INVARIANT DISTRIBUTIONS SUPPORTED ON THE NILPOTENT CONE OF A SEMISIMPLE LIE ALGEBRA

THIERRY LEVASSEUR

Abstract. Let 𝔤 be a semisimple complex Lie algebra with adjoint group G and D(𝔤) be the algebra of differential operators with polynomial coefficients on 𝔤. If 𝔤_{0} is a real form of 𝔤, we give the decomposition of the semisimple D(𝔤)^{G}-module of invariant distributions on 𝔤_{0} supported on the nilpotent cone.

0. Introduction

Let 𝔤 be a semisimple complex Lie algebra with adjoint group G. Choose a Cartan subalgebra 𝔥 of 𝔤 and let W be the associated Weyl group. Denote by W the set of isomorphism classes of irreducible W -modules and by H(𝔥^{∗}) the graded vector space of W -harmonic polynomials on 𝔥. For χ ∈ W , set

b(χ) = inf{j ∈ ℕ : [H^{j}(𝔥^{∗}) : χ] ≠ 0}

and choose a W -submodule V_{χ} ⊂ H^{b(χ)}(𝔥^{∗}) in the class of χ. Denote by d(χ) the dimension of V_{χ}.

Let S(𝔤^{∗}) be the algebra of polynomial functions on 𝔤 and D(𝔤) be the algebra of differential operators on 𝔤, with coefficients in S(𝔤^{∗}). The group G acts on 𝔤, via the adjoint action, and hence has an induced action on S(𝔤^{∗}), S(𝔤) and D(𝔤). Denote the differential of this action by τ : 𝔤 → D(𝔤). Let S_{+}(𝔤)^{G} and S_{+}(𝔤^{∗})^{G} be the set of invariant elements without constant term. Recall that N(𝔤), the nilpotent cone of 𝔤, is the variety of zeroes of the ideal S_{+}(𝔤^{∗})^{G}S(𝔤^{∗}).

Let 𝔤_{0} be a real form of 𝔤 with adjoint group G_{0} ⊂ G. Denote by Db(𝔤_{0}) the D(𝔤)-module of distributions on 𝔤_{0}. Then, the subspace of invariant distributions Db(𝔤_{0})^{G_{0}} = {T ∈ Db(𝔤_{0}) : τ(𝔤).T = 0} is a D(𝔤)^{G}-module, containing the submodule of invariant distributions supported on the nilpotent cone

Db(𝔤_{0})_{nil}^{G_{0}} = {Θ ∈ Db(𝔤_{0})^{G_{0}} : Supp Θ ⊂ N(𝔤_{0})}

where N(𝔤_{0}) = N(𝔤)∩𝔤_{0} is the nilpotent cone of 𝔤_{0}. The structure of Db(𝔤_{0})_{nil}^{G_{0}} as a vector space is well understood, see, for example, [1, 5]. Let [𝔥_{1}], …, [𝔥_{r}] be the conjugacy classes of Cartan subalgebras of 𝔤_{0}. For each j, let ε_{I,j} : W (𝔥_{j}) → {±1} be the imaginary signature of the real Weyl group W (𝔥_{j}). Then [5, Proposition 6.1.1] there exists a vector space isomorphism

⨁_{j=1}^{r} S(𝔥_{j,ℂ})^{ε_{I,j}} \underrightarrow{∼} Db(𝔤_{0})_{nil}^{G_{0}} (∗) where S(𝔥_{j,ℂ})^{ε_{I,j}} is the isotypic component of type ε_{I,j} in the W (𝔥_{j})-module S(𝔥_{j,ℂ}).

1991 Mathematics Subject Classification. 14L30, 16S32, 17B20, 22E46.

Key words and phrases. semisimple Lie algebra, invariant distribution, nilpotent orbit, Weyl group representation.

Research partially supported by EC TMR network “Algebraic Lie Representations”, Grant No. ERB FMRX-CT97-0100.

One aim of this note is to give a complete description of the D(𝔤)^{G}-module Db(𝔤_{0})_{nil}^{G_{0}}. This description is given in terms of the simple summands of the equivariant holonomic D(𝔤)-module

ℳ = D(𝔤)/(D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤^{∗})^{G}).

By [9], [18] or [13], it is known that we have a decomposition

ℳ = ⨁_{χ∈W}ˆ d(χ) ℳ_{χ}

where the ℳ_{χ} are pairwise non-isomorphic simple D(𝔤)-modules. Moreover, the support (in 𝔤) of ℳ_{χ} is the closure of a nilpotent orbit and ℳ_{χ}^{G} is a simple D(𝔤)^{G}- module. Then we have, see Corollary 3.6:

Theorem A. The D(𝔤)^{G}-module Db(𝔤_{0})_{nil}^{G_{0}} decomposes as

Db(𝔤_{0})_{nil}^{G_{0}} ≅ ⨁_{χ∈W}ˆ m_{χ}ℳ_{χ}^{G}

where m_{χ} = ∑_{j=1}^{r} dim V_{χ}^{ε_{I,j}}.

This theorem is proved by combining the isomorphism (∗) and the properties, established in [18, 11, 12, 13], of the Harish-Chandra homomorphism

δ : D(𝔤)^{G} ⟶ D(𝔥)^{W} .

In the particular case where 𝔤_{0} is a complex Lie algebra 𝔤_{1} (viewed as a real Lie algebra), Theorem A was proved by N. Wallach [18]. In this case, 𝔤 ≃ 𝔤_{1} × 𝔤_{1}, W ≃ W_{1} × W_{1} where W_{1} is the Weyl group of 𝔤_{1}. Then, each ℳ_{χ} occuring in the decomposition of Db(𝔤_{0})_{nil}^{G_{0}} is of the form ℳ_{ϕ} ⊠ ℳ_{ϕ} with χ = ϕ ⊠ ϕ, ϕ ∈ W_{1}, and one has m_{χ} = 1. Hence Db(𝔤_{0})_{nil}^{G_{0}} ≅ ⊕_{ϕ∈W_{ˆ1}}ℳ_{ϕ}^{G_{1}} ⊠ ℳ_{ϕ}^{G_{1}} as a D(𝔤)^{G}-module.

The next corollary is an easy consequence of Theorem A.

Corollary B. Let χ ∈ W . then, ℳ_{χ} ≅ D(𝔤).Θ for some Θ ∈ Db(𝔤_{0}) if, and only if, V_{χ}^{ε_{I,j}} ≠ 0 for some j ∈ {1, …, r}.

In Remark 3.7, we apply this result to give examples of modules ℳ_{χ} which cannot be generated by a distribution on any real form of 𝔤.

1. Preliminary results

We retain the notation of the introduction. Denote by Δ the root system of 𝔥 in 𝔤 and fix a system Δ^{+} of positive roots. Set n = dim 𝔤, ℓ = dim 𝔥 and ν = #Δ^{+}, hence n = 2ν + ℓ. Let π be the product of positive roots and recall that x ∈ 𝔤 is called generic if π(x) ≠ 0. If 𝔞 ⊂ 𝔤, we denote by 𝔞′ the set of generic elements in 𝔞.

