SEMI-SIMPLICITY OF INVARIANT HOLONOMIC SYSTEMS ON A REDUCTIVE LIE ALGEBRA

T. LEVASSEUR AND J. T. STAFFORD

Abstract. Let 𝔤 be a reductive, complex Lie algebra, with adjoint group G, let G act on the ring of differential operators D(𝔤) via the adjoint action and write τ : 𝔤 → D(𝔤) for the differential of this action. Fix λ ∈ 𝔥^{∗}. Generalizing work of Hotta and Kashiwara, we prove that the invariant holonomic system 𝒩_{λ} = D(𝔤)/(D(𝔤)τ(𝔤) +_{∑p∈S(𝔤)_{G}}D(𝔤)(p − p(λ))) is semisimple. The simple summands of 𝒩_{λ} are parametrized by the irreducible representations of W_{λ}, the stabilizer of λ in the Weyl group. Consequently, the subcategory generated by 𝒩_{λ} is equivalent to the category of finite dimensional representations of W_{λ}.

1. Introduction

Let G be a complex connected reductive algebraic group with 𝔤 = Lie(G). Let O(𝔤) = S(𝔤^{∗}) be the algebra of polynomial functions on 𝔤, and denote by D(𝔤) the algebra of differential operators on 𝔤, with coefficients in O(𝔤). We identify S(𝔤) ⊂ D(𝔤) with the algebra of differential operators with constant coefficients. 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). Fix a Cartan subalgebra 𝔥 of 𝔤 with associated Weyl group W.

Suppose that 𝔤_{0} is a real form of 𝔤, with adjoint group G_{0}, and recall that a distribution Θ ∈ Dist(𝔤_{0}) is a G_{0}-invariant eigendistribution if there exists λ ∈ 𝔥^{∗} such that τ(𝔤) · Θ = 0, and p · Θ = p(λ)Θ for all p ∈ S(𝔤)^{G}. As in [10], one defines a system of linear differential equations for invariant eigendistributions or invariant holonomic system on 𝔤 by

𝒩_{λ} = D(𝔤)/(D(𝔤)τ(𝔤) +_{∑p∈S(𝔤)_{G}}D(𝔤)(p − p(λ))) for λ ∈ 𝔥^{∗}.

This module is important, in part, because the invariant eigendistributions can be identified with Hom_{D(𝔤)}(𝒩_{λ}, Dist(𝔤_{0})). Its study is also of central importance in the representation theory of 𝔤 and Hotta and Kashiwara have obtained deep results on the system 𝒩_{λ}, and its Fourier transform, by means of the Riemann-Hilbert correspondence [10]. They showed, among other things, that the module 𝒩_{0} decomposes as

𝒩_{0} = ⨁_{χ∈W}ˆ 𝒩_{0,χ} ⊗_{ℂ} V_{χ}^{∗} (1.1)

1991 Mathematics Subject Classification. 22E47, 14L30, 16S32, 17B20.

Key words and phrases. Invariant differential operators, Weyl group representations, holonomic systems, invariant eigendistributions, semisimple Lie algebras.

The research of the second author was supported in part by an NSF grant.

(see [10, Theorem 5.3]). Here, W denotes the set of isomorphism classes of irreducible W -modules and V_{χ} is a representation in the class of χ. The modules 𝒩_{0,χ} are pairwise non-isomorphic, simple D(𝔤)-modules.

On the other hand, Wallach [19] has shown that the decomposition of the D(𝔥)^{W} - module S(𝔥^{∗}) is closely related to the study of invariant eigendistributions on 𝔤_{0}, and to the Springer correspondence. This connection is made possible by the existence of a homomorphism δ : D(𝔤)^{G} → D(𝔥)^{W} defined by Harish-Chandra [7]. The homomorphism δ is surjective with kernel (D(𝔤)τ(𝔤))^{G}, see [13, 14].

The aim of this paper is to prove the following theorems, which generalize (and provide different proofs of) the aforementioned results from [10] and [19]. The first theorem answers a question of M. Duflo, who asked for the decomposition of the D(𝔥)^{W} -module S(𝔥^{∗})e^{λ}.

Theorem A. Fix λ ∈ 𝔥^{∗} and let W_{λ} denote the stabilizer of λ in W.

