝔤) denotes the universal enveloping algebra of 𝔤, and we write U_{n}(𝔤) (n = 0, 1, …) for the natural increasing filtration of U(𝔤). This filtration gives rise to a graded (S(𝔤), K_{ℂ})-module M = gr X, annihilated by S(𝔨), as follows:

M = gr X = ⨁_{n=0}^{∞} M_{n} with M_{n} := X_{n}/X_{n−1} (X_{−1} := {0}). (2.2)

We note that

M_{n} = S^{n}(𝔤)V_{τ} = S^{n}(𝔭)V_{τ} and M_{0} = V_{τ} ,  (2.3)

where S^{n}(𝔳) is the homogeneous component of the symmetric algebra S(𝔳) of degree n. By definition, the associated variety V(X) of X is identified with the affine algebraic variety of 𝔤 given by the annihilator ideal Ann_{S(𝔤)}M in S(𝔤) of M:

V(X) = {Z ∈ 𝔤 | f(Z) = 0 for all f ∈ Ann_{S(𝔤)}M} ⊂ 𝔭,  (2.4)

where S(𝔤) is viewed as the ring of polynomial functions on 𝔤 by identifying 𝔤 with its dual space through the Killing form B of 𝔤.

Throughout this section, we assume that V(X) is irreducible. The Hilbert Nullstellensatz tells us that the radical of Ann_{S(𝔤)}M coincides with the prime ideal I = I(V(X)) defining the irreducible variety V(X): I = \sqrt{Ann_{S(𝔤)}M}. So we see I^{n}M = \{0\} for some positive integer n, and we write n_{0} for the smallest n of this nature. Then, one gets a strictly decreasing filtration of the (S(𝔤), K_{ℂ})-module M as

M = I^{0}M ⊋ I^{1}M ⊋ ⋯ ⊋ I^{n_{0}}M = {0}.  (2.5)

By the multiplicity mult_{I}(X) of X at I is meant the length as an S(𝔤)_{I}- module of the localization M_{I} of M = gr X at the prime ideal I. Then, the associated cycle C(X) of X turns to be

C(X) = mult_{I}(X) · [\overline{O}] with V(X) = \overline{O}.  (2.6)

Note that this cycle does not depend on the choice of a good filtration (2.1) of X.

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Now, let us explain how the multiplicity mult_{I}(X) can be interpreted as the dimension of an isotropy representation. For this, we take an element X in the open K_{ℂ}-orbit O ⊂ V(X). Set

K_{ℂ}(X) := {k ∈ K_{ℂ}| Ad(k)X = X},

the isotropy subgroup of K_{ℂ} at X. We write 𝔪(X) for the maximal ideal of S(𝔤) which defines the one point variety {X} in 𝔤:

𝔪(X) := ∑_{Y ∈𝔤}(Y − B(Y, X))S(𝔤) for X ∈ O.  (2.7)

For each j = 0, …, n_{0} −1, we introduce a finite-dimensional representation ϖ_{O}(j) of (S(𝔤), K_{ℂ}(X)) acting on

W(j) := I^{j}M/𝔪(X)I^{j}M,  (2.8)

in the canonical way, and we set

(ϖ_{O}, W) := ⨁_{j=0}^{n_{0}−1} (ϖ_{O}(j), W(j)).  (2.9)

We call ϖ_{O} the isotropy representation attached to the data (X, V_{τ} , O), where V_{τ} yields the filtration (2.1) of X. The following lemma (cf. [22, Corollary 2.7]; see also [31, Remark 2.2]) is essential for our succeeding discussion.

Lemma 2.1. Let N be a finitely generated (S(𝔤), K_{ℂ})-module such that IN = {0}. Then, the length of S(𝔤)_{I}-module N_{I} is equal to the dimension of the vector space N/𝔪(X)N for every X ∈ O.

This lemma tells us that the length of the localized S(𝔤)_{I}-module (I^{j}M/I^{j+1}M)_{I} equals dim ϖ_{O}(j), by noting that the ideal I annihilates the subquotient I^{j}M/I^{j+1}M of M. Together with the exactness of localization, we immediately get the following

Proposition 2.1. One has mult_{I}(X) = dim ϖ_{O}. Moreover, the equality

mult_{I}(X) = dim ϖ_{O}(0) = dim M/𝔪(X)M  (2.10)

holds if and only if the support of the S(𝔤)-module IM is contained in the boundary ∂O = \overline{O}\O.

Remark 2.1. The representation ϖ_{O}(0) in (2.10) never vanishes because the annihilator ideal Ann_{S(𝔤)}M/IM is equal to I (cf. [30, Lemma 3.4]). Moreover, the equality (2.10) holds for a number of unitary (𝔤, K_{ℂ})-modules

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X with unique extreme K_{ℂ}-types V_{τ}. See Example 2.1 and Theorem 4.2 (1).

2.2. Induced module Γ(Z)

We consider a finite-dimensional (S(𝔤), K_{ℂ}(X))-module (ϖ, Z) with X ∈ O, where K_{ℂ}(X) acts on Z holomorphically. Let Γ(Z) denote the space of all left K_{ℂ}-finite, holomorphic functions f : K_{ℂ} → Z satisfying

f(yh) = ϖ(h)^{−1}f(y) (y ∈ K_{ℂ}, h ∈ K_{ℂ}(X)).

Namely, Γ(Z) consists of all K_{ℂ}-finite, holomorphic cross sections of the K_{ℂ}-homogeneous vector bundle K_{ℂ} ×_{K_{ℂ}(X)} Z on K_{ℂ}/K_{ℂ}(X) ≃ O. Then, Γ(Z) has a structure of (S(𝔤), K_{ℂ})-module by the following actions:

(D · f)(y) := ϖ(Ad(y)^{−1}D)f(y), (k · f)(y) := f(k^{−1}y),

for D ∈ S(𝔤), k ∈ K_{ℂ} and f ∈ Γ(Z). We call Γ(Z) the (S(𝔤), K_{ℂ})-module induced from ϖ. We note that, if Z is annihilated by the maximal ideal 𝔪(X), the S(𝔤)-action on Γ(Z) turns to be the multiplication of functions on the orbit O:

(D · f)(y) = D(Ad(y)X)f(y).  (2.11)

In this case, the annihilator in S(𝔤) of any nonzero function f ∈ Γ(Z) coincides with the prime ideal I defining \overline{O}.

Let M be any (S(𝔤), K_{ℂ})-module. If ρ is a homomorphism from M to Z as (S(𝔤), K_{ℂ}(X))-modules, we define a function T_{m} : K_{ℂ} → Z for each m ∈ M by putting

T_{m}(y) := ρ(y^{−1} · m) (y ∈ K_{ℂ}).  (2.12)

Then it is standard to verify that T_{m} lies in Γ(Z) and that the map T : m ↦ T_{m} (m ∈ M) gives an (S(𝔤), K_{ℂ})-homomorphism from M to Γ(Z). More precisely, one readily obtains the following reciprocity law of Frobenius type.

Proposition 2.2. Under the above notation, the assignment ρ ↦ T sets up a linear isomorphism

Hom_{S(𝔤),K_{ℂ}(X)}(M, Z) ≃ Hom_{S(𝔤),K_{ℂ}}(M, Γ(Z)).  (2.13)

Here, for Ω-modules A and B, we denote by Hom_{Ω}(A, B) the space of Ωhomomorphisms from A to B.

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2.3. Homomorphism T = ⊕_{j} T̃(j)

We now return to our setting in Section 2.1, where M = gr X for an irreducible (𝔤, K_{ℂ})-module X with V(X) = \overline{O}. Take an integer j such that 0 ≤ j ≤ n_{0} − 1. Let ρ(j) denote the natural quotient map from I^{j}M to W(j) = I^{j}M/𝔪(X)I^{j}M. Correspondingly, we get an (S(𝔤), K_{ℂ})- homomorphism T (j) : I^{j}M → Γ(W(j)) by Proposition 2.2. It follows that

Ker T (j) = ⋂_{Y ∈O} 𝔪(Y)I^{j}M ⊃ I^{j+1}M,  (2.14)

by the definition of T (j) together with 𝔪(Y) ⊃ I (Y ∈ O).