For q ∈ S(𝔤), let ∂(q) ∈ D(𝔤) be the corresponding differential operator with constant coefficients. Let {e_{i}}_{1⩽i⩽n} be an orthonormal basis of 𝔤 with respect to the Killing form κ such that {e_{i}}_{1⩽i⩽ℓ} is a basis of 𝔥. Denote by x_{i} ∈ S(𝔤^{∗}), 1 ⩽ i ⩽ n, the associated coordinate functions; thus ∂(e_{i}) identifies with the partial derivative ∂_{i} = \frac{∂}{∂x_{i}}. Denote the Euler vector fields on 𝔤 and 𝔥 by E_{𝔤} = ∑_{i=1}^{n} x_{i}∂_{i} and E_{𝔥} = ∑_{i=1}^{ℓ} x_{i}∂_{i}.

We now give some notation and results from [11, 12, 13, 18]. Recall first that the algebra homomorphism, defined by Harish-Chandra,

δ : D(𝔤)^{G} ⟶ D(𝔥)^{W}

extends the Chevalley isomorphisms S(𝔤)^{G} \underrightarrow{∼} S(𝔥)^{W} and S(𝔤^{∗})^{G} \underrightarrow{∼} S(𝔥^{∗})^{W}. The map δ is surjective and its kernel is I = (D(𝔤)τ(𝔤))^{G}. This enables one to identify, through δ, modules over A(𝔤) := D(𝔤)^{G}/I with D(𝔥)^{W} -modules.

Lemma 1.1. Let D ∈ D(𝔤)^{G}. Then D = P + Q with P ∈ ℂ⟨S(𝔤)^{G}, S(𝔤^{∗})^{G}⟩ and Q ∈ I.

Proof. By [11], we know that D(𝔥)^{W} = ℂ⟨S(𝔥)^{W} , S(𝔥^{∗})^{W} ⟩. The lemma is therefore consequence of the properties of δ previously recalled.

Recall that the (D(𝔥)^{W} , W)-module S(𝔥^{∗}) decomposes as

S(𝔥^{∗}) ≅ ⨁_{χ∈W}ˆ V^{χ} ⊗_{ℂ} V_{χ} (1.1)

where V^{χ} = Hom_{W} (V_{χ}, S(𝔥^{∗})) is a simple D(𝔥)^{W} -module. Let {v_{χ}^{1}, …, v_{χ}^{d(χ)}} be a basis of V_{χ}, then V^{χ} ≅ D(𝔥)^{W} .v_{χ}^{j} for all j and (1.1) implies that

S(𝔥^{∗}) = ⨁_{χ∈W}ˆ ⨁_{j=1}^{d(χ)} D(𝔥)^{W} .v_{χ}^{j}.

Now, set 𝒩 = D(𝔤)/D(𝔤)τ(𝔤) ⊗_{A(𝔤)} S(𝔥^{∗}) and 𝒩_{χ} = D(𝔤)/D(𝔤)τ(𝔤) ⊗_{A(𝔤)} V^{χ}. We have

𝒩 = D(𝔤)/(D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤)^{G}) (1.2)

and, using (1.1),

𝒩 = ⨁_{χ∈W}ˆ 𝒩_{χ} ⊗_{ℂ} V_{χ}. (1.3)

Then each 𝒩_{χ} is a simple (holonomic) D(𝔤)-module [13] and, therefore, 𝒩 is a semisimple D(𝔤)-module (see also [9]). Let C(𝒩) be the full subcategory of finitely generated D(𝔤)-modules of the form ⊕_{χ∈W}ˆ m_{χ}𝒩_{χ}, m_{χ} ∈ ℕ. From [13] we know that the category C(𝒩) is equivalent to the category W -mod (of finite dimensional W -modules) via the functor

Sol : C(𝒩) ⟶ W -mod, Sol(N) = Hom_{D(𝔥)}^{W} (N^{G}, S(𝔥^{∗}))

where W acts on Sol(N) through its natural action on S(𝔥^{∗}).

The Killing form κ induces a G-isomorphism 𝔤 \underrightarrow{∼} 𝔤^{∗} and an algebra automorphism κ of D(𝔤), defined by κ(∂(v)) = κ(v, ), κ(κ(v, )) = −∂(v), for all v ∈ 𝔤. Hence, in coordinates, κ(∂_{j}) = x_{j}, κ(x_{j}) = −∂_{j}. Set i = \sqrt{−1} ∈ ℂ and denote by i the automorphism of D(𝔤) given by i(∂_{j}) = −i∂_{j}, i(x_{j}) = ix_{j}. Define then the “Fourier transformation” F_{𝔤} ∈ Aut D(𝔤) by F_{𝔤} = i ∘ κ = κ ∘ i^{−1}; thus F_{𝔤}(x_{j}) = i∂_{j}, F_{𝔤}(∂_{j}) = ix_{j}. One easily checks that κ(τ(x)) = F_{𝔤}(τ(x)) = τ(x) for all x ∈ 𝔤; moreover, κ and F_{𝔤} are G-equivariant. Similarly, since κ is non degenerate and W -invariant on 𝔥, one can define W -equivariant automorphisms κ and F_{𝔥} = i ∘ κ in Aut D(𝔥).

Lemma 1.2. One has δ ∘ F_{𝔤} = F_{𝔥} ∘ δ.

Proof. A direct computation shows that δ(F_{𝔤}(P)) = F_{𝔥}(δ(P)) when P belongs to S(𝔤)^{G} or S(𝔤^{∗})^{G}. Since δ is a homomorphism, it follows that δ(F_{𝔤}(P)) = F_{𝔥}(δ(P)) for all P ∈ ℂ⟨S(𝔤)^{G}, S(𝔤^{∗})^{G}⟩. Now, let D ∈ D(𝔤)^{G} and write D = P + Q as in Lemma 1.1. Then, since F_{𝔤}(I) = I, we have δ(F_{𝔤}(D)) = δ(F_{𝔤}(P)) = F_{𝔥}(δ(P)) = F_{𝔥}(δ(D)).

Recall that H(𝔥^{∗}) is the vector space of W -harmonic polynomials on 𝔥. Hence

H(𝔥^{∗}) = {f ∈ S(𝔥^{∗}) : ∂(q).f = 0 for all q ∈ S_{+}(𝔥)^{W} }

and, as W -module, H(𝔥^{∗}) identifies with the regular representation of W. The vector space H(𝔥^{∗}) is a graded subspace of S(𝔥^{∗}) and we set H^{j}(𝔥^{∗}) = S^{j}(𝔥^{∗}) ∩ H(𝔥^{∗}), 0 ⩽ j ⩽ ν. Define the harmonic elements of S(𝔥) by H(𝔥) = F_{𝔥}(H(𝔥^{∗})) = ⊕_{j=0}^{ν}H^{j}(𝔥). (We could as well have set H(𝔥) = κ(H(𝔥^{∗})), since H^{j}(𝔥^{∗}) is stable under i.)