(i) S(𝔥^{∗})e^{λ} is a semisimple (D(𝔥)^{W} , W_{λ})-module. Indeed:

S(𝔥^{∗})e^{λ} = ⨁_{χ∈W_{λ̂}} V^{χ} ⊗ V_{χ}^{∗},

where the V^{χ} are simple, pairwise non-isomorphic D(𝔥)^{W} -modules and V_{χ}^{∗} is the dual of the simple W -module V_{χ} in the class of χ.

(ii) In particular, if χ = triv is the trivial character, then V^{triv} = D(𝔥)^{W} e^{λ} = S(𝔥^{∗})^{W_{λ}}e^{λ} is a simple D(𝔥)^{W} -module.

(iii) Set p_{λ} = ∑_{p∈S(𝔥)^{W}} S(𝔥)^{W} (p − p(λ)). Then, S(𝔥^{∗})e^{λ} ≅ D(𝔥)^{W} /D(𝔥)^{W} p_{λ} as D(𝔥)^{W} -modules.

This theorem follows from Theorem 3.4 and Theorem 4.4, which give analogous results in more general circumstances. The main result of the paper, see Theorem 6.9, is the following:

Theorem B. Fix λ ∈ 𝔥^{∗}. As either a (D(𝔤), W_{λ})-module or a D(𝔤)-module, 𝒩_{λ} is semisimple. Indeed:

𝒩_{λ} ≅ ⨁_{χ∈W_{λ̂}} 𝒩_{λ,χ} ⊗_{ℂ} V_{χ}^{∗}, where 𝒩_{λ,χ} = (D(𝔤)/D(𝔤)τ(𝔤)) ⊗_{D(𝔤)^{G}} V^{χ}. (1.2)

Moreover, the 𝒩_{λ,χ}, for χ ∈ W_{λ}, are simple, pairwise nonisomorphic D(𝔤)-modules.

When λ = 0 this reduces to (1.1). Hotta and Kashiwara also prove the result in the case when λ is regular [10, Lemma 4.6.1]. The analog of Theorem B for differential operators over G has been raised as a conjecture in [9, Hypothesis 4.1].

The results and proofs in [10] rely on the fact that the modules 𝒩_{λ}, and their Fourier transforms, can be obtained via D-module constructions from O𝔤̃, where 𝔤̃ → 𝔤 is the Grothendieck-Springer resolution. Our approach to Theorem B is less geometric, but in a sense more elementary. The idea is as follows: Since 𝒩_{λ} ≅ (D(𝔤)/D(𝔤)τ(𝔤))⊗_{D(𝔥)^{W}} S(𝔥^{∗})e^{λ}, the decomposition (1.2) follows from Theorem A. Thus, the hard part of the proof is to show that the 𝒩_{λ,χ} are simple. To prove this, one first shows that 𝒩_{λ,χ} is torsion-free as an O(𝔤)-module (see Corollary 6.6). It follows that 𝒩_{λ,χ} is simple if and only if its restriction to the set of generic elements is also simple. As this second module is an integrable connection, it is easy to analyse and this enables one to prove the theorem.

One consequence of this argument is that:

every subfactor of 𝒩_{λ} is torsion-free as an O(𝔤)-module.

This generalizes work of Harish-Chandra in the sense that the essence of his result on the regularity of invariant eigendistributions [7, 8] is the assertion that 𝒩_{λ} has no factor that is {d^{n}}-torsion, where d ∈ O(𝔤) denotes the discriminant (see the discussion in [10, p.352]).

Let R_{λ} denote the full subcategory of finitely generated D(𝔤)-modules generated by 𝒩_{λ}. Then Theorem B has the following easy consequence:

Corollary C. The category R_{λ} is equivalent to W_{λ}-mod, through the functor

M → Hom_{D(𝔥)^{W}} (M^{G}, S(𝔥^{∗})e^{λ}).

As a consequence of these results, one obtains a “Springer correspondence” between representations of W_{λ}, local systems on G-orbits in N(λ) = {x ∈ 𝔤 : f(x) = f(λ) for all f ∈ S(𝔤^{∗})^{G}} and W_{λ}-harmonics in S(𝔥^{∗}). The details can be found in Sections 6 and 7.

