Associated variety, Kostant-Sekiguchi correspondence, and locally free U(𝔫)-action on Harish-Chandra modules

Dedicated to Professor Takeshi Hirai on his sixtieth birthday

By Akihiko Gyoja and Hiroshi Yamashita^{∗}

Introduction.

Let 𝔤 be a complex semisimple Lie algebra, and let 𝔤 = 𝔨 + 𝔭 be the symmetric decomposition of 𝔤 defined by an involutive automorphism θ of 𝔤. By a Harish-Chandra module associated to the pair (𝔤, 𝔨), we mean a U(𝔤)-module X of finite length on which the subalgebra U(𝔨) acts locally finitely. Here U(𝔮) denotes the universal enveloping algebra of a complex Lie algebra 𝔮.

The main purpose of this paper is to give for each irreducible Harish-Chandra module X a family of nilpotent Lie subalgebras 𝔫(O) of 𝔤 whose enveloping algebras U(𝔫(O)) act on X locally freely. The Lie subalgebras 𝔫(O) are parametrized by the nilpotent K_{C}^{ad}-orbits O contained in the associated variety V(X) ⊂ 𝔭 of X, where K_{C}^{ad} denotes the analytic subgroup of adjoint group G_{C}^{ad} = Int(𝔤) of 𝔤 corresponding to the Lie subalgebra 𝔨. We construct 𝔫(O) from a K_{C}^{ad}-orbit O through the Cayley transformation of normal 𝔰𝔩_{2}-triples that gives the Kostant-Sekiguchi correspondence of nilpotent orbits ([8]).

The Harish-Chandra modules are essentially related to infinite-dimensional representations of a real semisimple Lie group as follows. Let 𝔤_{0} be any real form of 𝔤, and let G be a connected linear Lie group with Lie algebra 𝔤_{0}. We can and do choose an involution θ of 𝔤 such that the real form 𝔤_{0} is θ-stable and that 𝔨_{0} := 𝔨 ∩ 𝔤_{0} coincides with the Lie algebra of a maximal compact subgroup K of G ([3, Ch.III, §4]). By fundamental results of Harish-Chandra ([2], see also [11, Ch. 3]), any admissible Hilbert representation (π, H) of G of finite length yields a Harish-Chandra module X by passing to the K-finite part of H through differentiation. The irreducibility is preserved by the assignment H → X. Accordingly, we may say that the present work reveals some new algebraic aspects of representations of the group G.

We now explain the results of this article in more detail.

(I) For a nonzero nilpotent K_{C}^{ad}-orbit O in 𝔭, take a normal 𝔰𝔩_{2}-triple (X, H, Y) ⊂ 𝔤 with X ∈ O (see 1.6), and define its Cayley transform (X′, H′, Y′) as in (1.2). Making use of the l-eigenspaces 𝔤(l) (l = 1, 2, ...) of 𝔤 with respect to ad(H′), we can construct a nilpotent Lie subalgebra 𝔫(O) = (𝔤_{1}^{1}(1)⊕𝔤_{3}^{3}(1))⊕(⊕_{l≥2}𝔤(l)) of 𝔤 with 𝔤_{1}^{1}(1)⊕𝔤_{3}^{3}(1) ⊂ 𝔤(1) (see 1.4 and 1.6 for the precise definition of subspaces 𝔤_{η}^{κ}(l) of 𝔤(l)) such that:

∗Supported in part by Grant-in-Aid for Scientific Research (C) (2) 08640001, The Ministry of Education, Science, Sports and Culture, Japan.

(i) dim 𝔫(O) = dim O, (ii) the Killing form B of 𝔤 is nondegenerate on ad(X)𝔨 × 𝔫(O). (See Theorem 1.2 and Lemma 3.1.) Up to K_{C}^{ad}-conjugacy, the Lie subalgebra 𝔫(O) is independent of the choice of an 𝔰𝔩_{2}-triple (X, H, Y). In addition, the ideal ⊕_{l≥2}𝔤(l) of 𝔫(O) becomes stable under the complex conjugation of 𝔤 with respect to the real form 𝔤_{0}, if we construct 𝔫(O) from a strictly normal 𝔰𝔩_{2}-triple (Proposition 3.1). We can describe concretely the Lie subalgebras 𝔫(O) associated to the holomorphic nilpotent orbits O in 𝔭 (Theorem 3.6), when 𝔤_{0} is a noncompact real simple Lie algebra of hermitian type. As we indicate below, the above two properties (i) and (ii) are crucial to establish the local freeness of the U(𝔫(O))-action on Harish-Chandra modules.

(II) Now let X be an irreducible Harish-Chandra module. Through the natural increasing filtlation U_{k}(𝔤) (k = 0, 1, ...) of U(𝔤), we attach to each nonzero vector v ∈ X a graded module M = gr(X; v) := ⨁_{k=0}^{∞} U_{k}(𝔤)v/U_{k−1}(𝔤)v over the symmetric algebra S(𝔤) ≃ ⊕_{k=0}^{∞}U_{k}(𝔤)/U_{k−1}(𝔤) of 𝔤, where U_{−1}(𝔤) := {0}. The associated variety V(X) of X is then defined to be the set of the common zeros of elements in the annihilator Ann_{S(𝔤)}(M) of M. Here, V(X) is independent of the choice of a vector v, and we identify S(𝔤) with the ring of polynomial functions on 𝔤 through the Killing form B.

As seen by Vogan [10], the variety V(X) associated to X is a union of finitely many nilpotent K_{C}^{ad}-orbits in 𝔭 (cf. Lemma 2.2). If O is a K_{C}^{ad}-orbit contained in V(X), the above properties (i) and (ii) imply that the natural projection p : 𝔤 → 𝔤/𝔫(O)^{⊥} induces a linear isomorphism from the tangent space ad(X)𝔨 of O at X onto 𝔤/𝔫(O)^{⊥}, where 𝔫(O)^{⊥} is the orthogonal of 𝔫(O) in 𝔤 with respect to B. This allows us to deduce that M = gr(X; v) is a torsion free S(𝔫(O))-module for every nonzero v ∈ X. As a consequence, we establish the main result of this article as follows.

Theorem. (Theorem 3.2) Let X be an irreducible Harish-Chandra module. The enveloping algebra U(𝔫(O)) of nilpotent Lie subalgebra 𝔫(O) acts on X locally freely for every nilpotent K_{C}^{ad}-orbit O ⊂ 𝔭 contained in the associated variety V(X) of X.

We remark that, by the Hilbert-Serre theorem, X is a torsion free U(𝔫)-module for a Lie subalgebra 𝔫 of 𝔤 only if dim 𝔫 ≤ dim V(X).

Bearing this remark in mind, we derive two interesting conclusions of the above theorem. First, we find that the nilpotent Lie subalgebra 𝔫(O_{max}) associated to a maximal K_{C}^{ad}-orbit O_{max} in V(X) realizes a maximal Lie subalgebra of 𝔤 among those having locally free action on X (Theorem 3.3). Second, let 𝔤 = 𝔨 + 𝔞 + 𝔫_{m} be a complexified Iwasawa decomposition of 𝔤_{0}. Then it can be shown that an irreducible Harish-Chandra module X is large i.e., dim V(X) = dim 𝔫_{m}, if and only if X is a torsion free U(𝔫_{m})- module (Theorem 3.4).

(III) The organization of this paper is as follows.

In section 1, we first study certain fine structure on finite-dimensional SL(2, C)- modules equipped with involutive linear transformations (see Proposition 1.2 and Theorem 1.1). The properties (i) and (ii) stated in I. for the nilpotent Lie subalgebra 𝔫(O) of 𝔤 are shown by applying Theorem 1.1 to the adjoint representation of 𝔰 := CX + CH + CY ≃ 𝔰𝔩(2, C) on 𝔤.

Section 2 is devoted to giving a simple criterion for X to be a torsion free U(𝔫)- module. More precisely the matters are discussed in much more general situation, where 𝔤 is an arbitrary complex Lie algebra, 𝔨 and 𝔫 are any two Lie subargebras of 𝔤, and X is a locally U(𝔨)-finte, irreducible U(𝔤)-module. Our criterion (Theorem 2.1) is given by means of the Lie subalgebras 𝔨, 𝔫 and the associated variety V(X) of X.

In section 3, the main result of this paper, Theorem 3.2, is established by using Theorems 1.2 and 2.1. Then we deduce two important consequences (Theorems 3.3 and 3.4) of Theorem 3.2. In addition, the Lie subalgebras 𝔫(O) associated to the holomorphic nilpotent K_{C}^{ad}-orbits O are described explicitly in 3.3.

1. SL(2, C)-modules with involution σ̃.

In this section, we begin with investigating in 1.1 - 1.5 certain fine structure on finite-dimensional SL(2, C)-modules V equipped with an involutive linear transformation σ̃ ∈ GL(V), compatible with a nontrivial involution σ of SL(2, C). The results are summarized as Proposition 1.2 and Theorem 1.1.

We then apply the results to Lie algebra case in 1.6, where V = 𝔤 is a complex semisimple Lie algebra with an involution σ̃ = θ, and SL(2, C) acts on 𝔤 through the adjoint representation of a θ-stable, simple Lie subalgebra 𝔰 ≃ 𝔰𝔩(2, C) of 𝔤. This gives us a new kind of decomposition of 𝔤 (Theorem 1.2(3)), which is, in a sense, comparable with the (complexified) generalized Iwasawa decompositions of 𝔤. The nilpotent Lie subalgebra 𝔫 of 𝔤 appearing in this decomposition will play an essential role in §3 for studying locally free U(𝔫)-action on Harish-Chandra modules.