Since V_{χ} ⊂ H^{b(χ)}(𝔥^{∗}), we have (E_{𝔥}−b(χ)).v_{χ}^{j} = 0. For all d ∈ L := ann_{D(𝔥)}^{W} (v_{χ}^{j}), we have [E_{𝔥} − b(χ), d] = [E_{𝔥}, d] ∈ L. It follows that L = ⊕_{k∈ℤ}L ∩ D^{k}(𝔥)^{W}, where D^{k}(𝔥) = {d ∈ D(𝔥) : [E_{𝔥}, d] = kd}. Equivalently, L is stable under the ℂ^{∗}-action on D(𝔥) given by f ↦ λf, ∂(v) ↦ λ^{−1}∂(v), f ∈ 𝔥^{∗}, v ∈ 𝔥. In particular, we see that F_{𝔥}(L) = κ(L).

Let R be a ring and α ∈ Aut(R). If M is an R-module, we define the R-module M^{α} to be the abelian group M with action of a ∈ R on x ∈ M given by a▪x = α(a)x. This applies to the modules 𝒩, 𝒩_{χ} and the automorphism α = F_{𝔤}^{−1}. Define

ℳ = 𝒩^{F_{𝔤}^{−1}}, ℳ_{χ} = 𝒩_{χ}^{F_{𝔤}^{−1}}.

Thus, from (1.2) and (1.3), we obtain

ℳ = D(𝔤)/(D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤^{∗})^{G}) ≅ ⨁_{χ∈W}ˆ ℳ_{χ} ⊗_{ℂ} V_{χ}.

Remark. In [13] one defines ℳ_{χ} to be 𝒩_{χ}^{κ^{−1}}, but the two definitions agree. Indeed, let V^{χ} ≅ D(𝔥)^{W} .v_{χ}^{j} = D(𝔥)^{W} /L be as above. Then,

𝒩_{χ} ≅ D(𝔤)/J, J = D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤)^{G} + D(𝔤)δ^{−1}(L).

Write 𝒩_{χ} = D(𝔤).(¯1⊗_{A(𝔤)} v_{χ}^{j}), where ¯1 is the canonical generator of D(𝔤)/D(𝔤)τ(𝔤). From δ(E_{𝔤}) = E_{𝔥} − ν, we get that (E_{𝔤} − (b(χ) − ν)).(¯1 ⊗_{A(𝔤)} v_{χ}^{j}) = 0. It follows (as above) that J is stable under the natural ℂ^{∗}-action on D(𝔤). Hence, F_{𝔤}(J) = κ(J) and we have 𝒩_{χ}^{κ^{−1}} = 𝒩_{χ}^{F_{𝔤}^{−1}}.

We can define the category C(ℳ) similar to C(𝒩). We clearly have M ∈ C(ℳ) if, and only if, N = M^{F_{𝔤}} ∈ C(𝒩). Moreover, by [13], this is equivalent to saying that M is a G-equivariant finitely generated D(𝔤)-module such that M = D(𝔤)M^{G} and Supp M ⊂ N(𝔤). This is also equivalent to: N is a G-equivariant finitely generated D(𝔤)-module such that N = D(𝔤)N^{G} and N is S_{+}-finite (meaning that each v ∈ N is killed by a power of S_{+}(𝔤)^{G}).

Recall that 𝒩_{χ}^{G} \underrightarrow{∼} V^{χ} through the identification of A(𝔤) with D(𝔥)^{W}.

Lemma 1.3. One has: ℳ_{χ}^{G} \underrightarrow{∼}(V^{χ})^{F_{𝔥}^{−1}}.

Proof. Write 𝒩_{χ} = D(𝔤)/J. Then, ℳ_{χ} = D(𝔤)/F_{𝔤}(J) and ℳ_{χ}^{G} = D(𝔤)^{G}/F_{𝔤}(J^{G}). By Lemma 1.2, δ(F_{𝔤}(J^{G})) = F_{𝔥}(δ(J^{G})), therefore ℳ_{χ}^{G} \underrightarrow{∼} D(𝔥)^{W} /F_{𝔥}(δ(J^{G})). Since V^{χ} ≅ D(𝔥)^{W} /δ(J^{G}), the lemma follows.

Let 𝔤_{0} be a real form of 𝔤 with adjoint group G_{0} ⊂ G. There exists a natural action of D(𝔤) on Db(𝔤_{0}) defined by

⟨∂(v).T, f⟩ = ⟨T, −∂(v).f⟩, ⟨ξ.T, f⟩ = ⟨T, ξf⟩

for all T ∈ Db(𝔤_{0}), f ∈ C_{c}^{∞}(𝔤_{0}), v ∈ 𝔤, ξ ∈ 𝔤^{∗}. This induces a structure of D(𝔤)^{G}- module on Db(𝔤_{0})^{G_{0}}. From I. Db(𝔤_{0})^{G_{0}} = 0, we obtain a natural A(𝔤)-module structure on Db(𝔤_{0})^{G_{0}}.

Fix a basis {u_{1}, …, u_{n}} of 𝔤_{0} such that κ(u_{j}, u_{k}) = ±δ_{jk} and denote by dy be the Lebesgue measure associated to this choice. Let S(𝔤_{0}) be the Schwartz space on 𝔤_{0}. Define, as in [18, Appendix 1], the Fourier transform of f ∈ S(𝔤_{0}) by

f̂(x) = \frac{1}{(2π)^{n/2}} ∫_{𝔤_{0}} f(y)e^{−iκ(y,x)}dy

Let T be a tempered distribution on 𝔤_{0}. The Fourier transform of T is defined by ⟨T̂, f⟩ = ⟨T, f̂⟩ for f ∈ C_{c}^{∞}(𝔤_{0}). Then we have

∀ D ∈ D(𝔤), ∀ T ∈ Db(𝔤_{0}), D̂.T = F_{𝔤}(D).T̂. (1.4)

Recall [2] that T ∈ Db(𝔤_{0}) is said to be homogeneous of degree d if, for all f ∈ C_{c}^{∞}(𝔤_{0}), t ∈ ℝ^{∗}, ⟨T, f_{t}⟩ = t^{d}⟨T, f⟩, where f_{t}(v) = t^{−n}f(t^{−1}v). Then, a homogeneous distribution of degree d is tempered and satisfies E_{𝔤}.T = dT. We will need the following well known result:

Lemma 1.4. Let T ∈ Db(𝔤_{0}) be tempered and set M = D(𝔤).T. Then M^{F_{𝔤}} ≅ D(𝔤).T̂.

Proof. By (1.4) we have ann_{D(𝔤)}(T̂) = F_{𝔤}^{−1}(ann_{D(𝔤)}(T)). Hence the result.

Let N(𝔤_{0}) be the set of nilpotent elements of 𝔤_{0}. Define D(𝔤)-submodules of Db(𝔤_{0}) by

Db(𝔤_{0})_{nil} = {Θ ∈ Db(𝔤_{0}) : Supp Θ ⊂ N(𝔤_{0})}
Db(𝔤_{0})_{S_{+}} = {T ∈ Db(𝔤_{0}) : ∃k ∈ ℕ, S_{+}(𝔤)^{k}.T = 0}.

The elements of Db(𝔤_{0})_{S_{+}} are called S_{+}-finite. Observe that Db(𝔤_{0})_{nil}^{G_{0}} and Db(𝔤_{0})_{S_{+}}^{G_{0}} are D(𝔤)^{G}-modules. The next theorem is consequence of the results proved in [18].