Proposition 2.3. Ker T (j) is the largest (S(𝔤), K_{ℂ})-submodule of I^{j}M among those N having the following two properties: (i) N ⊃ I^{j+1}M, and, (ii) the support of N/I^{j+1}M is contained in ∂O.

Proof. First, we show that Ker T (j) have two properties (i) and (ii). The inclusion (2.14) assures (i). As for (ii), we consider a short exact sequence of (S(𝔤), K_{ℂ})-modules:

0 → Ker T (j)/I^{j+1}M → I^{j}M/I^{j+1}M → I^{j}M/Ker T (j) → 0. (2.15)

Each module is annihilated by I. In view of Lemma 2.1, we find that the multiplicity of I^{j}M/Ker T (j) at I is equal to the dimension of vector space

(I^{j}M/Ker T (j))/𝔪(X)(I^{j}M/Ker T (j)) ≃ I^{j}M/(𝔪(X)I^{j}M + Ker T (j)) = W(j). (2.16)

Here, the last equality follows from Ker T (j) ⊂ 𝔪(X)I^{j}M (see (2.14)). This shows that the length of S(𝔤)_{I}-module I^{j}M/I^{j+1}M and that of I^{j}M/Ker T (j) coincide with one another. Hence (Ker T (j)/I^{j+1}M)_{I} vanishes by (2.15). This means that the support of Ker T (j)/I^{j+1}M is contained in ∂O.

Second, let N be any (S(𝔤), K_{ℂ})-submodule of I^{j}M with two properties (i) and (ii) in question. (2.14) tells us that T (j) naturally induces an (S(𝔤), K_{ℂ})-module map from I^{j}M/I^{j+1}M to Γ(W(j)) which we denote by T̃(j). Then, T̃(j)(N/I^{j+1}M) must vanish by virtue of (2.11) together with the property (ii) for N. This proves N ⊂ Ker T (j).

As for the injectivity of T (j), one gets the following consequence of Proposition 2.3.

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Corollary 2.1. The homomorphism T (j) : I^{j}M → Γ(W(j)) is injective if and only if Ann_{S(𝔤)} m = I for all m ∈ I^{j}M\{0}. In this case, one has I^{j+1}M = {0}, i.e., j = n_{0} − 1.

Example 2.1. We encounter the situation in the above corollary with j = 0, for example, if X is a unitary highest weight module of a simple hermitian Lie algebra 𝔤, and V_{τ} in (2.1) is the extreme K_{ℂ}-type of X. Note that the associated variety of such an X is the closure of a “holomorphic” nilpotent K_{ℂ}-orbit in 𝔭. See [30, Section 3.2] for details.

Summing up T̃(j)’s on I^{j}M/I^{j+1}M (j = 0, …, n_{0} − 1), we obtain an (S(𝔤), K_{ℂ})-homomorphism T := ⊕_{j}T̃(j):

M̂(I) := ⨁_{j} I^{j}M/I^{j+1}M ⟶^{T} ⨁_{j} Γ(W(j)) ≃ Γ(W),  (2.17)

where the support of the kernel Ker T is contained in ∂O.

Remark 2.2. By using the “microlocalization technique”, Vogan constructed a new K_{ℂ}-stable ℤ-gradation on X such that the corresponding graded module embeds into Γ(W) as a representation of K_{ℂ} (see [22, Theorem 4.2]). Thanks to this result, one always has X ↪ Γ(W) as K_{ℂ}- modules. Noting that M̂(I) ≃ X as K_{ℂ}-modules, we find that the above T : M̂(I) → Γ(W) must be an isomorphism if T is surjective.

2.4. Irreducibility of ϖ_{O}

The results in Sections 2.1–2.3 lead us to prove the following natural criterion for the irreducibility of isotropy representation (ϖ_{O}, W) of K_{ℂ}(X) (cf. [23, Proposition 7.6]; see also [31, Section 5]).

Theorem 2.1. The following two conditions on X are equivalent to each other.

(a) (ϖ_{O}, W) is irreducible as a K_{ℂ}(X)-module.

(b) If N is any (S(𝔤), K_{ℂ})-submodule of M = gr X, either the support of N or that of the quotient M/N is contained in ∂O.

In this case, we have ϖ_{O} = ϖ_{O}(0), or equivalently, the support of IM is contained in ∂O by Proposition 2.1.

Proof. The implication (a) ⇒ (b) is an easy consequence of the exactness of localization. In what follows let us prove (b) ⇒ (a). First, we note that

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the condition (b) together with Remark 2.1 implies that the support of IM is contained in ∂O. Thus one gets ϖ_{O} = ϖ_{O}(0), or,

W = W(0) = M/𝔪(X)M.

Now, suppose by contraries that W is not irreducible. Then, there exists a K_{ℂ}(X)-stable subspace C of M such that M ⊋ C ⊋ 𝔪(X)M and that Z := M/C is irreducible as a K_{ℂ}(X)-module. The condition C ⊃ 𝔪(X)M assures that C is S(𝔤)-stable. Thus Z becomes an (S(𝔤), K_{ℂ}(X))-module annihilated by 𝔪(X).

Next, we consider two induced (S(𝔤), K_{ℂ})-modules Γ(W) and Γ(Z). The quotient map W = M/𝔪(X)M ⟶ Z = M/C gives rise to an (S(𝔤), K_{ℂ})-homomorphism, say γ, from Γ(W) to Γ(Z) in the canonical way. Set T′ := γ ∘ T (0), where T (0) : M → Γ(W) is the (S(𝔤), K_{ℂ})- homomorphism defined in Section 2.3. Then, as shown in the proof of [23, Proposition 7.9], the image T′(M) of T′ is a finitely generated (S(𝔤), K_{ℂ})- submodule of Γ(Z) whose isotropy representation is isomorphic to Z. This combined with T′(M) ≃ M/Ker T′ tells us that the multiplicity of Ker T′ at the prime ideal I is equal to dim W − dim Z > 0. By the assumption (b), we find that the support of M/Ker T′ ≃ T′(M) is contained in ∂O. This necessarily implies Ker T′ = M, i.e., T′ = 0, because the S(𝔤)-module T′(M) (⊂ Γ(Z)) admits no embedded associated primes by (2.11). Finally, the resulting equality T′ = 0 means that

y^{−1} · m + 𝔪(X)M = T (0)_{m}(y) ∈ C/𝔪(X)M

for all y ∈ K_{ℂ} and m ∈ M. This contradicts C ≠ M.

2.5. Case of unitary highest weight representations

Let X be an irreducible unitary highest weight (𝔤, K_{ℂ})-module of a simple hermitian Lie algebra 𝔤, with extreme K_{ℂ}-type V_{τ}.

Example 2.2. In [30, Section 5], we have described the isotropy representation ϖ_{O} = ϖ_{O}(0) explicitly, when X is the theta lift of an irreducible representation of the compact groups G′ = O(k), U(k) and Sp(k) with respect to the reductive dual pairs (G, G′) = (Sp(n, ℝ), O(k)), (SU(p, q), U(k)) and (SO^{∗}(2n), Sp(k)), respectively. In particular, one finds that the representation ϖ_{O} is irreducible if the dual pair (G, G′) is in the stable range with smaller member G′. In this case, X ↔ ϖ_{O}^{∗} essentially gives the Howe duality correspondence ([10], [11], [8, Part II] etc.).

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An irreducible highest weight (𝔤, K_{ℂ})-module X is called singular if the Gelfand-Kirillov dimension dim V(X) is strictly smaller than one half of the dimension of the corresponding hermitian symmetric space. Recently, we have described the isotropy representations by using the projection onto the PRV-component (cf. Proposition 3.3), for all singular unitary highest weight modules which can not be obtained by the Howe duality correspondence. This work is in collaboration with Wachi (see [24] and [33] for details). As a result, we establish the following remarkable theorem.

Theorem 2.2. The isotropy representation is irreducible and explicitly described for every irreducible singular unitary highest weight representation of a simple Lie group of hermitian type.