1.1. 𝔰𝔩_{2}-triples and Cayley transformation.

Let 𝔰 = CX + CH + CY ≃ 𝔰𝔩(2, C) be a three-dimensional, complex simple Lie algebra with commutation relation:

[H, X] = 2X, [H, Y] = −2Y, [X, Y] = H. (1.1)

We denote by S ≃ SL(2, C) the simply connected Lie group with Lie algebra 𝔰. Setting

X′ = \frac{i}{2}(H − X + Y), H′ = X + Y, Y′ = −\frac{i}{2}(H + X − Y), (1.2)

one gets another 𝔰𝔩_{2}-triple (X′, H′, Y′) in 𝔰 which satisfies the same relation (1.1). If we identify 𝔰 with 𝔰𝔩(2, C) by

X = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , H = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , Y = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ,

then

X′ = \frac{i}{2} ( \begin{matrix}1 & −1 \\ 1 & −1 \end{matrix} ) , H′ = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , Y′ = \frac{i}{2} ( \begin{matrix}−1 & −1 \\ 1 & 1 \end{matrix} )

is a basis of the real form 𝔰𝔲(1,1) of 𝔰, and the Cayley transformation:

c̃: 𝔰 ∋ Z ↦ Ad(c)Z = cZc^{−1} ∈ 𝔰 with c = \frac{1}{1 + i} ( \begin{matrix}1 & −i \\ 1 & i \end{matrix} ) ∈ SL(2, C) (1.3)

sends the 𝔰𝔩_{2}-triple (X, H, Y) to (X′, H′, Y′). Note that the center of S contains a unique

nontrivial element ε = exp(πiH′) corresponding to the matrix ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ). Now let σ be the involutive automorphism of 𝔰 defined by

σX = −X, σH = H, σY = −Y. (1.4)

It then follows that σX′ = −Y′, σY′ = −X′ and σH′ = −H′. Extend σ to an automorphism of S through the exponential map, which we denote again by σ. Let

(1.5) w := exp \frac{π}{2} (X′ − Y′) = exp X′ · exp(−Y′) · exp X′

be an element of S corresponding to the matrix ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) which represents the nontrivial

element of the Weyl group of 𝔰 with respect to Cartan subalgebra CH′.

Direct computation in S ≃ SL(2, C) immediately gives the following lemma.

Lemma 1.1. One has the equalities:

(1) σ(w) = w, w^{2} = ε,

(2) σ(s) = wsw^{−1} (s ∈ S), and σ equals Ad(w) on 𝔰,

(3) Ad(exp(−iY′))X = iX′/2.

1.2. Irreducible S-modules.

For each nonnegative integer d, let (τ_{d}, V_{d}) be an irreducible S-module of dimension d + 1. The Lie algebra 𝔰 acts on V_{d} through differentiation. Take a nonzero highest weight vector v_{d}^{(d)} ∈ V_{d} such that

τ_{d}(H′)v_{d}^{(d)} = dv_{d}^{(d)}, τ_{d}(X′)v_{d}^{(d)} = 0 (1.6)

and set

v_{d−2j}^{(d)} = \frac{1}{j!}τ_{d}(Y′)^{j}v_{d}^{(d)} (j = 0, 1,…, d). (1.7)

Then the vectors v_{d−2j}^{(d)} (0 ≤ j ≤ d) form a basis of V_{d} for which the action of X′, H′, Y′ is described respectively as

\{ \begin{matrix}τ_{d}(X′)v_{d−2j}^{(d)} = (d + 1 − j)v_{d−2(j−1)}^{(d)}, \\ τ_{d}(H′)v_{d−2j}^{(d)} = (d − 2j)v_{d−2j}^{(d)}, \\ τ_{d}(Y′)v_{d−2j}^{(d)} = (j + 1)v_{d−2(j+1)}^{(d)}, \end{matrix} (1.8)

where v_{−2}^{(d)} = v_{d+2}^{(d)} = 0. We note that the element w ∈ S in (1.5) acts on V_{d} as

τ_{d}(w)v_{d−2j}^{(d)} = (−1)^{d−j}v_{−(d−2j)}^{(d)} (j = 0, 1,…, d). (1.9)

1.3. Extension σ̃ and S-homomorphism J.

Let (τ, V) be any finite-dimensional S-module (and so 𝔰-module). A map σ̃ : V → V is called an extension of σ to V if it is an involutive linear isomorphism on V satisfying

σ̃ τ(Z) σ̃^{−1} = τ(σZ) (Z ∈ 𝔰). (1.10)

The totality of such extensions will be denoted by E_{V}. If V = V_{d} is irreducible, σ̃ := iτ(w) for d ∈ 2Z + 1; σ̃ := τ(w) for d ∈ 2Z, gives an extension of σ to V, by Lemma 1.1(2).

In 1.6 we will meet extension σ̃ arising from an involutive automorphism of a semisimple Lie algebra 𝔤 = V, where 𝔰 is a Lie subalgebra of 𝔤 acting on V through the adjoint representation.

Let F_{V} denote the set of all S-homomorphisms J on V such that J^{2} = τ(ε), where w^{2} = ε is, as in Lemma 1.1, the nontrivial central element of S. Then,

Proposition 1.1. The assignment σ̃ ↦ J := σ̃ τ(w) gives a bijective correspondence from E_{V} onto F_{V}.

Proof. Let σ̃ be in E_{V}. Then Lemma 1.1 together with (1.10) yields that

Jτ(Z) = σ̃ τ(Ad(w)Z)τ(w) = σ̃ τ(σZ)τ(w) = τ(Z)J

for every Z ∈ 𝔰, and that

J^{2} = (σ̃ τ(w) σ̃)τ(w) = τ(σ(w))τ(w) = τ(w)^{2} = τ(ε).

We thus find that J ∈ F_{V}, for the group S is connected.

Conversely, if J is in F_{V}, then σ̃ := Jτ(w)^{−1} belongs to E_{V}. In fact, it follows from Lemma 1.1 that

σ̃^{2} = J^{2}τ(w)^{−2} = τ(ε)τ(ε)^{−1} = id_{V} ,

and that

σ̃ τ(Z) σ̃^{−1} = τ(w)^{−1}τ(Z)τ(w) = τ(σZ)

for Z ∈ 𝔰, where id_{V} denotes the identity operator on V. These two equalities show that σ̃ is an extension of σ to V. Q.E.D.

It should be noticed that J σ̃ = σ̃ J = τ(w), (1.11)

since σ̃ is involutive and it commutes with τ(w).

We fix once and for all an extension σ̃ of σ to V, and the corresponding S-homomorphism J = σ̃ τ(w).

1.4. The subspace U.

For an S-module (τ, V) with σ̃ ∈ E_{V} and the corresponding J ∈ F_{V} in 1.3, let

V = V (σ̃, +1) ⊕ V (σ̃, −1) with V (σ̃, ±1) := {v ∈ V | σ̃v = ±v} (1.12)

be the eigenspace decomposition of V with respect to σ̃. The semisimple element H′ ∈ 𝔰 gives a weight space decomposition of V:

V = ⨁_{l∈Z} V (l) with V (l) := {v ∈ V | τ(H′)v = lv}. (1.13)

We decompose the S-module V into irreducibles as

(1.14) V = ⨁_{d≥0} [m_{d}] · V_{d} with [m_{d}] · V ≃ V_{d} ⊕ ⋯ ⊕ V_{d} (m_{d}-copies),

where m_{d} denotes the multiplicity of simple S-module V_{d} (see 1.2) in V. Put

V^{(κ)} := ⨁_{d∈I(κ)}[m_{d}] · V_{d} ⊂ V, I(κ) := {κ + 4n| n = 0, 1,…}, (1.15)

for κ = 0, 1, 2, 3. Then V^{(κ)} is the S-submodule of V generated by all the maximal weight vectors in V with weight λ ≡ κ (mod 4). Clearly it holds that

(1.16) V = ⨁_{κ=0}^{3} V^{(κ)} as S-modules,

and that

V^{(κ)} = ⨁_{l∈Z} V^{(κ)}(l) with V^{(κ)}(l) := V^{(κ)} ∩ V (l), (1.17)

gives the weight space decomposition of V^{(κ)}, where V^{(κ)}(l) = {0} if κ − l ∉ 2Z. It should be remarked that any S-submodule W of V decomposes as

W = ⨁_{κ=0}^{3} W ∩ V^{(κ)}, (1.18)

since each irreducible constituent of W with highest weight d ∈ I(κ) is contained in V^{(κ)}.

Using the S-homomorphism J on V such that J^{4} = τ(ε)^{2} = τ(1) = id_{V}, we obtain another decomposition of the S-representation (τ, V) as

V = ⨁_{η=0}^{3} V_{(η)} with V_{(η)} := {v ∈ V | Jv = i^{η}v}. (1.19)

Denote by V_{(η)}(l) := V (l) ∩ V_{(η)} the l-weight subspace of V_{(η)}. We observe that V_{(η)}(l) = {0} if η − l ∉ 2Z, because J^{2} = τ(ε) = exp(πi τ(H′)) acts on V (l) by the scalar (−1)^{l}.