Theorem 1.5. (1) Db(𝔤_{0})_{nil}^{G_{0}} = {Θ ∈ Db(𝔤_{0})^{G_{0}} : D(𝔤).Θ ∈ C(ℳ)}.

(2) Db(𝔤_{0})_{S_{+}}^{G_{0}} = {T ∈ Db(𝔤_{0})^{G_{0}} : D(𝔤).T ∈ C(𝒩)}.

(3) Θ ∈ Db(𝔤_{0})_{nil}^{G_{0}} ⟺ Θ̂ ∈ Db(𝔤_{0})_{S_{+}}^{G_{0}}.

Proof. (1) follows from [18, Theorem 6.1], since D(𝔤).Θ ∈ C(ℳ) is equivalent to D(𝔤)^{G}.Θ ≅ ⊕_{χ∈W}ˆ m_{χ}ℳ_{χ}^{G}.

(2) and (3) are consequences of (1) and Lemma 1.4.

Remark 1.6. Let T ∈ Db(𝔤_{0})_{S_{+}}^{G_{0}}. Recall that by the Harish-Chandra regularity theorem, T is given by

⟨T, f⟩ = ∫_{𝔤′_{0}} F_{T} (y)f(y)dy

for some analytic function F_{T} on 𝔤′_{0}, locally integrable on 𝔤_{0}.

2. The distributions Θ_{u,Γ} and T_{p,Γ}

Let 𝔤_{0} be a real form of 𝔤, with adjoint group G_{0}, 𝔥_{0} a Cartan subalgebra and let H_{0} be the associated Cartan subgroup. Set 𝔥 = ℂ ⊗_{ℝ} 𝔥_{0} and adopt the notation of §1. Denote by W (𝔥_{0}) the real Weyl group, i.e. W (𝔥_{0}) = N_{G_{0}}(𝔥_{0})/Z_{G_{0}}(𝔥_{0}). Define

Δ_{R} = {α ∈ Δ : α(𝔥_{0}) ⊂ ℝ} (the real roots)
Δ_{I} = {α ∈ Δ : α(𝔥_{0}) ⊂ iℝ} (the imaginary roots).

A root which is neither real nor imaginary is called complex. Let Δ_{I}^{+} be a positive system of roots in Δ_{I} and set π_{I} = ∏_{α∈Δ_{I}^{+}} α. Then each w ∈ W (𝔥_{0}) permutes the imaginary roots and one can define a character of W (𝔥_{0}), the imaginary signature, by

ε_{I} : W (𝔥_{0}) → {±1}, w.π_{I} = ε_{I}(w)π_{I}.

If V is a W (𝔥_{0})-module we denote by V^{ε_{I}} the isotypic component of type ε_{I} in V.

In the sequel, we adopt the notation of [5] with the minor difference that we use e^{−iκ(x,y)} in the definition of the Fourier transform.

Let h ∈ 𝔥′_{0} and f ∈ C_{c}^{∞}(𝔤_{0}). Define [5, §3.1] the distribution μ_{G_{0}.h} by

⟨μ_{G_{0}.h}, f⟩ = | det ad_{𝔤_{0}/𝔥_{0}}(h)|^{\frac{1}{2}} ∫_{G_{0}/H_{0}} f(ġ.h)dġ

Then one defines the function J_{𝔤_{0}}(f), or simply J(f), on 𝔥′_{0} by

J_{𝔤_{0}}(f) = {h ↦ ⟨μ_{G_{0}.h}, f⟩}.

Set 𝔥_{0}^{reg} = {h ∈ 𝔥_{0} : π_{I}(h) ≠ 0} and fix a connected component Γ of 𝔥_{0}^{reg}. Let u ∈ S(𝔥); Harish-Chandra has shown, see [17, §8.1, p. 123], that one can define a tempered G_{0}-invariant distribution on 𝔤_{0} by

∀ f ∈ C_{c}^{∞}(𝔤_{0}), ⟨Θ_{u,Γ}, f⟩ = lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(u).J(f)](h). (2.1)

Furthermore Θ_{u,Γ} ∈ Db(𝔤_{0})_{nil}^{G_{0}} and, when u ∈ S^{b}(𝔥), Θ_{u,Γ} is homogeneous of degree −b − ν − ℓ.

Now let p ∈ S(𝔥^{∗}) and define T ∈ Db(𝔤_{0})_{S_{+}}^{G_{0}} by

T_{p,Γ} = Θ̂_{F_{𝔥}(p),Γ} = \{f ↦ lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(F_{𝔥}(p)).J(f̂)](h)\}. (2.2)

Then, T_{p,Γ} is tempered and is homogeneous of degree b − ν when p ∈ S^{b}(𝔥^{∗}).

Lemma 2.1. (1) Let φ ∈ S(𝔤^{∗})^{G}. Then, φT_{p,Γ} = T_{δ(φ)p,Γ}.

(2) Let q ∈ S(𝔤)^{G}. Then, ∂(q).T_{p,Γ} = T_{∂(δ(q)).p,Γ}.

Proof. Set u = F_{𝔥}(p), ϕ = δ(φ) ∈ S(𝔥^{∗})^{W} and s = δ(q) ∈ S(𝔥)^{W}. Let f ∈ C_{c}^{∞}(𝔤_{0}). (1) By definition, see (2.2), ⟨φT_{p,Γ}, f⟩ = lim\begin{matrix}h→0 \\ h∈Γ \end{matrix} [∂(u).J(φ̂f)](h). But, [17,

Lemma 3.2.7, p. 38], (1.4) and Lemma 1.2 imply that J(φ̂f) = ∂(F_{𝔥}(ϕ)).J(f̂). Hence,

⟨φT_{p,Γ}, f⟩ = lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(u)∂(F_{𝔥}(ϕ)).J(f̂)](h) = lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(F_{𝔥}(ϕp)).J(f̂)](h)
= ⟨T_{ϕp,Γ}, f⟩,

as desired.

(2) By (1.4), ∂(q).T_{p,Γ} is the Fourier transform of F_{𝔤}^{−1}(q)Θ_{u,Γ}, hence

⟨∂(q).T_{p,Γ}, f⟩ = lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(u).J(F_{𝔤}^{−1}(q)f̂)](h).

Set g = J(f̂). From [17, Lemma 3.2.7, p. 38] and Lemma 1.2 we obtain that J(F_{𝔤}^{−1}(q)f̂) = F_{𝔥}^{−1}(s)g. Therefore

⟨∂(q).T_{p,Γ}, f⟩ = lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(u).(F_{𝔥}^{−1}(s)g)](h).

Recall (see §1) that we have chosen a coordinate system {x_{j}; e_{j}}_{1⩽j⩽ℓ}. With standard notation, we write x^{α} = ∏_{k=1}^{ℓ} x_{k}^{α_{k}}, e^{μ} = ∏_{k=1}^{ℓ} e_{k}^{μ_{k}} and

p = ∑_{α∈ℕ^{ℓ}} p_{α}x^{α}, s = ∑_{μ∈ℕ^{ℓ}} s_{μ}e^{μ}.