The above result for DI and EVII gives a clear understanding of some multiplicity formulae obtained by Kato and Ochiai ([13], [14]). For EVII case, we get two distinguished series of isotropy representations which decompose the quasi-regular representations for the compact symmetric spaces S^{8} ≃ SO(9)/SO(8) and P^{2}(Cay) ≃ F_{4(−52)}/Spin(9), respectively. These decompositions can be related to tensor products of singular unitary representations in [4], by using the generalized Whittaker vectors (cf. [30]).

3. Utility of the dual (S(𝔤), K_{ℂ})-module

In this section, we do not assume a priori that the associated variety V(X) of X is irreducible. Let M = gr X be, as in (2.2), the graded (S(𝔤), K_{ℂ})- module attached to an irreducible (𝔤, K_{ℂ})-module X through the filtration (2.1). For any nilpotent element X ∈ 𝔭, we can define the maximal ideal 𝔪(X) of S(𝔤), the K_{ℂ}(X)-module W(0) = M/𝔪(X)M, and (S(𝔤), K_{ℂ})- homomorphism T (0) : M → Γ(W(0)), just as in Section 2. Here Γ(W(0)) = Ind_{K_{ℂ(X)}}^{K_{ℂ}}(W(0)) is the (S(𝔤), K_{ℂ})-module induced from W(0).

The purpose of this section is to make some simple observations concerning the associated cycle of X, in connection with the K_{ℂ}-finite dual M^{∗} of M realized as a space of V_{τ}^{∗}-valued polynomial functions on 𝔭. This is done by developing our arguments in [29], [30] in full generality. A sufficient condition is given in Proposition 3.1 for the annihilator Ann_{S(𝔤)}M of M being equal to the prime ideal of S(𝔤) defined by the K_{ℂ}-orbit O := Ad(K_{ℂ})X through X. Furthermore, we characterize M^{∗} as the kernel of a differential operator D on 𝔭 of gradient type (Proposition 3.2). Then the principal symbol of D allows us to describe the K_{ℂ}(X)-module W(0)^{∗} dual to W(0) (Proposition 3.3). The observations made in this section will be

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used effectively in Section 4, in order to describe the associated cycles for (𝔤, K_{ℂ})-modules X of discrete series.

3.1. K_{ℂ}-finite dual M^{∗} and its submodule Ψ

First, the tensor product S(𝔭)⊗V_{τ} admits a natural structure of (S(𝔤), K_{ℂ})- module so that 𝔨 annihilates the whole S(𝔭) ⊗ V_{τ}:

\begin{cases}D_{1} · (D ⊗ v) := D_{1}D ⊗ v & (D_{1} ∈ S(𝔭)), \\ D_{2} · (D ⊗ v) := 0 & (D_{2} ∈ S(𝔨)), \\ k · (D ⊗ v) := Ad(k)D ⊗ kv & (k ∈ K_{ℂ}), \end{cases}  (3.1)

where D ⊗ v ∈ S(𝔭) ⊗ V_{τ} with D ∈ S(𝔭) and v ∈ V_{τ}. Since M = S(𝔭)V_{τ} with V_{τ} = M_{0}, there exists a unique surjective (S(𝔤), K_{ℂ})-homomorphism

π : S(𝔭) ⊗ V_{τ} ⟶ M

such that π(1 ⊗ v) = v for v ∈ V_{τ}. We write N for the kernel of π. This is a graded (S(𝔤), K_{ℂ})-submodule of S(𝔭) ⊗ V_{τ}.

On the other hand, we identify S(𝔭^{∗}) ⊗ V_{τ}^{∗} canonically with the space of polynomial functions on 𝔭 with values in V_{τ}^{∗}, where U^{∗} denotes the dual space of a vector space U. S(𝔭^{∗}) ⊗ V_{τ}^{∗} also becomes an (S(𝔤), K_{ℂ})-module on which 𝔤 acts by directional differentiation through the quotient map 𝔤 → 𝔤/𝔨 ≃ 𝔭:

\begin{cases}(D_{1} · f)(Y) := (∂(D_{1})f)(Y) (D_{1} ∈ S(𝔭)), \\ (D_{2} · f)(Y) := 0  (D_{2} ∈ S(𝔨)), \\ (k · f)(Y) := k · f(Ad(k)^{−1}Y) (k ∈ K_{ℂ}), \end{cases}  (3.2)

for f ∈ S(𝔭^{∗}) ⊗ V_{τ}^{∗} and Y ∈ 𝔭. Here D_{1} ↦ ∂(D_{1}) denotes the algebra isomorphism from S(𝔭) onto the algebra of constant coefficient differential operators on 𝔭, defined by

∂(Z)f(Y) := \frac{d}{dt}f(Y + tZ)_{|t=0} for Z ∈ 𝔭.  (3.3)

Note that the action of S(𝔤) on S(𝔭^{∗})⊗V_{τ}^{∗} is locally finite. The assignment

𝔭 ∋ X ⟼ X^{∗} ∈ 𝔭^{∗} with X^{∗}(Y) := B(X, Y) (Y ∈ 𝔭).  (3.4)

gives a K_{ℂ}-isomorphism from 𝔭 onto 𝔭^{∗}.

Now it is standard to verify that

\begin{matrix}(S(𝔭) ⊗ V_{τ}) × (S(𝔭^{∗}) ⊗ V_{τ}^{∗}) ∋ (D ⊗ v, f) ⟼ ⟨D ⊗ v, f⟩ ∈ ℂ, \\ ⟨D ⊗ v, f⟩ := ((^{T} D · f)(0), v)_{V_{τ}^{∗}×V_{τ}} , \end{matrix}  (3.5)

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gives a nondegenerate (S(𝔤), K_{ℂ})-invariant pairing, where ^{T} denotes the principal automorphism of S(𝔭) such that ^{T} Y = −Y for Y ∈ 𝔭, and ( · , ·)_{V_{τ}^{∗}×V_{τ}} is the dual pairing on V_{τ}^{∗} × V_{τ}. Let M^{∗} denote the K_{ℂ}-finite dual space of M, viewed as an (S(𝔤), K_{ℂ})-module through the contragredient representation. We write N^{⊥} for the orthogonal of N in S(𝔭^{∗})⊗V_{τ}^{∗} with respect to ⟨ · , · ⟩. Then, (3.5) naturally induces a nondegenerate invariant pairing

⟨ · , · ⟩_{1} : M × N^{⊥} → ℂ,  (3.6)

which gives an isomorphism of (S(𝔤), K_{ℂ})-modules:

M^{∗} ≃ N^{⊥} ⊂ S(𝔭^{∗}) ⊗ V_{τ}^{∗}.  (3.7)

For an integer n ≥ 0, we denote by (N^{⊥})_{n} the homogeneous component of N^{⊥} of degree n: (N^{⊥})_{n} := N^{⊥} ∩ (S^{n}(𝔭^{∗}) ⊗ V_{τ}^{∗}).

Let X be a nilpotent element in 𝔭. Noting that M = V_{τ} + 𝔪(X)M, we have a natural K_{ℂ}(X)-homomorphism

V_{τ} ⟶ W(0) = M/𝔪(X)M ⟶ 0.

This induces an embedding of W(0)^{∗} into V_{τ}^{∗} as

W(0)^{∗} ≃ (V_{τ} /(V_{τ} ∩ 𝔪(X)M))^{∗} ↪ V_{τ}^{∗},  (3.8)

by passing to the dual. In this way, we regard W(0)^{∗} as a K_{ℂ}(X)-submodule of V_{τ}^{∗}.

For each integer n ≥ 0, let Ψ_{n} be the K_{ℂ}-submodule of S^{n}(𝔭^{∗}) ⊗ V_{τ}^{∗} generated by the vectors (X^{∗})^{n} ⊗ v^{∗} (v^{∗} ∈ W(0)^{∗}):

Ψ_{n} := ⟨(X^{∗})^{n} ⊗ v^{∗} | v^{∗} ∈ W(0)^{∗}⟩_{K_{ℂ}}.  (3.9)

Note that the V_{τ}^{∗}-valued polynomial function (X^{∗})^{n} ⊗ v^{∗} on 𝔭 is defined by

(X^{∗})^{n} ⊗ v^{∗} : 𝔭 ∋ Z ⟼ B(X, Z)^{n}v^{∗} ∈ ℂ.