Summarizing the above discussion, we immediately deduce the following lemma on the compatibility of two decompositions (1.16) and (1.19).

Lemma 1.2. (τ, V) admits the decomposition:

V = ⨁_{κ,η=0}^{3} V_{η}^{κ} with V_{η}^{κ} := V^{(κ)} ∩ V_{(η)} (1.20)

as S-modules, and V_{η}^{κ} equals {0} if κ − η ∉ 2Z.

This lemma shows that the even part V^{even} := ⊕_{l∈2Z}V (l) and the odd part V^{odd} := ⊕_{l∈2Z+1}V (l) of V decompose respectively as

\{ \begin{matrix}V^{even} = V^{(0)} ⊕ V^{(2)} = V_{(0)} ⊕ V_{(2)} = V_{0}^{0} ⊕ V_{2}^{0} ⊕ V_{0}^{2} ⊕ V_{2}^{2}, \\ V^{odd} = V^{(1)} ⊕ V^{(3)} = V_{(1)} ⊕ V_{(3)} = V_{1}^{1} ⊕ V_{3}^{1} ⊕ V_{1}^{3} ⊕ V_{3}^{3}. \end{matrix} (1.21)

We note that the involution σ̃ acts on V in the following way.

Lemma 1.3. For κ, η = 0, 1, 2, 3, and l ∈ Z, let V_{η}^{κ}(l) := V_{η}^{κ} ∩ V (l) denote the l-weight subspace of V_{η}^{κ}. Then it holds that

σ̃ V_{η}^{κ} = V_{η}^{κ}, σ̃ V (l) = V (−l), and so σ̃ V_{η}^{κ}(l) = V_{η}^{κ}(−l). (1.22)

Proof. It follows from (1.10) that, if W is any irreducible S-submodule of V, so is σ̃ W and dim W equals dim σ̃ W. This implies that each S-submodule V^{(κ)} is σ̃-stable. We get σ̃ V_{(η)} = V_{(η)} by virtue of the commutativity (1.11). Thus one finds the first equality in (1.22). The second one follows from σ H′ = −H′, and the third one is an immediate consequence of the former two. Q.E.D.

We now introduce a subspace U of V defined as follows:

U := (V_{1}^{1}(1) ⊕ V_{3}^{3}(1)) ⊕ (⊕_{l≥2}V (l)). (1.23)

This subspace U will provide us in §3 with a nilpotent Lie subalgebra 𝔫 of a semisimple Lie algebra 𝔤, admitting locally free action on Harish-Chandra modules for 𝔤.

The following proposition is one of the essential ingredients to establish our main result on locally free U(𝔫)-action on Harish-Chandra modules.

Proposition 1.2. Let (τ, V) be a finite-dimensional S-module with σ̃, J ∈ GL(V) in 1.3, and let U be the subspace of V defined above. Then V is expressed as a sum of three subspaces as

(1.24) V = V (σ̃, +1) + Ker τ(X) + U = σ̃U + Ker τ(X) + U,

where V (σ̃, +1) is the subspace of σ̃-fixed vectors as in (1.12), and Ker τ(X) = {v ∈ V | τ(X)v = 0} denotes the kernel of τ(X).

Proof. Set M := V (σ̃, +1) + Ker τ(X) + U. First we see easily from Lemma 1.3 together with the definition of U that the sum V (σ̃, +1) + U is decomposed as

V (σ̃, +1) + U = U + σ̃U + V (σ̃, +1)
 = {⊕_{|l|≥2}V (l)} ⊕ {V_{1}^{1}(1) ⊕ V_{1}^{1}(−1)} ⊕ {V_{3}^{3}(1) ⊕ V_{3}^{3}(−1)} ⊕
  {(V_{1}^{3}(1) ⊕ V_{1}^{3}(−1)) ∩ V (σ̃, +1)} ⊕ {(V_{3}^{1}(1) ⊕ V_{3}^{1}(−1)) ∩ V (σ̃, +1)}
  ⊕{V (0) ∩ V (σ̃, +1)}.

Hence, for the proof of (1.24) it is enough to show that three subspaces:

\{ \begin{matrix}Q_{η}^{κ} := (V_{η}^{κ}(1) ⊕ V_{η}^{κ}(−1)) ∩ V (σ̃; −1) with (κ, η) = (1, 3), (3, 1), and \\ R := V (0) ∩ V (σ̃, −1), \end{matrix} (1.25)

are contained in M. Before proving this, we remark that

(1.26) M ⊃ Ker τ(X) = (exp iτ(Y′)) · Ker τ(X′)

by Lemma 1.1(3), and that the subspace Ker τ(X′) is exactly the linear span of all the maximal weight vectors in V with respect to τ(H′).

Now let (κ, η) be (1, 3) or (3, 1), and let us show Q_{η}^{κ} ⊂ M. Decompose the S-module V_{η}^{κ} into irreducibles:

V_{η}^{κ} = ⨁_{p=1}^{s} V_{d_{p}}, (1.27)

where V_{d_{p}} is, as in 1.2, the irreducible S-module with highest weight d_{p} ≡ κ (mod 4).

For a while we fix any p ∈ {1, 2,…, s}, and take a nonzero highest weight vector v_{d_{p}}^{(d_{p})} ∈ V_{d_{p}}(d_{p}) ⊂ Ker τ(X′). Then one gets

(exp i τ(Y′)) · v_{d_{p}}^{(d_{p})} = ∑_{j=0}^{d_{p}} i^{j} · v_{d_{p}−2j}^{(d_{p})} ∈ M, (1.28)

by (1.26), where v_{d_{p}−2j}^{(d_{p})} = τ(Y′)^{j}v_{d_{p}}^{(d_{p})}/j! ∈ V_{d_{p}} is a weight vector with weight d_{p}−2j. Since ⊕_{|l|≥2}V (l) ⊂ M, we find that v_{1}^{(d_{p})} + iv_{−1}^{(d_{p})} lies in M, and this vector can be calculated as follows:

v_{1}^{(d_{p})} + iv_{−1}^{(d_{p})} = v_{1}^{(d_{p})} + (−1)^{(d_{p}+1)/2}i τ(w)v_{1}^{(d_{p})} by (1.9)
= v_{1}^{(d_{p})} + (−1)^{(d_{p}+1)/2}i σ̃Jv_{1}^{(d_{p})} by (1.11)
= v_{1}^{(d_{p})} + (−1)^{(κ+1)/2}i^{η+1}σ̃v_{1}^{(d_{p})} since v_{1}^{(d_{p})} ∈ V_{η}^{κ}
= v_{1}^{(d_{p})} − σ̃v_{1}^{(d_{p})}

for (κ, η) = (3, 1) or (1, 3). We thus conclude v_{1}^{(d_{p})} − σ̃v_{1}^{(d_{p})} ∈ M.

Varying p = 1, 2,…, s, we gain Q_{η}^{κ} ⊂ M since the vectors v_{1}^{(d_{p})} − σ̃v_{1}^{(d_{p})} (p = 1, 2,…, s) form a basis of Q_{η}^{κ} by (1.22).

The inclusion R ⊂ M can be shown in the same (even easier) way.

Thus the proposition has been completely proved. Q.E.D.

1.5. S-modules with JS-invariant form.

Let (τ, V) be, as in 1.3, a finite-dimensional S-module with extension σ̃ ∈ E_{V} and J = σ̃ τ(w) ∈ F_{V}. A bilinear form B on V is called J- and S-invariant, or JS-invariant for short, if it satisfies

B(Jv, Jv′) = B(τ(s)v, τ(s)v′) = B(v, v′) (s ∈ S), (1.29) or equivalently,

B(σ̃v, σ̃v′) = B(v, v′), and B(τ(Z)v, v′) + B(v, τ(Z)v′) = 0 (Z ∈ 𝔰) (1.30)

for all v, v′ ∈ V.

Now let us consider the subspaces V (σ̃, ±1), V (l) (l ∈ Z) and the S-submodules V^{(κ)}, V_{(η)} (κ, η = 0, 1, 2, 3) of V defined in (1.12), (1.13), and (1.15), (1.19) respectively. If B is any JS-invariant bilinear form on V, these subspaces have the following orthogonality relations with respect to B:

V (σ̃, ±1) ⊥ V (σ̃, ∓1), V (l) ⊥ V (l′) if l + l′ ≠ 0, (1.31)

V^{(κ)} ⊥ V^{(κ′)} if κ ≠ κ′, V_{(η)} ⊥ V_{(η′)} if η + η′ ≠ 0 or 4, (1.32) which can be checked easily by the JS-invariance of B. (For the third relation one may use the fact that the Casimir element H^{′2} +2(X′Y′ +Y′X′) for 𝔰 has distinct eigenvalues on each V^{(κ)}.) Here, for any subsets L_{1} and L_{2} of V, L_{1} ⊥ L_{2} stands for B(v_{1}, v_{2}) = 0 for every v_{1} ∈ L_{1} and v_{2} ∈ L_{2}.

We now derive an important consequence of Proposition 1.2 for (τ, V) with JSinvariant form, as follows.