Set ∂^{μ} = ∏_{j} ∂(e_{j})^{μ_{j}}; thus ∂(s) = ∑_{μ∈ℕ^{ℓ}} s_{μ}∂^{μ}. Order ℕ^{ℓ} by saying that μ ≤ α if μ_{j} ⩽ α_{j} for all j. Set α! = ∏_{j} α_{j}! and \binom{α}{μ} = ∏_{j} \binom{α_{j}}{μ_{j}}, when μ ≤ α. Then:

∂^{μ}(x^{α}) = \begin{cases}0 & if μ ≰ α, \\ \frac{α!}{(α−μ)!}x^{α−μ} & if μ ≤ α. \end{cases}

Now we have u = F_{𝔥}(p) = ∑_{α} p_{α}i^{|α|}∂^{α} and F_{𝔥}^{−1}(s) = ∑_{μ} q_{μ}i^{−|μ|}x^{μ}. Therefore, using the Leibniz formula, we get that

∂(u).(F_{𝔥}^{−1}(s)g) = ∑_{α} p_{α}i^{|α|}∂^{α}(F_{𝔥}^{−1}(s)g)
= ∑_{α} ∑_{μ} ∑_{β≤α} p_{α}s_{μ}i^{|α|−|μ|}\binom{α}{β}∂^{β}(x^{μ})∂^{α−β}(g).

But lim_{h→0} ∂^{β}(x^{μ})(h) = 0 unless β = μ, hence

lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(u).(F_{𝔥}^{−1}(s)g)](h) = ∑_{α} ∑_{μ≤α} p_{α}s_{μ}i^{|α|−|μ|}\binom{α}{μ}μ! lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂^{α−μ}(g)](h).

On the other hand, we have

⟨T_{∂(s).p,Γ}, f⟩ = lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂(F_{𝔥}(∂(s).p)).g](h).

Since ∂(s).p = ∑_{α} ∑_{μ≤α} \frac{α!}{(α−μ)!}s_{μ}p_{α}x^{α−μ}, we obtain that

⟨T_{∂(s).p,Γ}, f⟩ = ∑_{α} ∑_{μ≤α} \frac{α!}{(α − μ)!}s_{μ}p_{α}i^{|α|−|μ|} lim_{\begin{matrix}h→0 \\ h∈Γ \end{matrix}} [∂^{α−μ}(g)](h).

This proves the desired equality.

Theorem 2.2. Let p ∈ S(𝔥^{∗}) and D ∈ D(𝔤)^{G}. Then, D.T_{p,Γ} = T_{δ(D).p,Γ}.

Proof. Since T_{p,Γ} is G_{0}-invariant, we have I.T_{p,Γ} = 0. Let P ∈ ℂ⟨S(𝔤)^{G}, S(𝔤^{∗})^{G}⟩; by Lemma 2.1 and an obvious induction, we obtain that P.T_{p,Γ} = T_{δ(P).p,Γ}. The theorem then follows from Lemma 1.1.

Recall, see Remark 1.6, that Θ̂_{u,Γ} ∈ Db(𝔤_{0})_{S_{+}}^{G_{0}} is determined by a locally integrable function on 𝔤_{0}. We still denote this function by Θ̂_{u,Γ}.

Lemma 2.3. ([5, Lemme 6.1.2]) There exists c_{Γ} ∈ ℂ^{∗}, such that a_{Δ_{I}^{+}}(h)| det ad_{𝔤_{0}/𝔥_{0}}(h)|^{\frac{1}{2}} Θ̂_{F_{𝔥}(p),Γ}(h) = c_{Γ}p(h)

for all p ∈ S(𝔥^{∗})^{ε_{I}} and h ∈ 𝔥_{0}^{reg}.

Remark. In the notation of the lemma, if u = F_{𝔥}(p), the function ũ(ih) of [5] is replaced here by p(h) since we are using e^{−iκ(x,y)} in the definition of the Fourier transform.

Theorem 2.4. Let p ∈ S(𝔥^{∗})^{ε_{I}}. There exists a bijective map

ρ : D(𝔤)^{G}.T_{p,Γ} ⟶ D(𝔥)^{W} .p, ρ(D.T_{p,Γ}) = δ(D).p

which, through δ, yields an isomorphism

ρ : A(𝔤).T_{p,Γ} \underrightarrow{∼} D(𝔥)^{W} .p

Proof. We first need to show that ρ is well defined. Let D ∈ D(𝔤)^{G}; by Theorem 2.2 we have

D.T_{p,Γ} = T_{δ(D).p,Γ} = Θ̂_{F_{𝔥}(δ(D).p),Γ}. (†)

Suppose that D.T_{p,Γ} = 0. Then, the analytic function associated to T_{δ(D).p,Γ} ∈ Db(𝔤_{0})_{S_{+}}^{G_{0}} vanishes on 𝔥_{0}^{reg}. Notice that, since δ(D) is W -invariant, δ(D).p ∈ S(𝔥^{∗})^{ε_{I}}. Therefore Lemma 2.3 gives δ(D).p = 0 on 𝔥_{0}^{reg}. Thus δ(D).p = 0 on 𝔥 and ρ is well defined.

Now, it follows easily from (†) that ρ is a linear bijection. Since I.T_{p,Γ} = 0, the last assertion is clear.

Recall that we denote by V_{χ} ⊂ H^{b(χ)}(𝔥^{∗}) a simple W -module in the class of χ ∈ W .

Corollary 2.5. Let p ∈ S(𝔥^{∗})^{ε_{I}} such that ℂW.p is simple. Then there exists χ ∈ W such that V_{χ}^{ε_{I}} ≠ 0. We have:

1. D(𝔤).T_{p,Γ} \underrightarrow{∼} 𝒩_{χ} and D(𝔤)^{G}.T_{p,Γ} \underrightarrow{∼} V^{χ};

2. D(𝔤).Θ_{F_{𝔥}(p),Γ} \underrightarrow{∼} ℳ_{χ} and D(𝔤)^{G}.Θ_{F_{𝔥}(p),Γ} \underrightarrow{∼}(V^{χ})^{F_{𝔥}^{−1}}.

Proof. The first assertion follows from H(𝔥^{∗}) ≅ ℂW. Then, 1 and 2 are consequences of V^{χ} ≅ D(𝔥)^{W} .p, Lemma 1.3 and Theorem 2.4.

Remark 2.6. Let χ ∈ W be such that V_{χ}^{ε_{I}} ≠ 0. It follows obviously from the previous corollary that

𝒩_{χ} ≅ D(𝔤).T_{p,Γ}, ℳ_{χ} ≅ D(𝔤).Θ_{u,Γ}

where 0 ≠ p ∈ V_{χ}^{ε_{I}} ⊂ H^{b(χ)}(𝔥^{∗})^{ε_{I}} and u = F_{𝔥}(p) ∈ H^{b(χ)}(𝔥)^{ε_{I}}.

3. The decomposition of Db(𝔤_{0})_{S_{+}}^{G_{0}} and Db(𝔤_{0})_{nil}^{G_{0}}

Fix a real form 𝔤_{0} of 𝔤 and let [𝔥_{1}], …, [𝔥_{r}] be the conjugacy classes of Cartan subalgebras in 𝔤_{0}. For each j = 1, …, r we denote by

𝔥_{j,ℂ} = 𝔥_{j} ⊗_{ℝ} ℂ, W_{j} = W (𝔤, 𝔥_{j,ℂ}), Δ_{I,j}^{+} a set of positive imaginary roots,

ε_{I,j} : W (𝔥_{j}) = W (G_{0}, 𝔥_{j}) → {±1} the imaginary signature associated to 𝔥_{j}.