We set

Ψ := ⨁_{n=0}^{∞} Ψ_{n} ⊂ S(𝔭^{∗}) ⊗ V_{τ}^{∗}.

Then, we can show the following

Lemma 3.1. (1) Ψ is an (S(𝔤), K_{ℂ})-submodule of S(𝔭^{∗}) ⊗ V_{τ}^{∗} contained in N^{⊥}.

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(2) We write ^{⊥}Ψ for the orthogonal of Ψ in M with respect to the pairing ⟨ · , · ⟩_{1} on M × N^{⊥}. Let T (0) : M → Γ(W(0)) be the (S(𝔤), K_{ℂ})- homomorphism defined in Section 2.3. Then, one gets

Ker T (0) ∩ M_{n} = ^{⊥}Ψ ∩ M_{n} for every integer n ≥ 0,  (3.10)

In particular, ^{⊥}Ψ = ⊕_{n} ^{⊥}Ψ ∩ M_{n} is contained in Ker T (0).

Proof. (1) It is easy to see that Ψ is (S(𝔤), K_{ℂ})-stable by noting that

Y · (((Ad(y)X)^{∗})^{n} ⊗ y · v^{∗}) = nB(Ad(y)X, Y)((Ad(y)X)^{∗})^{n−1} ⊗ y · v^{∗}

lies Ψ_{n−1} for Y ∈ 𝔤, v^{∗} ∈ W(0)^{∗} and y ∈ K_{ℂ}. To prove Ψ ⊂ N^{⊥}, let ψ_{v^{∗}} denote the linear form on S(𝔭) ⊗ V_{τ}, which is the pull back of v^{∗} ∈ W(0)^{∗} through the quotient map

S(𝔭) ⊗ V_{τ} ⟶^{π} M ⟶ W(0) = M/𝔪(X)M.

Then ψ_{v^{∗}} is zero on the subspace 𝔪(X) ⊗ V_{τ} + N ⊂ S(𝔭) ⊗ V_{τ}.

If m̃ = ∑_{j} Y_{j}^{n} ⊗ v_{j} (Y_{j} ∈ 𝔭, v_{j} ∈ V_{τ}) is a homogeneous element of N of degree n, it follows that

⟨ m̃ , (X^{∗})^{n} ⊗ v^{∗}⟩ = ∑_{j}(−1)^{n}n!B(X, Y_{j})^{n}(v_{j}, v^{∗})_{V_{τ} ×V_{τ}^{∗}}
= (−1)^{n}n!ψ_{v^{∗}}(m̃) = 0,

by noting that Y_{j}^{n} − B(X, Y_{j})^{n} ∈ 𝔪(X). Hence one gets ((Ad(y)X)^{∗})^{n} ⊗ y · v^{∗} ∈ y · N^{⊥} = N^{⊥} for all v^{∗} ∈ W(0)^{∗}, y ∈ K_{ℂ} and n ≥ 0. Thus we obtain (1).

(2) Let m = ∑_{j} Y_{j}^{n} · v_{j} be an element of M_{n} with Y_{j} ∈ 𝔭 and v_{j} ∈ V_{τ}. Just as in the proof of (1), we see that m ∈ ^{⊥}Ψ if and only if

0 = ⟨ m , ((Ad(y)X)^{∗})^{n} ⊗ y · v^{∗}⟩_{1}
= ∑_{j}(−1)^{n}n!B(Y_{j}, Ad(y)X)^{n} (y^{−1} · v_{j}, v^{∗})_{V_{τ} ×V_{τ}^{∗}}
= (−1)^{n}n!(T (0)_{m}(y), v^{∗})_{W(0)×W(0)^{∗}},

for all v^{∗} ∈ W(0)^{∗} and y ∈ K_{ℂ}. This means m ∈ Ker T (0).

3.2. A sufficient condition for I = Ann_{S(𝔤)}M

Let O = Ad(K_{ℂ})X be the nilpotent K_{ℂ}-orbit through X. We write I for the prime ideal of S(𝔤) defining the Zariski closure \overline{O} of O, and let {D_{i}| i = 1, …, r} be a finite set of homogeneous elements of S(𝔭) which generates the ideal I. We set n(i) := deg D_{i} (1 ≤ i ≤ r) and n(0) := 0.

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Proposition 3.1. Assume that Ψ_{n(i)} = (N^{⊥})_{n(i)} for i = 0, …, r. Then we have I = Ann_{S(𝔤)}M. Therefore, X has the irreducible associated variety V(X) = \overline{O}, and the corresponding associated cycle C(X) of X turns to be

C(X) = dim W(0) · [\overline{O}].

Proof. We see for every v ∈ V_{τ} that

T (0)_{D_{i}v}(y) = (D_{i} · T (0)_{v})(y) = D_{i}(Ad(y)X) T (0)_{v}(y) = 0 (y ∈ K_{ℂ}),

since D_{i} ∈ I and Ad(y)X ∈ O. It then follows that D_{i}v = 0 by Lemma 3.1 (2) together with the assumption Ψ_{n(i)} = (N^{⊥})_{n(i)}, which is equivalent to ⊥Ψ∩M_{n(i)} = {0}. Hence, D_{i} annihilates V_{τ} and so the whole M = S(𝔭)V_{τ}. This shows I ⊂ Ann_{S(𝔤)}M.

Now, W(0)^{∗} cannot vanish by the assumption Ψ_{0} = (N^{⊥})_{0} ≃ V_{τ}^{∗}. This implies 𝔪(X) ⊃ Ann_{S(𝔤)}M. In reality, if 𝔪(X) ⊅ Ann_{S(𝔤)}M, one gets S(𝔤) = 𝔪(X) + Ann_{S(𝔤)}M by the maximality of 𝔪(X). This yields

M = S(𝔤)V_{τ} = 𝔪(X)V_{τ} ⊂ 𝔪(X)M, i.e., W(0) = {0}.

Hence we deduce

I = ⋂_{k∈K_{ℂ}} 𝔪(Ad(k)X) ⊃ Ann_{S(𝔤)}M,

because Ann_{S(𝔤)}M is stable under Ad(K_{ℂ}).

Thus we find I = Ann_{S(𝔤)}M and in particular V(X) = \overline{O}. The last assertion follows immediately from Proposition 2.1.

Under the assumption in Proposition 3.1, the (S(𝔤), K_{ℂ})-module Ψ is almost equal to N^{⊥} ≃ M^{∗}, in the sense that the support of the orthogonal ⊥Ψ ⊂ Ker T (0) is contained in the boundary ∂O by Proposition 2.3 and Lemma 3.1.

3.3. Differential operator of gradient type

We wish to characterize N^{⊥} ≃ M^{∗} as the kernel of a certain differential operator on 𝔭 of gradient type. For this, we first take an orthonormal basis (X_{1}, …, X_{s}) of 𝔭 with respect to the Killing form B_{|𝔭×𝔭}. Then, (X_{1}^{∗}, …, X_{s}^{∗}) (cf. (3.4)) gives a basis of 𝔭^{∗}, dual to (X_{1}, …, X_{s}). For a multi-index α = (α_{1}, …, α_{s}) of nonnegative integers α_{i} (1 ≤ i ≤ s), we set

D^{α} := X_{1}^{α_{1}} ⋯ X_{s}^{α_{s}}, (D^{∗})^{α} := (X_{1}^{∗})^{α_{1}} ⋯ (X_{s}^{∗})^{α_{s}}.

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Then, the elements D^{α} (resp. (D^{∗})^{α}) with |α| = n form a basis of S^{n}(𝔭) (resp. S^{n}(𝔭^{∗})) for every integer n ≥ 1, where |α| := α_{1} + ⋯ + α_{s} denotes the length of α. Note that

∂(D^{α})(D^{∗})^{β} = α! · δ_{α,β} (Kronecker’s δ_{α,β}) and

∂(D^{α})(Z^{∗})^{n} = n!(D^{∗})^{α}(Z) (3.11)

for all α, β of length n, and Z ∈ 𝔭. Here, we set α! = α_{1}! ⋯ α_{s}!. Note that the above functions ∂(D^{α})(D^{∗})^{β} and ∂(D^{α})(Z^{∗})^{n} are constant on 𝔭.