Theorem 1.1. Assume that the S-module (τ, V) admits a JS-invariant, nondegenerate symmetric bilinear form B on V. Then it holds that

(1) dim U = dim τ(X)V (σ̃, +1) = dim τ(X)V (σ̃, −1) = (dim τ(X)V)/2.

(2) B is nondegenerate on τ(X)V (σ̃, +1) × U.

(3) V = (V (σ̃, +1) + Ker τ(X)) ⊕ U (direct sum). Here V (σ̃, ±1) and U are the subspaces of V defined by (1.12) and in (1.23), respectively.

Proof. (1) We first prove the equality dim τ(X)V (σ̃, +1) = dim τ(X)V (σ̃, −1). This can be done just in the same way as in the proof of [6, Prop.5].

In fact, set (v, v′)_{X} := B(τ(X)v, v′) (v, v′ ∈ V). Then it is immediate to verify that (·, ·)_{X} gives an skew-symmetric bilinear form on V with kernel Ker τ(X). Hence this (·, ·)_{X} naturally induces a nondegenerate, skew-symmetric bilinear form on V/Ker τ(X) which we denote again by (·, ·)_{X}. Noting that the operator τ(X) sends V (σ̃, ±1) to V (σ̃, ∓1) by σX = −X, we can (and do) identify V/Ker τ(X) with the direct sum:

V (σ̃, +1)/(Ker τ(X) ∩ V (σ̃, +1)) ⨁ V (σ̃, −1)/(Ker τ(X) ∩ V (σ̃, −1)),

in the canonical way, and each constituent V (σ̃, ±1)/(Ker τ(X) ∩ V (σ̃, ±1)) is totally isotropic with respect to (·, ·)_{X} by the first orthogonality in (1.31). This shows that the bilinear form (·, ·)_{X} gives a nondegenerate pairing on

V (σ̃, +1)/(Ker τ(X) ∩ V (σ̃, +1)) × V (σ̃, −1)/(Ker τ(X) ∩ V (σ̃, −1)).

We thus obtain

dim τ(X)V (σ̃, +1) = dim V (σ̃, +1)/(Ker τ(X) ∩ V (σ̃, +1))
= dim V (σ̃, −1)/(Ker τ(X) ∩ V (σ̃, −1)) = dim τ(X)V (σ̃, −1),

which is equal to (dim τ(X)V)/2.

Second, let us prove the first equality in (1), or equivalently dim U = (dim τ(X)V)/2 by the above result. Keeping (1.26) in mind, we can calculate dim τ(X)V as

dim τ(X)V = dim V − dim Ker τ(X′) = dim V − (dim V (0) + dim V (1))
= dim V (1) + 2 ∑_{l≥2} dim V (l).

Here we used the fact that dim Ker τ(X′) coincides with the number of (linearly independent) maximal weight vectors in the S-module V, which is given by dim V (0)+dim V (1) (see 1.2). In view of the definition of U, it is sufficient for us to show

(1.33) dim V_{1}^{1}(1) + dim V_{3}^{3}(1) = \frac{1}{2} dim V (1).

But this is true because the bilinear form B is nondegenerate on the pairings

V_{1}^{1}(1) × V_{3}^{1}(−1) and V_{1}^{3}(1) × V_{3}^{3}(−1), (1.34)

thanks to (1.31) and (1.32). One thus finishes the proof of (1).

(2) For any subset L of V, let L^{⊥} denote the orthogonal of L in V with respect to B. Observe that (Ker τ(X))^{⊥} = τ(X)V and that V (σ̃, ±1)^{⊥} = V (σ̃, ∓1), then one deduces

(V (σ̃, +1) + Ker τ(X))^{⊥} = τ(X)V (σ̃, +1), (1.35)

with τ(X)V (σ̃, ±1) ⊂ V (σ̃, ∓1) in mind. Proposition 1.2 combined with this equality (1.35) shows that U^{⊥} ∩ τ(X)V (σ̃, +1) = {0}. We thus get the assertion, because the subspaces U and τ(X)V (σ̃, +1) have the same dimension as shown in (1).

(3) Because of (1) and (1.35), it suffices to show that the intersection of the two direct summands of the right hand side is equal to zero. If v belongs to both summands, then it follows that v ∈ U, that v ∈ (τ(X)V (σ̃, +1))^{⊥} by (1.35), and hence that v = 0 by (2). Q.E.D.

Remark. Since τ(Y′)V_{η}^{κ}(1) = V_{η}^{κ}(−1), it follows from (1.31) and (1.32) that

B(τ(Y′)v, v′) = 0 for v, v′ ∈ U ∩ V (1) = V_{1}^{1}(1) ⊕ V_{3}^{3}(1). (1.36)

This combined with (1.33) shows that U∩V (1) is a maximally totally isotropic subspace for the skew-symmetric bilinear form V (1) × V (1) ∋ (v_{1}, v_{2}) → B(τ(Y′)v_{1}, v_{2}) ∈ C on V (1).

1.6. An application of Theorem 1.1.

We finish this section by giving an application of Theorem 1.1 of particular importance, to the case where 𝔤 = V is a semisimple Lie algebra and (τ, V) is the adjoint representation on 𝔤 of a Lie subalgebra 𝔰 ≃ 𝔰𝔩(2, C) ⊂ 𝔤.

To be more precise, let 𝔤 be a complex semisimple Lie algebra, and θ be an involutive automorphism of 𝔤. We denote by 𝔤 = 𝔨 + 𝔭 the eigenspace decomposition of 𝔤 with respect to θ, where 𝔨 := 𝔤(θ, +1) and 𝔭 := 𝔤(θ, −1) are as in (1.12) with σ̃ = θ.

Let (X, H, Y) be an 𝔰𝔩_{2}-triple in 𝔤 with commutation relation (1.1). Such a triple is called normal (cf. [6]) if σ = θ acts on the elements X, H, and Y as in (1.4). Take an arbitrary normal 𝔰𝔩_{2}-triple (X, H, Y) in 𝔤, and set 𝔰 := CX + CH + CY = CX′ + CH′ + CY′ ≃ 𝔰𝔩(2, C). Here (X′, H′, Y′) is the Cayley transform of (X, H, Y) defined by (1.2).

We consider (τ, V) = (ad|𝔰, 𝔤), the adjoint representation of 𝔰 on 𝔤. Put J := θ Ad(w), where w is defined by (1.5). Then it should be mentioned that the involution θ on 𝔤 is actually an extension of θ|𝔰 (= θ restricted to 𝔰) to 𝔤 in the sense of (1.10), and that the Killing form B of 𝔤 gives a nondegenerate JS-invariant form on 𝔤 (see 1.5 for the definition). Let 𝔫 = 𝔫_{𝔰} := (𝔤_{1}^{1}(1) ⊕ 𝔤_{3}^{3}(1)) ⊕ (⊕_{l≥2}𝔤(l)) (1.37) denote the subspace U of 𝔤 defined in (1.23) for V = 𝔤. Then it is easily seen that 𝔫 is a nilpotent Lie subalgebra of 𝔤.

Applying Theorem 1.1 to the above setting, we now gain

Theorem 1.2. Let 𝔤 = 𝔨+𝔭 be the symmetric decomposition of a complex semisimple Lie algebra 𝔤 with respect to an involution θ of 𝔤, and let (X, H, Y) be a normal 𝔰𝔩_{2}-triple in 𝔤. Then one gets the following properties (1)-(3) for the nilpotent Lie subalgebra 𝔫 = 𝔫_{𝔰} of 𝔤 in (1.37):

(1) dim 𝔫 = dim ad(X)𝔨 = dim ad(X)𝔭 = (dim ad(X)𝔤)/2,

(2) the Killing form B of 𝔤 is nondegenerate on ad(X)𝔨 × 𝔫,

(3) 𝔤 = (𝔨 + 𝔷(X)) ⊕ 𝔫 as vector spaces. Here 𝔷(X) := Ker ad(X) denotes the centralizer of X in 𝔤.

Remarks. (1) Set 𝔫̃ := ⊕_{l≥1}𝔤(l), then 𝔫̃ is a nilpotent Lie subalgebra of 𝔤 containing 𝔫 as its ideal. The Remark in 1.5 implies that our 𝔫 is a polarizing subalgebra (see e.g., [1, p.28]) of 𝔫̃ for the linear form:

ξ_{Y′} : 𝔫̃ ∋ Z ↦ B(Y′, Z) ∈ C

on 𝔫̃, defined by the nilpotent element Y′ ∈ 𝔤(−2) through the Killing form. In particular ξ_{Y′} gives a one-dimensional representation of 𝔫.

(2) Let G_{C} be a complex semisimple Lie group with Lie algebra 𝔤, and N_{C} = exp 𝔫 be the analytic subgroup of 𝔤 with Lie algebra 𝔫. Then, the character ξ_{Y′} of 𝔫 gives rise to an induced G_{C}-module Ind_{N_{C}}^{G_{C}}(exp ξ_{Y′}), called the generalized Gelfand-Graev representation of G_{C} associated to the nilpotent orbit Ad(G_{C})Y′. ([4], see also [12, §1].)