For each j we fix a connected component Γ_{j} of 𝔥_{j}^{reg}. The results of §2 then apply to 𝔥_{0} = 𝔥_{j}, Γ = Γ_{j} etc.

Remark 3.1. Recall that the 𝔥_{j,ℂ} are G-conjugate. Therefore, if 1 ⩽ j, k ⩽ r, the algebras D(𝔥_{j,ℂ})^{W_{j}} and D(𝔥_{k,ℂ})^{W_{k}} are naturally isomorphic. Denote this isomorphism by γ_{jk} and let δ_{j} be the Harish-Chandra isomorphism from A(𝔤) onto D(𝔥_{j,ℂ})^{W_{j}}. One can check that δ_{k} = γ_{jk} ∘ δ_{j}. Therefore, we can choose an “abstract” Cartan subalgebra 𝔥 and identify δ_{j} with the homomorphism δ : D(𝔤)^{G} → D(𝔥)^{W}, where W = W (G, 𝔥). Then, if χ ∈ W , we have an irreducible W -module V_{χ} ⊂ H^{b(χ)}(𝔥^{∗}) and a simple D(𝔥)^{W} -module V^{χ}.

For each χ ∈ W , choose a simple W -module V_{χ,j} ⊂ H^{b(χ)}(𝔥_{j,ℂ}^{∗}), V_{χ,j} ≅ V_{χ}. Write V_{χ,j} = V_{χ,j}^{ε_{I,j}} ⊕ E_{χ,j} with E_{χ,j} stable under W (𝔥_{j}). Let {v_{χ,j}^{k}}_{1⩽k⩽d(χ)} be a basis of V_{χ,j} such that

V_{χ,j}^{ε_{I,j}} = ⨁_{k=1}^{n_{j}(χ)} ℂv_{χ,j}^{k}, E_{χ,j} = ⨁_{k=n_{j}(χ)+1}^{d(χ)} ℂv_{χ,j}^{k}

(hence n_{j}(χ) = dim V_{χ}^{ε_{I,j}}).

Lemma 3.2. The D(𝔥_{j,ℂ})^{W_{j}}-module S(𝔥_{j,ℂ}^{∗})^{ε_{I,j}} decomposes as

S(𝔥_{j,ℂ}^{∗})^{ε_{I,j}} = _{⨁χ∈W}ˆ ⨁_{k=1}^{n_{j}(χ)} D(𝔥_{j,ℂ})^{W_{j}}.v_{χ,j}^{k}

with D(𝔥_{j,ℂ})^{W_{j}}.v_{χ,j}^{k} ≅ V^{χ}.

Proof. Clearly, we can drop the index j and write 𝔥_{0} = 𝔥_{j}, 𝔥 = 𝔥_{j,ℂ}, v_{χ}^{k} = v_{χ,j}^{k} etc. Since D(𝔥)^{W} .v_{χ}^{k} ⊂ S(𝔥^{∗})^{ε_{I}} for 1 ⩽ k ⩽ n(χ) = dim V_{χ}^{ε_{I}}, one has

S(𝔥^{∗})^{ε_{I}} ⊃ ⨁_{χ∈W}ˆ ⨁_{k=1}^{n(χ)} D(𝔥)^{W} .v_{χ}^{k}.

Recall from §1 that S(𝔥^{∗}) = ⊕_{χ}S(𝔥^{∗})[χ] with S(𝔥^{∗})[χ] = ⊕_{k=1}^{d(χ)}D(𝔥)^{W} .v_{χ}^{k}. Write S(𝔥^{∗})[χ] = E_{1} ⊕ E_{2}, where E_{1} = ⊕_{k=1}^{n(χ)}D(𝔥)^{W} .v_{χ}^{k} and E_{2} = ⊕_{k=n(χ)+1}^{d(χ)}D(𝔥)^{W} .v_{χ}^{k}. Notice that E_{1}, E_{2} are stable under W (𝔥_{0}) and that we have S(𝔥^{∗})[χ]^{ε_{I}} = E_{1}⊕E_{2}^{ε_{I}}.

We now show that E_{2}^{ε_{I}} = 0. This will prove that

S(𝔥^{∗})^{ε_{I}} = ⨁_{χ∈W}ˆ ⨁_{k=1}^{n(χ)} D(𝔥)^{W} .v_{χ}^{k}.

Let D ∈ D(𝔥)^{W} and v ∈ V_{χ}. Notice first that if D.v ≠ 0, the operator D yields an isomorphism of W -modules V_{χ} \underrightarrow{∼} D.V_{χ}. Therefore, if V_{χ} = ⊕_{k}S_{k} with S_{k} irreducible W (𝔥_{0})-module, we get that D.V_{χ} = ⊕_{k}D.S_{k}, D.S_{k} ≅ S_{k}. It follows that if v ∈ E_{χ} (the W (𝔥_{0})-stable complement of V_{χ}^{ε_{I}}) then D.v ∈ D.E_{χ} with D.E_{χ} ∩ S(𝔥^{∗})^{ε_{I}} = 0. Let p = ∑_{k=n(χ)+1}^{d(χ)} D_{k}.v_{χ}^{k} ∈ E_{2}. Then, ℂW (𝔥_{0}).p ⊂ ∑_{k>n(χ)} ℂW (𝔥_{0}).(D_{k}.v_{χ}^{k}) and, by the previous remarks, (ℂW (𝔥_{0}).(D_{k}.v_{χ}^{k}))^{ε_{I}} = 0. Thus (ℂW (𝔥_{0}).p)^{ε_{I}} = 0, which shows that E_{2}^{ε_{I}} = 0.

Recall the following result:

Proposition 3.3. ([5, Proposition 6.1.1]) (1) The linear map

T : ⨁_{j=1}^{r} S(𝔥_{j,ℂ}^{∗})^{ε_{I,j}} ⟶ Db(𝔤_{0})_{S_{+}}^{G_{0}}, T(p_{1}, …, p_{r}) = ∑_{j=1}^{r} T_{p_{j},Γ_{j}}

is an isomorphism of vector spaces.

(2) The map T induces an isomorphism:

⨁_{j=1}^{r} H(𝔥_{j,ℂ}^{∗})^{ε_{I,j}} \underrightarrow{∼} \{T ∈ Db(𝔤_{0})_{S_{+}}^{G_{0}} : S_{+}(𝔤)^{G}.T = 0\}.

Proof. (2) follows from the proof of [5, Proposition 6.1.1].

Theorem 3.4. Set T(𝔥_{j}) = ∑_{p∈S(𝔥_{j,ℂ}^{∗})^{ε_{I,j}}} ℂT_{p,Γ_{j}}. Then we have the following

decomposition of D(𝔤)^{G}-modules:

Db(𝔤_{0})_{S_{+}}^{G_{0}} = ⨁_{j=1}^{r} T(𝔥_{j})

with

T(𝔥_{j}) = ⨁_{χ∈W}ˆ ⨁_{k=1}^{n_{j}(χ)} D(𝔤)^{G}.T_{v_{χ,j}^{k},Γ_{j}}

and D(𝔤)^{G}.T_{v_{χ,j}^{k},Γ_{j}} ≅ 𝒩_{χ}^{G}.