We now introduce a gradient map ∇^{n} of order n by

(∇^{n}f)(Y) := ∑_{|α|=n} \frac{1}{α!}(D^{∗})^{α} ⊗ ∂(D^{α})f(Y),  (3.12)

for f ∈ S(𝔭^{∗}) ⊗ V_{τ}^{∗}. It is easy to observe that ∇^{n}f is independent of the choice of an orthonormal basis (X_{i})_{1≤i≤s}. Furthermore, ∇^{n} gives an (S(𝔤), K_{ℂ})-homomorphism

S(𝔭^{∗}) ⊗ V_{τ}^{∗} ∋ f ⟼^{∇^{n}} ∇^{n}f ∈ S(𝔭^{∗}) ⊗ (S^{n}(𝔭^{∗}) ⊗ V_{τ}^{∗}),

where S(𝔭^{∗}) ⊗ (S^{n}(𝔭^{∗}) ⊗ V_{τ}^{∗}) is looked upon as the space of polynomial functions on 𝔭 with values in S^{n}(𝔭^{∗}) ⊗ V_{τ}^{∗}.

Lemma 3.2. It holds that ∇^{n}f = 1⊗f for every f ∈ S^{n}(𝔭^{∗})⊗V_{τ}^{∗}, where 1 denotes the identity element of S(𝔭^{∗}). Namely, ∇^{n}f is the constant function on 𝔭 with the value f ∈ S^{n}(𝔭^{∗}) ⊗ V_{τ}^{∗}.

Proof. It is enough to prove the lemma for f = (D^{∗})^{β} ⊗ v^{∗} with |β| = n and v^{∗} ∈ V_{τ}^{∗}. In view of (3.11), ∇^{n}f turns to be

∇^{n}f(Y) = ∑_{|α|=n} \frac{1}{α!}(D^{∗})^{α} ⊗ (∂(D^{α})(D^{∗})^{β})(Y)v^{∗} = (D^{∗})^{β} ⊗ v^{∗} = f,

which proves the lemma.

We note that our submodule N of S(𝔭) ⊗ V_{τ} is finitely generated over S(𝔤), since the ring S(𝔤) is Noetherian and since S(𝔭) ⊗ V_{τ} = S(𝔭) · V_{τ}. Hence, there exist a finite number of homogeneous K_{ℂ}-submodules W_{u} ⊂ N (u = 1, …, q) which generate N over S(𝔤):

N = S(𝔤) · W_{1} + ⋯ + S(𝔤) · W_{q} with W_{u} ⊂ S^{i_{u}}(𝔭) ⊗ V_{τ}  (3.13)

for some integers i_{u} ≥ 0 arranged as i_{1} < ⋯ < i_{q}. For each u = 1, …, q, let P_{u} denote the K_{ℂ}-homomorphism from S^{i_{u}}(𝔭^{∗}) ⊗ V_{τ}^{∗} to W_{u}^{∗} defined by

P_{u}(h)(w) := ⟨ w, h ⟩ (w ∈ W_{u})  (3.14)

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for h ∈ S^{i_{u}}(𝔭^{∗}) ⊗ V_{τ}^{∗}. Here, ⟨ · , · ⟩ is the dual pairing in (3.5).

We now set

W^{∗} := ⨁_{u=1}^{q} W_{u}^{∗},

and let us introduce an (S(𝔤), K_{ℂ})-homomorphism

D : S(𝔭^{∗}) ⊗ V_{τ}^{∗} ⟶ S(𝔭^{∗}) ⊗ W^{∗},  (3.15)

by putting

(Df)(Y) := ∑_{u=1}^{q} P_{u}(∇^{i_{u}}f(Y)) (Y ∈ 𝔭; f ∈ S(𝔭^{∗}) ⊗ V_{τ}^{∗}).  (3.16)

Definition 3.1. We call D the differential operator of gradient type associated with (V_{τ}^{∗}, W^{∗}).

The space of solutions of the differential equation Df = 0 is characterized as follows.

Proposition 3.2. One gets N^{⊥} = Ker D. Hence, the kernel of the differential operator D is isomorphic to the K_{ℂ}-finite dual M^{∗} of M ≃ (S(𝔭) ⊗ V_{τ})/N, as (S(𝔤), K_{ℂ})-modules.

Proof. Let f be a homogeneous element of S(𝔭^{∗})⊗V_{τ}^{∗} of degree n. We are going to show that f lies in N^{⊥} if and only if Df = 0. This will prove the proposition because both Ker D and N^{⊥} are graded (S(𝔤), K_{ℂ})-submodules of S(𝔭^{∗}) ⊗ V_{τ}^{∗}.

First, the condition f ∈ N^{⊥} is written as

⟨Dw_{u}, f⟩ = 0  (3.17)

for all D ∈ S^{n−i_{u}}(𝔭) and w_{u} ∈ W_{u} (u = 1, …, q), by noting that S^{m}(𝔭) ⊗ V_{τ} is orthogonal to f ∈ S^{n}(𝔭^{∗}) ⊗ V_{τ}^{∗} if m ≠ n. Here S^{n−i_{u}}(𝔭) should be understood as {0} if n − i_{u} < 0. Since the pairing ⟨ · , · ⟩ is S(𝔤)-invariant, (3.17) is equivalent to

P_{u}(∂(D)f) = 0 for all D ∈ S^{n−i_{u}}(𝔭) (u = 1, …, q).  (3.18)

We set D_{u}f := P_{u}((∇^{i_{u}}f)(·)) ∈ S^{n−i_{u}}(𝔭^{∗}) ⊗ W_{u}^{∗}. Then, in view of Lemma 3.2, the left hand side of (3.18) turns to be

P_{u}(∂(D)f) = P_{u}((∇^{i_{u}}∂(D)f)(Y)) = (∂(D)(D_{u}f))(Y) (Y ∈ 𝔭).

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We thus find that f lies in N^{⊥} if and only if

∂(D)D_{u}f = 0 for all D ∈ S^{n−i_{u}}(𝔭) (u = 1, …, q),

or equivalently, D_{u}f = 0 (u = 1, …, q). This means Df = 0 as desired.

Let us define a map σ from 𝔭^{∗} × V_{τ}^{∗} to W^{∗} by

σ(X^{∗}, v^{∗}) := ∑_{u=1}^{q} P_{u}((X^{∗})^{i_{u}} ⊗ v^{∗}) for (X^{∗}, v^{∗}) ∈ 𝔭^{∗} × V^{∗}, (3.19)

which we call the symbol map of D.

For any fixed X ∈ 𝔭, we get the following characterization of W(0)^{∗}.

Proposition 3.3. The K_{ℂ}(X)-submodule W(0)^{∗} of V_{τ}^{∗} (see (3.8)) is described as

W(0)^{∗} = Ker σ(X^{∗}, ·) := {v^{∗} ∈ V_{τ}^{∗} | σ(X^{∗}, v^{∗}) = 0},  (3.20)

where X^{∗} ∈ 𝔭^{∗} corresponds to X ∈ 𝔭 by (3.4).

This proposition can be proved just as in the proof of [30, Lemma 3.10] (see also the proof of Lemma 3.1 (1)). We omit the proof here.

4. Isotropy representation attached to discrete series

In this section, we assume that 𝔤 = 𝔨 ⊕ 𝔭 is an equi-rank algebra (cf. [17]), i.e., rank 𝔤 = rank 𝔨. By using our results in Section 3, we study the isotropy representations attached to irreducible (𝔤, K_{ℂ})-modules of discrete series. This develops our work in [29].

4.1. Discrete series

We begin with a quick review on the discrete series representations, and let us fix our notation. As is well known, the complex Lie algebra 𝔤 has a θ-stable real form 𝔤_{0} such that

𝔤_{0} = 𝔨_{0} ⊕ 𝔭_{0} with 𝔨_{0} := 𝔨 ∩ 𝔤_{0}, 𝔭_{0} := 𝔭 ∩ 𝔤_{0},

gives a Cartan decomposition of 𝔤_{0}. Such a real form 𝔤_{0} is unique up to K_{ℂ}-conjugacy. Take a maximal abelian subalgebra 𝔱_{0} of 𝔨_{0}, and we write 𝔱 for the complexification of 𝔱_{0} in 𝔨. Since 𝔤 is an equi-rank algebra, 𝔱 turns to be a Cartan subalgebra of 𝔤. We write Δ for the root system of (𝔤, 𝔱). The subset of compact (resp. noncompact) roots will be denoted by Δ_{c} (resp. Δ_{n}).