2. Associated variety and a criterion for locally free U(𝔫)-action.

Let 𝔤 be any finite-dimensional complex Lie algebra, and U(𝔤) be the universal enveloping algebra of 𝔤. We now consider two Lie subalgebras 𝔨 and 𝔫 of 𝔤. In this section we give a simple criterion (Theorem 2.1) for a locally U(𝔨)-finite, irreducible U(𝔤)-module X to be a torsion free U(𝔫)-module. Our criterion is described by means of the Lie subalgebras 𝔨, 𝔫 and the associated variety V(X) of X. It has, as we show in §3, an interesting application when X is a Harish-Chandra module for a semisimple Lie algebra 𝔤.

2.1. Associated variety for U(𝔤)-modules.

First of all we introduce the associated variety for finitely generated U(𝔤)-modules which is one of the principal objects in the present article, and review after [10] some fundamental properties for this variety.

Denote by (U_{k}(𝔤))_{k=0,1,..} the natural increasing filtration of U(𝔤), where U_{k}(𝔤) is the subspace of U(𝔤) generated by elements X_{1} ⋯ X_{m} (m ≤ k) with X_{j} ∈ 𝔤 (1 ≤ j ≤ m). By the Poincaré-Birkhoff-Witt theorem, we can and do identify the associated graded ring

gr U(𝔤) = ⨁_{k≥0} U_{k}(𝔤)/U_{k−1}(𝔤) (U_{−1}(𝔤) := (0))

with the symmetric algebra S(𝔤) = ⨁_{k≥0} S^{k}(𝔤) of 𝔤 in the canonical way. Here S^{k}(𝔤) denotes the homogeneous component of S(𝔤) of degree k.

Let X be a finitely generated U(𝔤)-module. Take a finite-dimensional subspace X_{0} of X such that X = U(𝔤)X_{0}. Setting X_{k} = U_{k}(𝔤)X_{0} (k = 1, 2, ...), one gets an increasing filtration (X_{k})_{k} of X and correspondingly a finitely generated, graded S(𝔤)-module

(2.1) M = gr(X; X_{0}) := ⨁_{k≥0} M_{k} with M_{k} = X_{k}/X_{k−1}.

The annihilator Ann_{S(𝔤)}M := {D ∈ S(𝔤)| Dv = 0 (∀v ∈ M)} of M is a graded ideal of S(𝔤), and it defines an algebraic cone in the dual space 𝔤^{∗} of 𝔤:

V(M) := {λ ∈ 𝔤^{∗}| D(λ) = 0 (∀D ∈ Ann_{S(𝔤)}M)}, (2.2)

as the set of common zeros of elements of Ann_{S(𝔤)}M. Here S(𝔤) is looked upon as the polynomial ring over 𝔤^{∗} in the canonical way. It is then easily seen that the variety V(M) does not depend on the choice of a generating subspace X_{0}. So, hereafter we write V(X) for this invariant V(M) of X.

Definition.(Cf. [10], see also [14]) For a finitely generated U(𝔤)-module X, the variety V(X) ⊂ 𝔤^{∗} and its dimension d(X) := dim V(X) are called respectively the associated variety and the Gelfand-Kirillov dimension of X.

Remark. By the Hilbert-Serre theorem (cf. [14, Th.1.1]), the map k → dim X_{k} coincides with a polynomial in k of degree d(X), for sufficiently large k.

Let G_{C}^{ad} := Int(𝔤) be the adjoint group of 𝔤. We write I(𝔤) for the graded subalgebra of S(𝔤) consisting of all G_{C}^{ad}-fixed elements in S(𝔤). Then I(𝔤) has a unique maximal graded ideal I(𝔤)_{+} := ⊕_{k>0}I(𝔤) ∩ S^{k}(𝔤).

By making use of the Schur lemma [11, Lemma 0.5.2] for irreducible U(𝔤)-modules, one can deduce the following

Lemma 2.1. (Cf. [10, Cor.5.4]) Suppose that X is a U(𝔤)-module of finite length. Then its associated variety V(X) is contained in the cone N^{∗} defined by I(𝔤)_{+} :

N^{∗} := {λ ∈ 𝔤^{∗}| D(λ) = 0 (∀D ∈ I(𝔤)_{+})}. (2.3)

It should be noticed that, if 𝔤 is semisimple, the cone N^{∗} turns out to be the totality of nilpotent elements in 𝔤 under the identification of 𝔤^{∗} with 𝔤 through the Killing form.

2.2. The variety V(X) for (𝔤, 𝔨)-module X.

Now let 𝔨 be a Lie subalgebra of 𝔤. A U(𝔤)-module X is said to be locally U(𝔨)-finite if the U(𝔨)-submodule U(𝔨)v is finite-dimensional for every v ∈ X. By a (𝔤, 𝔨)-module is meant a locally U(𝔨)-finite, finitely generated U(𝔤)-module. Hereafter we concentrate on such (𝔤, 𝔨)-modules.

Let K̃_{C} denote the connected, simply connected Lie group with Lie algebra 𝔨. The natural inclusion i : 𝔨 ↪ 𝔤 gives rise to a Lie group homomorphism:

Ad : K̃_{C} ∋ k ↦ Ad(k) ∈ G_{C}^{ad} ⊂ GL(𝔤), (2.4)

from K̃_{C} into the group G_{C}^{ad} of all inner automorphisms of 𝔤, in the canonical way. We notice that, since K̃_{C} is simply connected, any (𝔤, 𝔨)-module X admits a K̃_{C}-module structure compatible with the U(𝔤)-action in the following sense:

(exp Z) · v = ∑_{j=0}^{∞} \frac{1}{j!}Z^{j}v (Z ∈ 𝔨), (2.5)

k · (Dv) = (Ad(k)D) · kv (D ∈ U(𝔤), k ∈ K̃_{C}), (2.6)

for every v ∈ X. Here the sum in (2.5) converges because Z^{j}v stay in a finite-dimensional subspace U(𝔨)v for all j ≥ 0.

By making use of this K̃_{C}-action, it is an easy task to deduce the following lemma on an orbital structure of the associated variety of a (𝔤, 𝔨)-module.

Lemma 2.2. (Cf. [10, Cor.5.13]) Let X be a (𝔤, 𝔨)-module. Then the associated variety V(X) of X is a union of K_{C}^{ad}-orbits contained in the orthogonal 𝔨^{⊥} := {λ ∈ 𝔤^{∗}| λ(Z) = 0 (∀Z ∈ 𝔨)} of 𝔨 in 𝔤^{∗}. Here K_{C}^{ad} := Ad(K̃_{C}) ⊂ G_{C}^{ad} denotes the analytic subgroup of G_{C}^{ad} with Lie algebra 𝔨, and it acts on 𝔤^{∗} through the coadjoint representation.

2.3. A criterion for locally free U(𝔫)-action.

Let 𝔨 and K_{C}^{ad} be as in 2.2. Take another Lie subalgebra 𝔫 of 𝔤 (not necessarily the one given by (1.37)). We are going to give a criterion for an irreducible (𝔤, 𝔨)-module to have locally free U(𝔫)-action.

To do this, let p^{∗} be the surjective linear map from 𝔤^{∗} to 𝔫^{∗} defined by the restriction to 𝔫 of each linear form on 𝔤. We say that an element λ ∈ 𝔤^{∗} satisfies the condition (P_{𝔨,𝔫}) if the projection p^{∗} carries the subspace ad^{∗}(𝔨)λ := {ad^{∗}(Z)λ| Z ∈ 𝔨} onto 𝔫^{∗}, i.e., (P_{𝔨,𝔫}) p^{∗}(ad^{∗}(𝔨)λ) = 𝔫^{∗}. Here ad^{∗}(Z)λ := (d/dt)(exp tZ ·λ)|_{t=0}, and ad^{∗}(𝔨)λ can be identified naturally with the tangent space of K_{C}^{ad}-orbit K_{C}^{ad} ·λ at the point λ.

This condition (P_{𝔨,𝔫}) for λ has a geometric interpretation as follows.

Lemma 2.3. If λ ∈ 𝔤^{∗} fulfills the condition (P_{𝔨,𝔫}), then the image p^{∗}(K_{C}^{ad} · λ) of K_{C}^{ad}-orbit K_{C}^{ad} ·λ under p^{∗} contains an open neighbourhood of p^{∗}(λ) in 𝔫^{∗}.

Proof. The condition (P_{𝔨,𝔫}) ensures that the differential of

K_{C}^{ad} ∋ k ↦ p^{∗}(k ·λ) ∈ 𝔫^{∗}

is surjective at the origin e ∈ K_{C}^{ad}. This immediately proves the claim. Q.E.D.

Proposition 2.1. Let X be a cyclic (𝔤, 𝔨)-module generated by a vector v_{0} ∈ X : X = U(𝔤)v_{0}. For a Lie subalgebra 𝔫 of 𝔤, the annihilator Ann_{U(𝔫)}(v_{0}) vanishes if there exists an element λ ∈ V(X) satisfying the condition (P_{𝔨,𝔫}).

Proof. Let M = gr(X; X_{0}) = ⨁_{k≥0} M_{k}, with X_{0} = Cv_{0}, be the graded S(𝔤)-module in (2.1). Write ṽ_{0} for the vector v_{0} viewed as an element of M_{0} ⊂ M. We remark that, since M = S(𝔤)ṽ_{0}, an element D ∈ S(𝔤) lies in the annihilator Ann_{S(𝔤)}M of M if and only if Dṽ_{0} = 0.

In order to prove the proposition, it suffices to show that the annihilator Ann_{S(𝔫)}(ṽ_{0}) vanishes by passing from X to the graded module M. This can be done as follows.