Proof. The decomposition of T(𝔥_{j}), as a D(𝔤)^{G}-module, is consequence of Theorem 2.4, Lemma 3.2 (using the isomorphism δ_{j} : A(𝔤) \underrightarrow{∼} D(𝔥_{j,ℂ})^{W_{j}}) and Proposition 3.3. The decomposition of Db(𝔤_{0})_{S_{+}}^{G_{0}} follows from Proposition 3.3.

Using the Fourier transform, we obtain the following:

Corollary 3.5. The D(𝔤)^{G}-module Db(𝔤_{0})_{nil}^{G_{0}} decomposes as

Db(𝔤_{0})_{nil}^{G_{0}} = ⨁_{j=1}^{r} ⨁_{χ∈W}ˆ ⨁_{k=1}^{n_{j}(χ)} D(𝔤)^{G}.Θ_{F_{𝔥}^{−1}(v_{χ,j}^{k}),Γ_{j}}

with D(𝔤)^{G}.Θ_{F_{𝔥}^{−1}(v_{χ,j}^{k}),Γ_{j}} ≅ ℳ_{χ}^{G}.

The next corollary follows from Theorem 3.4 and Corollary 3.5.

Corollary 3.6. We have:

Db(𝔤_{0})_{S_{+}}^{G_{0}} ≅ ⨁_{χ∈W}ˆ m_{χ}𝒩_{χ}^{G}, Db(𝔤_{0})_{nil}^{G_{0}} ≅ ⨁_{χ∈W}ˆ m_{χ}ℳ_{χ}^{G}

where m_{χ} = ∑_{j=1}^{r} dim V_{χ}^{ε_{I,j}}.

Remark 3.7. Let χ ∈ W . It is not always possible to “realize” the modules 𝒩_{χ} and ℳ_{χ} as D(𝔤).T for some T ∈ Db(𝔤_{0}), where 𝔤_{0} is a real form of 𝔤. By the previous results, this statement is equivalent to the existence of a Cartan subalgebra 𝔥_{j} ⊂ 𝔤_{0} such that V_{χ}^{ε_{I,j}} ≠ 0. D. Renard has observed that, using the results of W. Rossmann [15], this can be translated to a question about centralizers of nilpotent elements. Fix a real form 𝔤_{ℝ} of 𝔤 with adjoint group G_{ℝ}. If x ∈ 𝔤_{ℝ} is nilpotent one defines a subgroup of the component group A(G.x) (see §4 for notation) by

A(G_{ℝ}.x) = G_{ℝ}^{x}/G_{ℝ}^{x} ∩ (G^{x})^{0}.

Recall that χ ∈ W can be written σ(O, ψ) via the Springer correspondence, where O ⊂ 𝔤 is a nilpotent orbit and ψ : A(O) → GL(E) is an irreducible representation. Then, by [15, Corollary 3.2 & Theorem 3.3], there exists a Cartan subalgebra 𝔥_{0} ⊂ 𝔤_{ℝ} such that V_{χ}^{ε_{I}} ≠ 0 if, and only if, there exists a nilpotent element x ∈ 𝔤_{ℝ} such that O = G.x and E^{A(G_{ℝ}.x)} ≠ 0.

Let 𝔤 = 𝔰𝔭(ℓ, ℂ) and let ϕ ∈ W be the long sign character, i.e. V_{ϕ} = ℂπ_{l} where π_{l} is the product of the long roots. Then, see [6, §13.3], ϕ = σ(O, ψ) where O = G.x is the subregular nilpotent orbit with partition [2ℓ − 2, 2] and ψ is the non trivial character of A(O) ≅ {±1}. The real forms of 𝔤 are 𝔰𝔭(ℓ, ℝ) and the 𝔰𝔭(p, q), p+q = ℓ. Assume now that ℓ ⩾ 3. By the classification of nilpotent orbits in 𝔰𝔭(p, q), see [7, Theorem 9.2.5], we know that O∩𝔰𝔭(p, q) = ∅. Hence, by Rossmann’s results, V_{ϕ}^{ε_{I,j}} = 0 for each Cartan subalgebra 𝔥_{j} ⊂ 𝔰𝔭(p, q). On the other hand, if G_{ℝ} is the adjoint group of 𝔰𝔭(ℓ, ℝ), one can show that A(G_{ℝ}.x) = A(G.x). Thus, with the above notation, E^{A(G_{ℝ}.x)} = 0 and it follows that V_{ϕ}^{ε_{I,j}} = 0 for each Cartan subalgebra 𝔥_{j} ⊂ 𝔰𝔭(ℓ, ℝ). For instance, when 𝔤 = 𝔰𝔭(3, ℝ) there are six conjugacy classes of Cartan subalgebras and one can directly verify (without using [15]) that V_{ϕ}^{ε_{I,j}} = 0 for j = 1, …, 6. We thank D. Renard for showing this computation to us.

Let x ∈ N(𝔤_{0}) and denote by β_{x} the Liouville (Kostant-Kirillov) measure on G_{0}.x. By [14] one can define Θ_{x} ∈ Db(𝔤_{0})_{nil}^{G_{0}} by ⟨Θ_{x}, f⟩ = ∫_{G_{0}.x} fdβ_{x} for all f ∈ C_{c}^{∞}(𝔤_{0}). Set O = G.x. Then, see [9], [10] or [18], Θ_{x} is homogeneous of degree λ_{O} = \frac{1}{2} dim O − dim 𝔤 and satisfies

D(𝔤).Θ_{x} ≅ ℳ_{χ_{O}} (3.1)

for some χ_{O} ∈ W such that λ_{O} = ν − n − b(χ_{O}).

Corollary 3.8. There exists j ∈ {1, …, r} and u ∈ F_{𝔥}^{−1}(V_{χ_{O},j})^{ε_{I,j}} such that

D(𝔤)^{G}.Θ_{x} ≅ D(𝔤)^{G}.Θ_{u,Γ_{j}}.

Proof. Since D(𝔤)^{G}.Θ_{x} ≅ ℳ_{χ_{O}}^{G} is a simple submodule of Db(𝔤_{0})_{nil}^{G_{0}}, the claim follows from Corollary 3.5.

Remark 3.9. It is proved in [1], see also [5], that Θ_{x} can be written as ∑_{j=1}^{r} Θ_{a_{j},Γ_{j}} with a_{j} ∈ H^{b(χ_{O})}(𝔥_{j,ℂ})^{ε_{I,j}}. It is easily seen that we may assume ℂW.a_{j} ≅ V_{χ_{O}} for all j such that a_{j} ≠ 0. W. Rossmann [15] has given conditions to ensure that Θ_{x} = Θ_{a_{j},Γ_{j}} for some j.