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Let G be a connected Lie group with Lie algebra of 𝔤_{0} such that K_{ℂ} is the complexification of a maximal compact subgroup K of G. An irreducible unitary representation σ of G is called a member of discrete series if the matrix coefficients of σ are square-integrable on G. We are concerned with the irreducible (𝔤, K_{ℂ})-modules X of discrete series, consisting of K-finite vectors for such σ’s. For example, we refer to [9], [18], and also [26, I, Section 1] for the parametrization and realization of discrete series representations.

Now, let X = X_{Λ} be the (𝔤, K_{ℂ})-module of discrete series with Harish-Chandra parameter Λ ∈ 𝔱^{∗}. Since the parameter Λ is regular and real on \sqrt{−1}𝔱_{0}, there exists a unique positive system Δ^{+} of Δ for which Λ is dominant:

Δ^{+} := {α ∈ Δ | (Λ, α) > 0}.  (4.1)

We denote by (τ, V_{τ}) the unique lowest K_{ℂ}-type of X which occurs in X with multiplicity one. Set Δ_{c}^{+} := Δ^{+} ∩ Δ_{c} (resp. Δ_{n}^{+} := Δ^{+} ∩ Δ_{n}). The Δ_{c}^{+}-dominant highest weight λ (say) for τ is called the Blattner parameter of X. Then, λ is expressed as λ = Λ − ρ_{c} + ρ_{n} with ρ_{c} := (1/2) · ∑_{α∈Δ_{c}^{+}} α and ρ_{n} := (1/2) · ∑_{β∈Δ_{n}^{+}} β.

4.2. Results of Hotta-Parthasarathy

In what follows, we always assume that the Blattner parameter λ of X is far from the walls (defined by compact roots) in the sense of [26, I, Definition 1.7]. Let M = gr X = ⊕_{n≥0}M_{n} be the graded (S(𝔤), K_{ℂ})-module defined through the lowest K_{ℂ}-type V_{τ}. As in Section 3, we have a natural quotient map π : S(𝔭) ⊗ V_{τ} → M with N = Ker π. This subsection explains the structure of graded modules M, N, and M^{∗} ≃ N^{⊥} by interpreting the results of Hotta-Parthasarathy in [9].

For this, we first decompose the tensor product 𝔭 ⊗ V_{τ} as

𝔭 ⊗ V_{τ} = V_{τ}^{+} ⊕ V_{τ}^{−} as K_{ℂ}-modules,

where V_{τ}^{±} denotes the sum of irreducible K_{ℂ}-submodules of 𝔭 ⊗ V_{τ} with highest weights of the form λ ± β (β ∈ Δ_{n}^{+}), respectively. The inclusion V_{τ}^{−} ↪ 𝔭 ⊗ V_{τ} naturally induces a quotient map of K_{ℂ}-modules:

P : 𝔭^{∗} ⊗ V_{τ}^{∗} = (𝔭 ⊗ V_{τ})^{∗} ⟶ (V_{τ}^{−})^{∗}.  (4.2)

Hereafter, we replace 𝔭^{∗} by 𝔭 through the identification 𝔭 = 𝔭^{∗} by the Killing form of 𝔤 restricted to 𝔭 × 𝔭.

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Let B_{c} be the Borel subgroup of K_{ℂ} with Lie algebra 𝔟_{c} = 𝔱⊕∑_{α∈Δ_{c}^{−}} 𝔤_{α}, where 𝔤_{α} is the root subspace of 𝔤 corresponding to a root α. We set

𝔭_{±} := ⨁_{β∈Δ_{n}^{+}} 𝔤_{±β}.  (4.3)

Then, we have 𝔭 = 𝔭_{+} ⊕ 𝔭_{−} as vector spaces, and 𝔭_{−} is stable under the action of B_{c}.

If U is a holomorphic representation of B_{c}, the i-th cohomology space H^{i}(K_{ℂ}/B_{c}; U) of K_{ℂ}/B_{c} with coefficients in the sheaf of holomorphic sections of the vector bundle K_{ℂ} ×_{B_{c}} U has a structure of K_{ℂ}-module.

The following theorem can be read off from the proof of [9, Theorem 1] by taking into account the Blattner multiplicity formula [7] for discrete series. (See also [18]; [29].)

Theorem 4.1. (1) One has N = S(𝔭)V_{τ}^{−}.

(2) The orthogonal N^{⊥} of N in S(𝔭)⊗V_{τ}^{∗} coincides with the kernel of the differential operator D on 𝔭 of gradient-type associated with (V_{τ}^{∗}, (V_{τ}^{−})^{∗}) :

(Df)(Y) := P (∑_{i=1}^{s} X_{i} ⊗ (∂(X_{i})f)(Y)) (Y ∈ 𝔭),  (4.4)

for f ∈ S(𝔭)⊗V_{τ}^{∗}. Here {X_{i}}_{1≤i≤s} is an orthonormal basis of 𝔭 with respect to the Killing form.

(3) For every integer n ≥ 0, the dual M_{n}^{∗} of M_{n} is isomorphic to the cohomology space:

H^{q}(K_{ℂ}/B_{c}; S^{n}(𝔭_{−}) ⊗ ℂ_{−λ−2ρ_{c}}) with q := dim K_{ℂ}/B_{c},

as a K_{ℂ}-module. Here ℂ_{−λ−2ρ_{c}} denotes the one dimensional B_{c}-module corresponding to −λ − 2ρ_{c} ∈ 𝔱^{∗}.

Note that the claim (2) can be deduced from claim (1) by Proposition 3.2.

Remark 4.1. The maximal globalization of dual (𝔤, K_{ℂ})-module X^{∗} of discrete series can be realized as the kernel space of an invariant differential operator D̃ of gradient type (on the Riemannian symmetric space for (𝔤_{0}, 𝔨_{0})). The operator D in the above theorem gives the “polynomialization” of D̃.

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4.3. Description of associated cycle

We are going to apply Theorem 4.1 in order to describe the associated cycles for (𝔤, K_{ℂ})-modules of discrete series. For a positive number c, we say that a linear form μ on 𝔱 satisfies the condition (FFW(c)) if

(μ, α) ≥ c for all α ∈ Δ_{c}^{+}.  (FFW(c))

Theorem 4.1 coupled with the Borel-Weil Bott theorem for the group K_{ℂ} leads us to the following proposition(cf. [29, Section 6.1]), which is crucial to describe the associated cycle of X.

Proposition 4.1. (1) Let v_{λ}^{∗} be a nonzero lowest weight vector of V_{τ}^{∗} of weight −λ. Then, N^{⊥} = Ker D contains the K_{ℂ}-submodule ⟨S(𝔭_{−}) ⊗ v_{λ}^{∗}⟩_{K_{ℂ}} generated by S(𝔭_{−}) ⊗ v_{λ}^{∗}.

(2) For any integer n ≥ 0, there exists a positive constant c_{n} such that

(N^{⊥})_{n} = ⟨S^{n}(𝔭_{−}) ⊗ v_{λ}^{∗}⟩_{K_{ℂ}}  (4.5)

holds if the Blattner parameter λ satisfies the condition (FFW(c_{n})).

Now, let O be the unique nilpotent K_{ℂ}-orbit in 𝔭 which intersects 𝔭_{−} densely. Then one sees that \overline{O} = Ad(K_{ℂ})𝔭_{−}. As before, we write I for the prime ideal of S(𝔤) defining \overline{O}. It follows from the the claim (1) in Proposition 4.1 that Ann_{S(𝔤)}M ⊂ I, i.e., V(X) ⊃ \overline{O}. Also, the same claim shows 𝔭_{−} ⊗ v_{λ}^{∗} ∈ Ker P, which can be easily verified by noting that −λ − β (β ∈ Δ_{n}^{+}) cannot be a weight of (V_{τ}^{−})^{∗}.