Suppose that a λ ∈ V(X) satisfies the condition (P_{𝔨,𝔫}), and that an element D ∈ S(𝔫) annihilates the vector ṽ_{0}. Then one sees from the above remark that

D(p^{∗}(μ)) = D(μ) = 0 for all μ ∈ V(X).

This combined with Lemmas 2.2 and 2.3 shows that the function D on 𝔫^{∗} is identically zero on an open neighbourhood of p^{∗}(λ). We thus get D = 0 because D ∈ S(𝔫) is a polynomial on 𝔫^{∗}. This completes the proof. Q.E.D.

By focusing our attention on irreducible modules X, we immediately deduce from Proposition 2.1 a criterion (sufficient condition) for X to admit locally free U(𝔫)-action, as follows.

Theorem 2.1. Let 𝔨, 𝔫 be two Lie subalgebras of 𝔤, and let X be an irreducible (𝔤, 𝔨)- module. Then, the action of the enveloping algebra U(𝔫) on X is locally free, that is, X is a torsion free U(𝔫)-module, provided that the associated variety V(X) of X contains a point λ with the condition (P_{𝔨,𝔫}).

Remark. In view of the Remark in 2.1, one necessarily deduces

(2.7) dim 𝔫 ≤ d(X) = dim V(X),

if a finitely generated U(𝔤)-module X enjoys a locally free U(𝔫)-action.

3. Locally free U(𝔫(O))-action on Harish-Chandra modules.

From now on, let 𝔤 be a complex semisimple Lie algebra, and θ be an involutive automorphism of 𝔤. The associated symmetric decomposition of 𝔤 is denoted by 𝔤 = 𝔨+𝔭 with 𝔨 := 𝔤(θ, +1) and 𝔭 := 𝔤(θ, −1) as in 1.6. Then there exists a θ-stable real form 𝔤_{0} of 𝔤 such that the θ on 𝔤_{0} gives a Cartan involution of 𝔤_{0} (see [3, Ch.III, Lemma 4.1]). We fix once and for all such a real form 𝔤_{0}, and let 𝔤_{0} = 𝔨_{0} + 𝔭_{0} denote the corresponding Cartan decomposition of 𝔤_{0}, where 𝔨_{0} := 𝔨 ∩ 𝔤_{0} and 𝔭_{0} := 𝔭 ∩ 𝔤_{0}.

By a Harish-Chandra module, we mean in this paper a (𝔤, 𝔨)-module X (see 2.2) of finite length, associated with the symmetric pair (𝔤, 𝔨). As shown by Harish-Chandra, the category of such (𝔤, 𝔨)-modules has an essential role in studying representations of a real semisimple Lie group with Lie algebra 𝔤_{0}.

For each irreducible Harish-Chandra module X, we construct in this section a family of nilpotent Lie subalgebras 𝔫(O) of 𝔤 for which X is locally free as a U(𝔫(O))-module, by using the associated variety V(X) of X and the Cayley transformation of normal 𝔰𝔩_{2}-triples. The Lie subalgebras 𝔫(O) are parametrized by the K_{C}^{ad}-orbits O contained in V(X).

The main result of this paper is given as Theorem 3.2 whose proof is carried out by combining Theorems 1.2 and 2.1. We shall give in 3.3 a concrete description of Lie subalgebras 𝔫(O) associated to holomorphic orbits O when the real form 𝔤_{0} is a simple Lie algebra of hermitian type.

3.1. Lie subalgebras 𝔫(O) associated with a nilpotent K_{C}^{ad}-orbit O.

We denote by N_{𝔭} the totality of nilpotent elements of 𝔤 contained in 𝔭. By [6, Th.2], the variety N_{𝔭} is a union of finitely many K_{C}^{ad}-orbits, where K_{C}^{ad} is as in 2.2 the connected Lie subgroup of G_{C}^{ad} = Int(𝔤) with Lie algebra 𝔨.

Let O be a K_{C}^{ad}-orbit in N_{𝔭}. We are attaching to O a K_{C}^{ad}-conjugacy class of nilpotent Lie subalgebras 𝔫(O) of 𝔤 in the following fashion.

Suppose that O ≠ {0}, and take any element X ∈ O. A strengthened version of the Jacobson-Morozov theorem [6, Prop.4] assures that X can be embedded to a unique, up to K_{C}^{ad}-conjugacy, normal 𝔰𝔩_{2}-triple (X, H, Y) in 𝔤 (see 1.6), where H ∈ 𝔨 and X, Y ∈ 𝔭. Set 𝔰 := CX + CH + CY ⊂ 𝔤 and define a nilpotent Lie subalgebra 𝔫 = 𝔫_{𝔰} = (𝔤_{1}^{1}(1) ⊕ 𝔤_{3}^{3}(1)) ⊕ (⊕_{l≥2}𝔤(l)) just as in (1.37), through the Cayley transform (X′, H′, Y′) of (X, H, Y) defined by (1.2). Then it is immediate to check that, up to K_{C}^{ad}-conjugacy, the Lie subalgebra 𝔫 is uniquely determined by O, independent of the choice of an X in O and that of an 𝔰𝔩_{2}-triple (X, H, Y). So we can and do write 𝔫(O) for this 𝔫.

We attach 𝔫(O) = {0} for the zero orbit O = {0}.

From Theorem 1.2(1), one immediately deduces

Lemma 3.1. It holds that dim 𝔫(O) = dim O.

Now let 𝔤 ∋ Z → \overline{Z} ∈ 𝔤 be the complex conjugation of 𝔤 with respect to the real form 𝔤_{0}. Sekiguchi’s result [8] enables us to choose a nice representative 𝔫(O) which is compatible with this conjugation except the 𝔤(1)-part.

More precisely speaking, take a normal 𝔰𝔩_{2}-triple (X, H, Y) in 𝔤 with X ∈ O. By virtue of [8, Lemma 1.4], there exists a k ∈ K_{C}^{ad} such that (X_{1}, H_{1}, Y_{1}) := (k·X, k·H, k·Y) is a strictly normal 𝔰𝔩_{2}-triple in the following sense:

(3.1) \overline{X_{1}} = Y_{1}, \overline{H_{1}} = −H_{1}, or equivalently X_{1} + Y_{1}, i(X_{1} − Y_{1}) ∈ 𝔭_{0}, iH_{1} ∈ 𝔨_{0}.

Then, as checked immediately, the Cayley transform (X′_{1}, H′_{1}, Y′_{1}) of (X_{1}, H_{1}, Y_{1}) (see (1.2)) lies in 𝔤_{0}.

Theorem 3.1. (Kostant-Sekiguchi, see [8, Th.1.9]) Under the above notation, the assignment O = K_{C}^{ad} · X ↦ O′ := G^{ad} · X′_{1} (3.2) gives a bijection (Kostant-Sekiguchi correspondence) between the set of nilpotent K_{C}^{ad}- orbits in 𝔭 and that of nilpotent G^{ad}-orbits in 𝔤_{0}. Here G^{ad} ⊂ C_{C}^{ad} denotes the adjoint group of 𝔤_{0}.

As for our Lie subalgebra 𝔫(O), one gains the following advantage by choosing a strictly normal 𝔰𝔩_{2}-triple (X_{1}, H_{1}, Y_{1}).

Proposition 3.1. Let 𝔫(O) = (𝔤_{1}^{1}(1) ⊕ 𝔤_{3}^{3}(1)) ⊕ (⊕_{l≥2}𝔤(l)) be a Lie subalgebra of 𝔤 constructed as above from a strictly normal 𝔰𝔩_{2}-triple (X_{1}, H_{1}, Y_{1}). Then one has

\overline{𝔤(l)} = 𝔤(l) (l ∈ Z), 𝔤(1) = \{𝔫(O) ∩ 𝔤(1)\} ⊕ \{\overline{𝔫(O) ∩ 𝔤(1)}\},

where 𝔤(l) denotes as in 1.6 the l-eigensubspace of 𝔤 for ad(H′_{1}) with H′_{1} = X_{1} + Y_{1} ∈ 𝔭_{0}. In particular, 𝔫(O) is stable under the complex conjugation \overline{·} if and only if 𝔤(1) = \{0\}, i.e., O is an even nilpotent orbit in 𝔭.

Proof. Note that the real form 𝔤_{0} is stable under the linear operators ad(H′_{1}) and stable also under J = θ Ad(w) in 1.2 with V = 𝔤, σ̃ = θ. Here w is defined as in (1.5) with X′ and Y′ replaced by X′_{1} and Y′_{1} respectively. Then the claim immediately follows from (1.33) and from the definition of subspaces 𝔤(l) and 𝔤_{η}^{κ}(l) (κ, η = 0, 1, 2, 3; l ∈ Z) in 1.4. Q.E.D.

3.2. Main result.

By virtue of Lemmas 2.1 and 2.2 one finds that the associated variety V(X) of each Harish-Chandra module X is a K_{C}^{ad}-stable algebraic cone in N_{𝔭}, by identifying the dual space 𝔤^{∗} with 𝔤 itself through the Killing form B of 𝔤.

We are now in a position to give the following theorem which establishes locally free U(𝔫(O))-action on Harish-Chandra modules.