4. Example: the complex case

We assume in this section that 𝔤_{0} = 𝔤_{1}^{ℝ} is a complex semisimple Lie algebra, 𝔤_{1}, viewed as a real Lie algebra. Then, 𝔤 can be identified with 𝔤_{1} × 𝔤_{1} and 𝔤_{0} with the diagonal {(a, a) ∈ 𝔤_{1} × 𝔤_{1}}. Let 𝔥_{1} be a Cartan subalgebra of 𝔤_{1}. Recall the following well known facts, see [17] or [18]:

▪ 𝔥_{0} = {(a, a) : a ∈ 𝔥_{1}} is a Cartan subalgebra of 𝔥_{0} and 𝔥 = 𝔥_{0} ⊗_{ℝ} ℂ = 𝔥_{1} × 𝔥_{1}; ▪ W (𝔤, 𝔥) = W_{1} × W_{1}, where W_{1} = W (𝔤_{1}, 𝔥_{1}), and W (𝔥_{0}) = {(w, w) ∈ W } is isomorphic to W_{1}; ▪ there is a unique conjugacy class [𝔥_{0}] of Cartan subalgebras and 𝔥′_{0} is connected; ▪ the roots in Δ(𝔤, 𝔥) are complex and, therefore, ε_{I} = 1; ▪ the irreducible representations of W are of the form χ = ϕ ⊠ μ, ϕ, μ ∈ W_{1}; ▪ one has ϕ = ϕ^{∗} for all ϕ ∈ W_{1}, where ϕ^{∗} is the dual representation.

Observe that D(𝔤) = D(𝔤_{1}) ⊠ D(𝔤_{1}) and D(𝔤)^{G} = D(𝔤_{1})^{G_{1}} ⊠ D(𝔤_{1})^{G_{1}}.

Lemma 4.1. Let χ ∈ W . Then, the simple D(𝔤)-module ℳ_{χ} is of the form

ℳ_{ϕ} ⊠ ℳ_{μ} for some ϕ, μ ∈ W_{1}.

Proof. The claim follows easily from the definition of the category C(ℳ) and the decomposition of the W -module S(𝔥^{∗}) = S(𝔥_{1}^{∗}) ⊠ S(𝔥_{1}^{∗}).

Corollary 4.2. ([18, Theorem 6.11]) We have

Db(𝔤_{0})_{nil}^{G_{0}} ≅ ⨁_{ϕ∈W_{ˆ1}}ℳ_{ϕ}^{G_{1}} ⊠ ℳ_{ϕ}^{G_{1}}

as a D(𝔤)^{G}-module.

Proof. Let χ = ϕ ⊠ μ ∈ W . Then, V_{χ}^{ε_{I}} = (V_{ϕ} ⊠ V_{μ})^{W_{1}} ≠ 0 if, and only if, ϕ = μ and therefore n(χ) = 1. The assertion now follows from Corollary 3.5.

Recall the following general results from [13]. Since the module ℳ_{χ} is irreducible and G-equivariant, its support is the closure of a nilpotent orbit O = G.x. Furthermore, if ı : O ↪ 𝔤 is the inclusion, ℳ_{χ} is uniquely determined by its (Dmodule) inverse image ℒ_{χ} := ı^{!}ℳ_{χ}. The D_{O}-module ℒ_{χ} is an irreducible integrable connection associated to an irreducible representation ψ of the component group A(O) := G^{x}/(G^{x})^{0} (where (G^{x})^{0} is the connected component of the centralizer G^{x}). Therefore, since χ is uniquely determined by O and ψ, we set χ = σ(O, ψ).

In our situation, i.e. in the complex case, we have O = O_{1}^{1} × O_{1}^{2} with O_{1}^{j} nilpotent orbits in 𝔤_{1} for j = 1, 2. Then, χ = σ(O, ψ) = ϕ_{1} ⊠ ϕ_{2}, ℒ_{χ} = ℒ_{ϕ_{1}} ⊠ ℒ_{ϕ_{2}}, ϕ_{j} = σ(O_{1}^{j}, ψ_{j}), ψ = ψ_{1} ⊠ ψ_{2}. Note that b(χ) = b(ϕ_{1}) + b(ϕ_{2}) and λ_{O} = λ_{O_{1}^{1}} + λ_{O_{1}^{2}}.

Let x ∈ N(𝔤_{0}); set x = (x_{1}, x_{1}), x_{1} ∈ N(𝔤_{1}), O_{1} = G_{1}.x_{1}, O = G.x = O_{1} × O_{1}. The inclusion ı : O ↪ 𝔤 is equal to ı_{1} × ı_{1}, where ı_{1} : O_{1} ↪ 𝔤_{1}. By (3.1) and Corollary 4.2 there exist χ ∈ W , χ_{1} ∈ W_{1} such that χ = χ_{1} ⊠ χ_{1} and D(𝔤).Θ_{x} ≅ ℳ_{χ_{1}} ⊠ ℳ_{χ_{1}}.

It is known (Harish-Chandra) that Θ_{x} = Θ_{u,𝔥′_{0}} for some u ∈ S(𝔥_{1}) ⊠ S(𝔥_{1}). The following result has been proved by various authors; see [2, 3] (when O_{1} is “special”), [8], [9], [16].

Theorem 4.3. One has χ_{1} = σ(O_{1}, triv), and there exists p ∈ (V_{χ_{1}} ⊠ V_{χ_{1}})^{W_{1}} such that Θ_{x} = Θ_{F_{𝔥}(p),𝔥′_{0}}.

Proof. Recall from [9] or [10] that χ = χ_{1} ⊠ χ_{1} = σ(O, triv). This means that

ℒ_{χ} = ℒ_{χ_{1}} ⊠ ℒ_{χ_{1}} = O_{O} = O_{O_{1}} ⊠ O_{O_{1}}

(where we denote by O_{X} the structural sheaf of an algebraic variety X). This yields ℒ_{χ_{1}} = O_{O_{1}} and χ_{1} = σ(O_{1}, triv).

Set T_{x} = Θ̂_{x}; then D(𝔤).T_{x} = 𝒩_{χ_{1}} ⊠𝒩_{χ_{1}} (see Lemma 1.4). Since S_{+}(𝔤^{∗})^{G}.Θ_{x} = 0 we have S_{+}(𝔤)^{G}.T_{x} = 0. It follows from Proposition 3.3(2) that we can write T_{x} = T_{p,𝔥′_{0}} for some p ∈ (H(𝔥_{1}^{∗}) ⊠ H(𝔥_{1}^{∗}))^{W_{1}} or, equivalently, Θ_{x} = Θ_{F_{𝔥}(p),𝔥′_{0}}. Now, by Theorem 2.4, D(𝔥)^{W} .p = V^{χ_{1}} ⊠V^{χ_{1}} and therefore ℂW.p ≅ V_{χ_{1}} ⊠V_{χ_{1}}. Moreover, T_{x} = T_{p,𝔥′_{0}} is homogeneous of degree b(χ_{O}) − 2ν = 2b(χ_{1}) − 2ν = deg p − 2ν. Thus deg p = 2b(χ_{1}) and, by definition of V_{χ_{1}}, p ∈ (V_{χ_{1}} ⊠ V_{χ_{1}})^{W_{1}}.

Acknowledgment

This work was partially completed while the author was visiting the Weizmann Institute of Science. He thanks this institution, and A. Joseph, for their hospitality.

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Département de Mathématiques, Université de Brest, 29285 Brest, France.

E-mail address: Thierry.Levasseur@univ-brest.fr