Take an element X ∈ O ∩ 𝔭_{−}. By Proposition 3.3, we find that the K_{ℂ}(X)-module W(0)^{∗} = (M/𝔪(X)M)^{∗} ⊂ V_{τ}^{∗} consists exactly of all the vectors v^{∗} ∈ V_{τ}^{∗} satisfying P (X ⊗ v^{∗}) = 0. Let N_{K_{ℂ}}(X, 𝔭_{−}) be the totality of elements k ∈ K_{ℂ} such that Ad(k)X ∈ 𝔭_{−} (cf. [3]). For any subset R of N_{K_{ℂ}}(X, 𝔭_{−}), we denote by U_{λ}(R) the K_{ℂ}(X)-submodule of V_{τ}^{∗} generated by R^{−1} · v_{λ}^{∗}:

U_{λ}(R) := ⟨ R^{−1} · v_{λ}^{∗} ⟩_{K_{ℂ}(X)}.

Then, we readily find from 𝔭_{−} ⊗ v_{λ}^{∗} ∈ Ker P that

U_{λ}(R) ⊂ W(0)^{∗}, and so ⟨ X^{n} ⊗ U_{λ}(R) ⟩_{K_{ℂ}} ⊂ Ψ_{n} ⊂ (N^{⊥})_{n}, (4.6)

for every n ≥ 0. Moreover one gets the equality

⟨ X^{n} ⊗ U_{λ}(R) ⟩_{K_{ℂ}} = ⟨ S^{n}(𝔭_{−}) ⊗ v_{λ}^{∗} ⟩_{K_{ℂ}},  (4.7)

if Ad(R)X ⊂ 𝔭_{−} is Zariski dense in 𝔭_{−}. This is true when R equals the whole N_{K_{ℂ}}(X, 𝔭_{−}), because Ad(N_{K_{ℂ}}(X, 𝔭_{−}))X = O ∩ 𝔭_{−} is dense in 𝔭_{−}.

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As in Section 3, we take homogeneous generators D_{i} (i = 1, …, r) of the ideal I such that deg D_{i} = n(i). We set c(I) := max_{i}(c_{n(i)}). By virtue of Proposition 3.1 together with (4.5), (4.6) and (4.7), we come to the following conclusion.

Theorem 4.2. Assume that the Blattner parameter λ of discrete series X is far from the walls in the sense of [26, I, Definition 1.7] and that it also satisfies the condition (FFW(c)) with c = c(I).

(1) One gets I = Ann_{S(𝔤)}M and so V(X) = Ad(K_{ℂ})𝔭_{−} = \overline{O}. Moreover, the K_{ℂ}(X)-module W^{∗} contragredient to the isotropy representation (ϖ_{O}, W) is described as

W^{∗} = W(0)^{∗} = {v^{∗} ∈ V_{τ}^{∗} | P (X ⊗ v^{∗}) = 0},  (4.8)

where X ∈ O ∩ 𝔭_{−}, and P : 𝔭 ⊗ V_{τ}^{∗} → (V_{τ}^{−})^{∗} is the K_{ℂ}-homomorphism in (4.2).

(2) Let R be a subset of N_{K_{ℂ}}(X, 𝔭_{−}) such that Ad(R)X is Zariski dense in 𝔭_{−}. Then, the K_{ℂ}(X)-submodule U_{λ}(R) = ⟨ R^{−1} · v_{λ}^{∗}⟩_{K_{ℂ}(X)} ⊂ W^{∗} is exhaustive in the following sense: for every integer n ≥ 0, one has

⟨X^{n} ⊗ W^{∗}⟩_{K_{ℂ}} = ⟨X^{n} ⊗ U_{λ}(R)⟩_{K_{ℂ}}  (4.9)

if λ satisfies FFW(c_{n}), where c_{n} is the positive constant in Proposition 4.1.

Remark 4.2. (1) The assertions I = Ann_{S(𝔤)}M and V(X) = Ad(K_{ℂ})𝔭_{−} have been obtained in [29]. But, in that paper, we did not discuss the possibility of applying the results to describe the isotropy representation.

(2) One should get a result similar to Theorem 4.2, more generally for the derived functor modules A_{𝔮}(λ).

(3) Compare Theorem 4.2 (1) with Chang’s result [3, Proposition 1.4] established by means of the localization theory of Harish-Chandra modules.

4.4. Submodule U_{λ}(Q_{c})

In this subsection, we give a natural choice of R ⊂ N_{K_{ℂ}}(X, 𝔭_{−}) for which we expect to have the property (4.9). Let Π be the set of simple roots in Δ^{+}. We write S = Π ∩ Δ_{c} for the totality of compact simple roots. Then, there exists a unique element H_{S} ∈ 𝔱 such that

α(H_{S}) = \begin{cases}0 & if α ∈ S, \\ 1 & if α ∈ Π\S. \end{cases}

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The adjoint action of H_{S} yields a gradation on the Lie algebra 𝔤 as

𝔤 = ⨁_{j} 𝔤(j) with 𝔤(j) := {Z ∈ 𝔤| (ad H_{S})Z = jZ}.

Here j runs through the integers such that |j| ≤ δ(H_{S}) with the highest root δ. Note that

𝔨 = ⨁_{j:even} 𝔤(j), 𝔭 = ⨁_{j:odd} 𝔤(j) with 𝔭_{±} = ⨁_{j>0,odd} 𝔤(±j).

Now, we set

𝔮 := ⨁_{j≤0} 𝔤(j), 𝔩 := 𝔤(0) ⊂ 𝔨 and 𝔲 := ⨁_{j<0} 𝔤(j).

Then, 𝔮 = 𝔩 ⊕ 𝔲 gives the Levi decomposition of the standard parabolic subalgebra 𝔮 of 𝔤 associated with the subset S of Π. We write Q (resp. Q_{c}) for the parabolic subgroup of G_{ℂ} := G_{ℂ}^{ad} (resp. of K_{ℂ}) with Lie algebra 𝔮 (resp. 𝔮 ∩ 𝔨). The group Q (resp. Q_{c}) admits the Levi decomposition Q = LU (resp. Q_{c} = L_{c}U_{c}), where L and U (resp. L_{c} and U_{c}) are the connected subgroups of Q (resp. Q_{c}) with Lie algebras 𝔩 and 𝔲 (resp. 𝔩 and 𝔲 ∩ 𝔨) respectively. Note that Ad(L_{c}) = L. The parabolic subgroup Q acts on its nilradical 𝔲, and so Q_{c} acts on 𝔭_{−} = 𝔭 ∩ 𝔲 by the adjoint action. Thus, Q_{c} is contained in N_{K_{ℂ}}(X, 𝔭_{−}) for all X ∈ 𝔭_{−}, and the corresponding K_{ℂ}(X)-submodule U_{λ}(Q_{c}) of V_{τ}^{∗} turns to be

U_{λ}(Q_{c}) = ⟨(V_{λ}^{L_{c}})^{∗}⟩_{K_{ℂ}(X)}.  (4.10)

Here, (V_{λ}^{L_{c}})^{∗} = U(𝔩)v_{λ}^{∗} denotes the irreducible L_{c}-submodule of V_{τ}^{∗} generated by the lowest weight vector v_{λ}^{∗}.

We can now apply Theorem 4.2 to deduce

Corollary 4.1. Under the assumption in Theorem 4.2, the K_{ℂ}(X)- submodule U_{λ}(Q_{c}) of W^{∗} is exhaustive in the sense of (4.9), if 𝔭_{−} is a prehomogeneous vector space under the adjoint action of the group Q_{c}, and if X ∈ O ∩ 𝔭_{−} lies in the open Q_{c}-orbit in 𝔭_{−}.

4.5. Relation to the Richardson orbit

We end this article by looking at the condition for 𝔭_{−} in Corollary 4.1, and also some related conditions, in relation to the Richardson G_{ℂ}-orbit associated with the parabolic subalgebra 𝔮.