Theorem 3.2. Let X be an irreducible Harish-Chandra module. The action of enveloping algebra U(𝔫(O)) of 𝔫(O) on X is locally free for every nilpotent K_{C}^{ad}-orbit O ⊂ 𝔭 contained in the associated variety V(X) of X. Here 𝔫(O) is the nilpotent Lie subalgebra of 𝔤 constructed in 3.1.

This is the main result of this paper.

Proof of Theorem 3.2. Take an element X ∈ O ⊂ V(X), and construct the Lie subalgebra 𝔫(O) as in 3.1. By Theorem 1.2(2), the Killing form B of 𝔤 is nondegenerate on [𝔨, X] × 𝔫(O). This shows that X ∈ V(X) satisfies the condition (P_{𝔨,𝔫}) in 2.3 with 𝔫 = 𝔫(O). Hence X admits a locally free U(𝔫(O))-action by Theorem 2.1. Any K_{C}^{ad}-conjugate k · 𝔫(O) of 𝔫(O) also has locally free action on X, because the universal covering group K̃_{C} of K_{C}^{ad} acts on X as in (2.5), (2.6). One thus establishes the result, as desired. Q.E.D.

We now deduce two important consequences of the above main result.

First, Theorem 3.2 together with the Remark in 2.3 allows us to derive the following theorem by concentrating our attention on a K_{C}^{ad}-orbit of V(X) of maximal dimension:

Theorem 3.3. Let X be as in Theorem 3.2, and let O_{max} be a nilpotent K_{C}^{ad}-orbit in V(X) of maximal dimension, that is, dim O_{max} = dim V(X). Then the corresponding 𝔫(O_{max}) is maximal (with respect to the inclusion relation) among the Lie subalgebras 𝔫 of 𝔤 having locally free U(𝔫)-action on X.

Second, a Harish-Chandra module X is called large if its associated variety V(X) contains an open K_{C}^{ad}-orbit in N_{𝔭}, or dim V(X) = dim N_{𝔭}. Our main result yields the following characterization of irreducible large Harish-Chandra modules.

Theorem 3.4. Let 𝔫_{m,0} be a maximal nilpotent Lie subalgebra of real form 𝔤_{0} appearing in an Iwasawa decomposition of 𝔤_{0}. An irreducible Harish-Chandra module X is large if and only if X is a locally free U(𝔫_{m})-module. Here we write 𝔫_{m} for the complexification of 𝔫_{m,0} in 𝔤.

Proof. Assume that an irreducible Harish-Chandra module X is large, and take a K_{C}^{ad}-orbit O_{max} in V(X) of maximal dimension. Then one finds that the Lie subalgebra 𝔫(O_{max}) is conjugate to 𝔫_{m} under the action of K_{C}^{ad} on 𝔤, because the nilpotent G^{ad}-orbit O′_{max} in 𝔤_{0} corresponding to O_{max} in Theorem 3.1 is a principal nilpotent orbit in 𝔤_{0} with dimension 2 dim 𝔫_{m,0}. Thus we can apply Theorem 3.2 to deduce the locally free U(𝔫_{m})-action on X. The converse follows immediately from the Remark in 2.3. Q.E.D.

Remark. The largeness of an irreducible Harish-Chandra module X is characterized also by the existence of Whittaker vectors for X. See for example [5, Th.K] and [7, Cor.2.2].

3.3. Lie subalgebras 𝔫(O_{t}) for holomorphic orbits O_{t}.

Now suppose that 𝔤_{0} = 𝔨_{0} + 𝔭_{0} is a noncompact real simple Lie algebra of hermitian type. We denote by ω the unique (up to sign) 𝔨_{0}-invariant complex structure on 𝔭_{0}. Extending ω to 𝔭 by complex linearity, one gets a triangular decomposition

𝔤 = 𝔭_{−} ⊕ 𝔨 ⊕ 𝔭_{+} with 𝔭_{±} := {Z ∈ 𝔭| ωZ = ±iZ}, (3.3)

of 𝔤 such that

[𝔨, 𝔭_{±}] ⊂ 𝔭_{±}, [𝔭_{+}, 𝔭_{−}] ⊂ 𝔨, [𝔭_{+}, 𝔭_{+}] = [𝔭_{−}, 𝔭_{−}] = {0}. (3.4)

It then follows that the subspaces 𝔭_{±} are included in the nilpotent variety N_{𝔭} of 𝔭, since (ad Z)^{3} = 0 for every Z ∈ 𝔭_{±}. A K_{C}^{ad}-orbit O contained in 𝔭_{+} is called holomorphic, as 𝔭_{+} is naturally identified with the holomorphic tangent space at the origin of the hermitian symmetric space G/K with 𝔤_{0} = Lie(G) and 𝔨_{0} = Lie(K).

We end this article by describing the nilpotent Lie subalgebras 𝔫(O) of 𝔤 associated with holomorphic K_{C}^{ad}-orbits O.

3.3.1. In order to do this, we prepare after [12, 3.1] and [13, 9.1] refined structure theorems for 𝔤, originally due to Harish-Chandra and Moore. Now let 𝔱_{0} be a compact Cartan subalgebra of 𝔤_{0} which is contained in 𝔨_{0}. We denote by Δ the root system of 𝔤 with respect to the complexification 𝔱 of 𝔱_{0}. For γ ∈ Δ, the corresponding root subspace is denoted by 𝔤(𝔱; γ). A root γ ∈ Δ is called compact (resp. noncompact) if 𝔤(𝔱; γ) ⊂ 𝔨 (resp. 𝔤(𝔱; γ) ⊂ 𝔭), and Δ_{c} (resp. Δ_{n}) stands for the set of all compact (resp. noncompact) roots. To each γ ∈ Δ we can and do attach a nonzero vector X_{γ} ∈ 𝔤(𝔱; γ) satisfying X_{γ} − X_{−γ}, i(X_{γ} + X_{−γ}) ∈ 𝔨_{0} + i𝔭_{0}, [X_{γ}, X_{−γ}] = H_{γ}. (3.5)

Here H_{γ} is the element of i𝔱_{0} corresponding to the coroot γ^{∨} := 2γ/(γ, γ) through the identification 𝔱^{∗} = 𝔱 by the Killing form B.

Take a positive system Δ^{+} of Δ compatible with the decomposition (3.3):

𝔭_{±} = ⨁_{γ∈Δ_{n}^{+}} 𝔤(𝔱; ±γ) with Δ_{n}^{+} := Δ^{+} ∩ Δ_{n},

and fix a lexicographic order on i𝔱_{0}^{∗} which yields Δ^{+}. Using this order we define a fundamental sequence (γ_{1}, γ_{2},…, γ_{r}) of strongly orthogonal (i.e., γ_{i} ± γ_{j} ∉ Δ ∪ {0} for i ≠ j) noncompact positive roots in such a way that γ_{k} is the maximal element of Δ^{+}, which is strongly orthogonal to γ_{k+1},…, γ_{r}.

Now, put 𝔱^{−} := ∑_{k=1}^{r} CH_{γ_{k}} ⊂ 𝔱, and denote by π(γ) ∈ (𝔱^{−})^{∗} the restriction to 𝔱^{−} of a linear form γ ∈ 𝔱^{∗}. For integers k, m with 1 ≤ m < k ≤ r, we define subsets P_{km}, P_{k}, P_{0} of Δ_{n}^{+} and subsets C_{km}, C_{k}, C_{0} of Δ_{c}^{+} respectively by

P_{km} := \{γ ∈ Δ_{n}^{+}| π(γ) = \frac{π(γ_{k}) + π(γ_{m})}{2}\}, (3.6)

C_{km} := \{γ ∈ Δ_{c}^{+}| π(γ) = \frac{π(γ_{k}) − π(γ_{m})}{2}\}, (3.7)

P_{k} := \{γ ∈ Δ_{n}^{+}| π(γ) = \frac{π(γ_{k})}{2}\}, C_{k} := \{γ ∈ Δ_{c}^{+}| π(γ) = \frac{π(γ_{k})}{2}\}, (3.8)

P_{0} := {γ_{1}, γ_{2},…, γ_{r}}, C_{0} := {γ ∈ Δ_{c}^{+}| π(γ) = 0}. (3.9)

Then, by Harish-Chandra the subsets Δ_{n}^{+} and Δ_{c}^{+} are expressed as

Δ_{n}^{+} = ( ⋃_{1≤k≤r} P_{k}) ⋃ P_{0} ⋃( ⋃_{1≤m<k≤r} P_{km}), (3.10)

Δ_{c}^{+} = C_{0} ⋃( ⋃_{1≤k≤r} C_{k}) ⋃( ⋃_{1≤m<k≤r} C_{km}), (3.11)

(3.12)

where the unions are disjoint.

We set H_{k} := X_{γ_{k}} + X_{−γ_{k}} ∈ 𝔭_{0} for 1 ≤ k ≤ r. Then

𝔞_{p,0} := ∑_{k=1}^{r} RH_{k} (3.13)

turns out to be a maximal abelian subspace of 𝔭_{0}. Let Ψ denote the root system of 𝔤_{0} with respect to 𝔞_{p,0}, and for each k let ψ_{k} ∈ 𝔞_{p,0}^{∗} be the linear form on 𝔞_{p,0} defined by ψ_{k}(H_{m}) = 2δ_{km} (m = 1,…, r; with Kronecker’s δ_{km}). Moore’s restricted root theorem describes Ψ as follows.