2. The structure of the D(V) ∗ W -module generated by e^{λ}.

For the next four sections, we fix an algebraically closed field 𝕜 of characteristic zero and a 𝕜-vector space V of dimension ℓ. Set: R = O(V) = S(V^{∗}), the algebra of functions on V; R̂ the algebra of formal power series on V; S = S(V), the symmetric algebra on V, which will be identified with the algebra of constant coefficients differential operators on V; A = D(V), the algebra of differential operators on V. The action of d ∈ A on f ∈ R̂ will be denoted by d · f ∈ R̂.

Fix a finite subgroup W of GL(V). Then W and its subgroups have natural induced actions on the algebras R, R̂, S and A. Given λ ∈ V^{∗}, we regard λ as a character λ : S(V) → 𝕜 and set m_{λ} = Ker λ and n_{λ} = m_{λ} ∩ S(V)^{W_{λ}}, where W_{λ} is the stabilizer of λ. An element g ∈ W acts on λ by (g.λ)(v) = λ(g^{−1}.v), v ∈ V. The element e^{λ} can be viewed as an element of R̂ and the A-submodule it generates is the simple A-module A · e^{λ} = Re^{λ} ≅ A/Am_{λ}.

The aim of the first three sections is to describe the structure of Re^{λ} as a module over A^{W}, the subring of W-invariant elements in A. The description is given in Theorem 3.4. When W is generated by pseudo-reflections a further description is given in Theorem 4.4.

The way we prove these results is to use the Morita equivalence (see §3) between A^{W} and the skew group ring A ∗ W. This ring is defined as follows: If d ∈ A and w ∈ W the (left) action of w on d is denoted by w.d and we set d^{w} = w^{−1}.d. The skew group ring A ∗ W is the free (left and right) A-module with basis {w}_{w∈W} and multiplication defined by dw = wd^{w} for d ∈ A and w ∈ W. We always identify A with the subring Aϵ of A ∗ W, where ϵ is the identity element of W.

Notice that R̂ is in a natural way a left A ∗ W -module. Moreover, if g ∈ W, then g.e^{λ} = e^{g.λ} and so Re^{λ} is a A ∗ W_{λ}-submodule of R̂. To analyze this module we need to introduce the categories of modules on which m_{λ} or n_{λ} acts locally nilpotently. Let B-mod denote the category of finitely generated left B-modules over a ring B and set

S_{λ} = {M ∈ A ∗ W_{λ}-mod | ∀ x ∈ M, ∃ k ∈ ℕ, such that m_{λ}^{k}x = 0}
C_{λ} = {M ∈ A^{W_{λ}}-mod | ∀ x ∈ M, ∃ k ∈ ℕ, such that n_{λ}^{k}x = 0}.

Observe that Re^{λ} ∈ S_{λ} as an A ∗ W_{λ}-module, while, as an A^{W_{λ}}-module, Re^{λ} ∈ C_{λ}. The next two lemmas give some elementary properties of Re^{λ} and (A ∗ W_{λ}) ⊗_{A} Re^{λ}. The first is standard.

Lemma 2.1. (i) Let N be a finitely generated left A-module that is locally finite as a left S(V)-module. Then N is a finite direct sum of simple A-modules of the form Re^{μ}, for μ ∈ V^{∗}.

(ii) If Γ is a finite group of automorphisms of A and N is a semi-simple left A-module, then (A ∗ Γ) ⊗_{A} N is a semi-simple left A ∗ Γ-module.

Proof. (i) The hypothesis ensures that N is a D(V^{∗})-module supported on a finite number of points in V^{∗}. The claim therefore follows from Kashiwara’s equivalence [2, Theorem 7.11, p.264].

(ii) This is [16, Theorem 7.6(iv)].

Now we introduce a functor from the category of left A-modules, to the category of 𝕜-vector spaces by setting

Φ(M) = {x ∈ M | m_{λ}x = 0}.

Note that Φ(Re^{λ}) = 𝕜e^{λ}. Furthermore, Φ(M) identifies with Hom_{A}(Re^{λ}, M) under the map f ↦ f(e^{λ}), for f ∈ Hom_{A}(Re^{λ}, M). Moreover, if M is an A ∗ W_{λ}-module, then Φ(M) is a W_{λ}-module. Denote by W_{λ}-mod the category of finite dimensional representations of W_{λ}.