First, let us recall some basic facts on the Richardson orbit (cf. [12, Chapter 5]). The G_{ℂ}-stable subset G_{ℂ} · 𝔲 (⊂ 𝔤) forms an irreducible affine

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variety of 𝔤 whose dimension is equal to 2 dim 𝔲. Noting that G_{ℂ} ·𝔲 consists of nilpotent elements only, there exists a unique G_{ℂ}-orbit Õ such that

\overline{Õ} = G_{ℂ} · 𝔲,

by the finiteness of the number of nilpotent G_{ℂ}-orbits in 𝔤. Õ is called the Richardson G_{ℂ}-orbit associated with 𝔮. The parabolic subgroup Q acts on 𝔲 prehomogenenously, and Õ∩𝔲 turns to be a single Q-orbit in 𝔲. Moreover, the centralizer in 𝔤 of any element X ∈ Õ ∩ 𝔲 is contained in 𝔮.

Now, let O be the nilpotent K_{ℂ}-orbit in Section 4.3. Then we have two nilpotent G_{ℂ}-orbits G_{ℂ} · O and Õ with the closure relation G_{ℂ} · O ⊂ \overline{Õ}. By virtue of a result of Kostant-Rallis [15, Proposition 5], this relation implies that

dim O = \frac{1}{2} dim G_{ℂ} · O ≤ \frac{1}{2} dim Õ = dim 𝔲.  (4.11)

In particular, we find that the Gelfand-Kirillov dimension dim V(X) = dim O of discrete series X cannot exceed dim 𝔲. The following proposition tells us when these two orbits turn to be equal.

Proposition 4.2. The following three conditions (a), (b) and (c) on the positive system Δ^{+} = {α| (Λ, α) > 0} are equivalent with each other :

(a) G_{ℂ} · O = Õ, (b) dim O = dim 𝔲, (c) Õ ∩ 𝔭_{−} ≠ ∅.

In this case, O ∩ 𝔭_{−} is a single open Q_{c}-orbit in 𝔭_{−}, and so one gets the conclusion of Corollary 4.1.

Proof. The equivalence (a) ⇔ (b) is a direct consequence of (4.11). The condition (a) immediately implies (c), since O (⊂ G_{ℂ} · O = Õ) contains an element of 𝔭_{−}. Conversely, if Õ ∩ 𝔭_{−} ≠ ∅, this is a nonempty open subset of 𝔭_{−}, since Õ ∩ 𝔭_{−} = (Õ ∩ 𝔲) ∩ 𝔭_{−} with Õ ∩ 𝔲 open in 𝔲. Hence, Õ ∩ 𝔭_{−} intersects O. We thus get (c) ⇒ (a). This proves the equivalence of three conditions in question.

Next, we assume the condition (b) (⇔ (a) ⇔ (c)), and let X be any element of O ∩ 𝔭_{−}. We write 𝔷_{𝔰}(X) for the centralizer of X in a Lie subalgebra 𝔰 of 𝔤. By noting that 𝔷_{𝔤}(X) ⊂ 𝔮, the dimension of the Q_{c}-orbit Ad(Q_{c})X is calculated as

dim Ad(Q_{c})X = dim 𝔮 ∩ 𝔨 − dim 𝔷_{𝔮∩𝔨}(X)
= (dim 𝔨 − dim 𝔲 ∩ 𝔨) − dim 𝔷_{𝔨}(X)
= dim O − dim 𝔲 ∩ 𝔨 = dim 𝔲 − dim 𝔲 ∩ 𝔨 = dim 𝔭_{−},

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where we used the condition (b) for the forth equality. This shows that the orbit Ad(Q_{c})X is open in 𝔭_{−} for every X ∈ O ∩ 𝔭_{−}. We thus find that O ∩ 𝔭_{−} forms a single Q_{c}-orbit, because of the uniqueness of the open Q_{c}-orbit in 𝔭_{−}.

Remark 4.3. Each of the conditions (a), (b) and (c) in Proposition 4.2 is equivalent to Assumption 2.5 in [2] concerning the generically finiteness of the moment map defined on the conormal bundle T_{Z_{1}}^{∗}(G_{ℂ}/Q), where Z_{1} is a closed K_{ℂ}-orbit in G_{ℂ}/Q through the origin eQ.

Suggested by Corollary 4.1 and Proposition 4.2, let us consider the following three conditions on 𝔭_{−} which depends on the choice of a positive system Δ^{+}:

Õ ∩ 𝔭_{−} ≠ ∅ (⇔ dim O = dim 𝔲 ⇔ G_{ℂ} · O = Õ),  (C1)
O ∩ 𝔭_{−} is a single Q_{c}-orbit,  (C2)
𝔭_{−} is a prehomogeneous vector space under Ad(Q_{c}).  (C3)

Proposition 4.2 says (C1) ⇒ (C2), and the implication (C2) ⇒ (C3) is obvious.

As for the conditions (C2) and (C3), we can show the following

Proposition 4.3. One gets (C3) if O ∩ 𝔤(−1) ≠ ∅. Moreover, the equality Ad(Q_{c})(O ∩ 𝔤(−1)) = O ∩ 𝔭_{−} assures (C2).

Proof. Let X ∈ O ∩ 𝔤(−1). Since O = Ad(K_{ℂ})X contains a nonempty open subset of 𝔭_{−}, we find that [𝔨, X] ⊃ 𝔭_{−}. We set 𝔨_{+} := ⊕_{j>0}𝔤(2j). Then 𝔨 = 𝔨 ∩ 𝔮 ⊕ 𝔨_{+} is a direct sum of vector spaces. Then it follows from the assumption X ∈ 𝔤(−1) that [𝔨_{+}, X] ⊂ 𝔭_{+} and [𝔨 ∩ 𝔮, X] ⊂ 𝔭_{−}. We thus obtain

𝔭_{−} = [𝔨, X] ∩ 𝔭_{−} = [𝔨 ∩ 𝔮, X].

Hence Ad(Q_{c})X is open in 𝔭_{−}, and one gets (C3).

The above argument shows that any element X ∈ O ∩ 𝔤(−1) lies in the unique open Q_{c}-orbit in 𝔭_{−}. This proves the latter claim, too.

Following Gross-Wallach [5], we say that a discrete series (𝔤, K_{ℂ})-module X is small if δ(H_{S}) ≤ 2, or equivalently, 𝔤(j) = {0} if |j| ≥ 3. Here δ is the highest root of Δ^{+} as before. In this case, one has 𝔭_{−} = 𝔤(−1), and so the above proposition implies

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Corollary 4.2. The subspace 𝔭_{−} corresponding to a small discrete series admits the property (C2).

Remark 4.4. By case-by-case analysis, Chang [3] proved the property (C2) for any discrete series representations of simple Lie groups of ℝ-rank one.

4.6. Condition (C1) for SU(p, q)

It should be important to study when 𝔭_{−} admits the properties (C1), (C2) and (C3), respectively. Toward this direction, we end this paper by giving an explicit, combinatorial criterion for the condition (C1) in case of G = SU(p, q) with n = p + q. In this case, we have 𝔤 = 𝔰𝔩(n, ℂ), and let

α_{1} = ε_{1} − ε_{2}, α_{2} = ε_{2} − ε_{3}, …, α_{n−1} = ε_{n−1} − ε_{n}

be the simple roots for (𝔤, 𝔱) by the standard notation of Bourbaki. Then 𝔭_{−}’s in question are in one-one correspondence to the set of noncompact positive roots {α_{n_{1}}, α_{n_{1}+n_{2}}, …, α_{n_{1}+···+n_{t−1}}}, where n_{1}, …, n_{t−1}, and n_{t} := n − (n_{1} + ⋯ + n_{t−1}) are positive integers such that

∑_{j:odd} n_{j} = p or q.

Theorem 4.3. 𝔭_{−} satisfies the condition (C1) if and only if the corresponding partition (n_{1}, n_{2}, …, n_{t}) of n is unimodal, that is, there exists a positive integer k (1 ≤ k ≤ t) such that

n_{1} ≤ ⋯ ≤ n_{k−1} ≤ n_{k} ≥ n_{k+1} ≥ ⋯ ≥ n_{t}.

If p, q ≤ 3, every partition (n_{1}, …, n_{t}) is unimodal, and we always get the property (C1). On the contrary, the partition (2, 1, 2) for SU(4, 1) is not unimodal, and (C1) fails in this case, where dim O = 7 < 8 = dim 𝔲.

We will discuss the detail elsewhere.

Acknowledgments

The author thanks Hiroyuki Ochiai for stimulating discussion.

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