Theorem 3.5. (Moore) The elements ψ_{k} (1 ≤ k ≤ r) form a basis of 𝔞_{p,0}^{∗}, and there exist only two possibilities for the root system Ψ :

Ψ = \{±(\frac{ψ_{k} − ψ_{m}}{2} )| 1 ≤ m < k ≤ r\} ∪ \{±(\frac{ψ_{k} + ψ_{m}}{2} )| 1 ≤ m ≤ k ≤ r\},

if the subsets P_{k} and C_{k} are empty for every k, or otherwise

Ψ = \{±(\frac{ψ_{k} − ψ_{m}}{2} )| 1 ≤ m < k ≤ r\} ∪ \{±(\frac{ψ_{k} + ψ_{m}}{2} )| 1 ≤ m ≤ k ≤ r\}
  ∪\{±\frac{ψ_{k}}{2} | 1 ≤ k ≤ r\}.

The former possibility occurs exactly when the corresponding hermitian symmetric space is analytically equivalent to a tube domain.

3.3.2. For each restricted root ψ ∈ Ψ, let 𝔤(𝔞_{p}; ψ) denote the complexified root subspace of 𝔤 corresponding to ψ. We can now write down a basis of each 𝔤(𝔞_{p}; ψ) by means of the vectors X_{γ} ∈ 𝔤(𝔱; γ) (γ ∈ Δ) defined in (3.5), as follows.

Proposition 3.2. (Hashizume, cf [13, Lemmas 9.1 and 9.2]) (1) For 1 ≤ m < k ≤ r, the vectors

E_{γ}^{±} := X_{γ} + [X_{−γ_{k}}, X_{γ}] ± [X_{−γ_{m}}, X_{γ}] ± [X_{−γ_{m}}, [X_{−γ_{k}}, X_{γ}]] (3.14)

form a basis of the root subspace 𝔤(𝔞_{p}; (ψ_{k} ± ψ_{m})/2), where γ runs over the elements of P_{km} in (3.6).

(2) The element E_{k} := i(H_{γ_{k}} − X_{γ_{k}} + X_{−γ_{k}})/2 (3.15)

lies in 𝔤(𝔞_{p}; ψ_{k}) and it holds that dim 𝔤(𝔞_{p}; ψ_{k}) = 1 for every 1 ≤ k ≤ r.

(3) The subspace 𝔤(𝔞_{p}; ψ_{k}/2) has a basis:

E_{γ}^{1} := X_{γ} + [X_{−γ_{k}}, X_{γ}] (γ ∈ P_{k} ∪ C_{k}) (3.16)

for every 1 ≤ k ≤ r, where P_{k} and C_{k} are as in (3.8).

3.3.3. Set X(t) := ∑_{t<k≤r} X_{γ_{k}} ∈ 𝔭_{+} (X(r) := 0) for 0 ≤ t ≤ r, and let O_{t} ⊂ 𝔭_{+} be the holomorphic K_{C}^{ad}-orbit through X(t). The following well-known proposition parametrizes such K_{C}^{ad}-orbits in N_{𝔭}.

Proposition 3.3. The subspace 𝔭_{+} splits into a disjoint union of r + 1 number of K_{C}^{ad}-orbits O_{t} (0 ≤ t ≤ r): 𝔭_{+} = ∐_{0≤t≤r} O_{t}, and the closure \overline{O_{t}} of orbit O_{t} is equal to ∪_{s≥t}O_{s} for every t.

Suggested by this proposition, we want to describe the nilpotent Lie subalgebra 𝔫(O_{t}) in terms of root vectors E_{γ}^{±}, E_{k} and E_{γ}^{1} in Proposition 3.2, for every 0 ≤ t ≤ r.

This is achieved in the following way. Put

H(t) := ∑_{t<k≤r} H_{γ_{k}}, Y (t) := ∑_{t<k≤r} X_{−γ_{k}}. (3.17)

Then it follows from (3.5) together with the strong orthogonality of γ_{k}’s that (X(t), H(t), Y (t)) is a strictly normal 𝔰𝔩_{2}-triple in 𝔤. We denote by (X′(t), H′(t), Y′(t)) the Cayley transform of (X(t), H(t), Y (t)) defined in (1.2). Noting that H′(t) = ∑_{t<k≤r} H_{k}, one deduces from Theorem 3.5 the following

Lemma 3.2. The Lie algebra 𝔤 decomposes into a direct sum of the eigensubspaces for ad H′(t) as: 𝔤 = 𝔤(−2) ⊕ 𝔤(−1) ⊕ 𝔤(0) ⊕ 𝔤(1) ⊕ 𝔤(2), (3.18)

where

𝔤(±2) = ⨁_{t<m≤k≤r} 𝔤(𝔞_{p}; ±(\frac{ψ_{k} + ψ_{m}}{2} )),

𝔤(±1) = ⨁_{1≤m≤t<k≤r} (𝔤(𝔞_{p}; ±(\frac{ψ_{k} + ψ_{m}}{2} )) ⊕ 𝔤(𝔞_{p}; ±(\frac{ψ_{k} − ψ_{m}}{2} ))) ⨁ ( ⨁_{t<k≤r} 𝔤(𝔞_{p}; ±\frac{ψ_{k}}{2} )) ,

𝔤(0) = 𝔷_{𝔤}(𝔞) ⨁ ( ⨁_{1≤m≤k≤t} (𝔤(𝔞_{p}; \frac{ψ_{k} + ψ_{m}}{2} ) ⊕ 𝔤(𝔞_{p}; \frac{ψ_{k} − ψ_{m}}{2} )))
  ⨁ ( ⨁_{1≤m≤k≤t} (𝔤(𝔞_{p}; −\frac{ψ_{k} + ψ_{m}}{2} ) ⊕ 𝔤(𝔞_{p}; −\frac{ψ_{k} − ψ_{m}}{2} ))) ,

and 𝔷_{𝔤}(𝔞) denotes the centralizer of 𝔞 in 𝔤. In particular, 𝔤_{η}^{3}(l) = {0} for all η and l.

By utilizing Proposition 3.2 and Lemma 3.2, we obtain the following complete description of Lie subalgebra 𝔫(O_{t}) associated to the orbit O_{t}.

Theorem 3.6. Let O_{t} = K_{C}^{ad} ·X(t) with 0 ≤ t ≤ r be a holomorphic K_{C}^{ad}-orbit in 𝔭_{+}, and let 𝔫(O_{t}) be the Lie subalgebra of 𝔤 constructed as in 3.1 from the Cayley transform of (X(t), H(t), Y (t)). Then 𝔫(O_{t}) is expressed as

𝔫(O_{t}) = 𝔤_{1}^{1}(1) ⊕ 𝔤(2), (3.19)

with 𝔤(2) as in Lemma 3.2, and 𝔤_{1}^{1}(1) is the subspace of 𝔤(1) having a basis:

E_{γ}^{+} − E_{γ}^{−} (γ ∈ P_{km}; 1 ≤ m ≤ t < k ≤ r), E_{γ}^{1} (γ ∈ C_{k}; t < k ≤ r). (3.20)

Here E_{γ}^{±} and E_{γ}^{1} are as in Proposition 3.2.

Proof. The first claim (3.19) follows immediately from Lemma 3.2 since 𝔤(j) = {Z ∈ 𝔤| [H′(t), Z] = jZ} equals {0} for |j| ≥ 3. Notice that the subspace 𝔤_{1}^{1}(1) of 𝔤(1) is given as 𝔤_{1}^{1}(1) = {Z ∈ 𝔤(1)| JZ = iZ}, (3.21)

where J = θw ∈ GL(𝔤) with

w = exp \frac{π}{2} (X′(t) − Y′(t)) = exp \frac{πi}{2}H(t) ∈ G_{C}^{ad}, (3.22)

by definition (see 1.4 and 1.6). Then one finds by direct computation that the operator J acts on vectors E_{γ}^{±}, E_{γ}^{1} respectively as

JE_{γ}^{±} = −iE_{γ}^{∓} if γ ∈ P_{km} with k > t ≥ m, (3.23) JE_{γ}^{1} = −iE_{γ}^{1} if γ ∈ P_{k} with k > t, (3.24) JE_{γ}^{1} = iE_{γ}^{1} if γ ∈ C_{k} with k > t. (3.25)

This combined with Proposition 3.2 and Lemma 3.2 shows that the vectors in (3.20) form a basis of the subspace 𝔤_{1}^{1}(1). Q.E.D.

Theorem 3.2 coupled with Theorem 3.6 implies that the (at most) two-step nilpotent Lie subalgebras 𝔫(O_{t}) enjoy locally free action on the Harish-Chandra module of a holomorphic discrete series for every t, because its associated variety coincides with the whole 𝔭_{+} (cf. [15]). More generally, the associated variety of any irreducible highest weight Harish-Chandra module X is contained in 𝔭_{+}, and so X admits locally free U(𝔫(O_{t}))-action for some t’s specified according as the rank of X.

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Akihiko Gyoja Hiroshi Yamashita Division of Mathematics Department of Mathematics Faculty of Integrated Human Studies Faculty of Science Kyoto University Hokkaido University Kyoto 606-01, Japan Sapporo 060, Japan E-mail: gyoja@math.h.kyoto-u.ac.jp E-mail: yamasita@math.hokudai.ac.jp