Proposition 2.2. Define 𝕜_{λ} = S/m_{λ}. Then Φ induces an equivalence of categories

S_{λ} ≈ W_{λ}-mod

whose inverse functor is Ψ : E → Re^{λ} ⊗_{𝕜_{λ}} E. Here, E is considered as a trivial left 𝕜_{λ}-module and W_{λ} acts on Re^{λ} ⊗ E via its diagonal action: w.(d ⊗ x) = w.d ⊗ w.x.

Proof. Clearly, E ≅ ΦΨ(E), for any W_{λ}-module E. Let M be in S_{λ}. By Lemma 2.1(i), M ≅ (Re^{λ})^{(m)} as a left A-module and so, if E = Φ(M), then M = AE = RE. Thus, there is a surjection of A-modules ψ : Ψ(E) ↠ M given by re^{λ} ⊗ ε ↦ rε for r ∈ R and ε ∈ E. Using the fact that w.e^{λ} = e^{λ}, for w ∈ W_{λ}, it is easily checked that ψ is actually an A ∗ W_{λ}-module map. Since Ψ(E) is a semisimple right A-module, Φ(Ker ψ) = 0 and so ψ is an isomorphism. The rest of the proof is an easy verification.

Let W_{λ} denote the set of isomorphism classes of irreducible representations of W_{λ}. For χ ∈ W_{λ}, fix a module V_{χ} in the class of χ and set m_{χ} = dim V_{χ}.

Lemma 2.3. Let χ ∈ W_{λ}.

(i) As an A-module, (A∗W_{λ})⊗_{A} Re^{λ} is a direct sum of |W_{λ}| copies of the simple module Re^{λ}. Moreover, Re^{λ} ≅ R as W_{λ}-modules.

(ii) Fix a left transversal {h_{1} = ϵ, h_{2}, …, h_{s}} of W_{λ} in W. Then Re^{h_{i}.λ} ≅ Re^{h_{j}.λ} as A-modules if and only if h_{i} = h_{j}.

Proof. (i) As left A-modules, (A ∗ W_{λ}) ⊗_{A} Re^{λ} = ⊕_{g∈W_{λ}}A(g ⊗ e^{λ}). If v ∈ V ⊂ S = S(V) and g ∈ W_{λ}, then λ(v^{g}) = (g.λ)(v) = λ(v). Thus,

v(g ⊗ e^{λ}) = g ⊗ v^{g} · e^{λ} = g ⊗ λ(v^{g})e^{λ} = g ⊗ v · e^{λ}.

Therefore, A(g ⊗ e^{λ}) ≅ Ae^{λ} = Re^{λ}. The final assertion follows immediately from the fact that W_{λ} acts trivially on e^{λ}.

(ii) It suffices to recall that every element of Re^{h_{i}.λ} is killed by a power of the maximal ideal m_{h_{i}.λ}.

By combining the previous results we obtain the following minor generalization of [19, Theorem 2.7].

Proposition 2.4. As a left A ∗ W_{λ}-module,

(A ∗ W_{λ}) ⊗_{A} Re^{λ} ≅ ⨁_{χ∈W_{λ̂}} (Re^{λ} ⊗_{𝕜_{λ}} V_{χ})^{(m_{χ})} .

Moreover, the modules Re^{λ} ⊗_{𝕜_{λ}} V_{χ} are simple and pairwise non-isomorphic.

Proof. Let 𝕜[W_{λ}] denote the group ring of W_{λ}. Under the natural embedding, we have 𝕜[W_{λ}] ⊗_{𝕜} Re^{λ} = (A ∗ W_{λ}) ⊗_{A} Re^{λ}. Hence,

Φ((A ∗ W_{λ}) ⊗_{A} Re^{λ}) = 𝕜[W_{λ}] ⊗ e^{λ} ≅ 𝕜[W_{λ}],

thought of as a W_{λ}-module under the left regular representation. Since 𝕜[W_{λ}] ≅ ⨁ V_{χ}^{(m_{χ})}, the result follows from the equivalence of categories given by Proposition 2.2.

We can now state the main result of this section.

Theorem 2.5. Let λ ∈ V^{∗} and set Z_{χ} = (A ∗ W) ⊗_{A∗W_{λ}} (Re^{λ} ⊗_{𝕜_{λ}} V_{χ}). Then, as an A ∗ W -module,

(A ∗ W) ⊗_{A} Re^{λ} ≅ ⨁_{χ∈W_{λ̂}} Z_{χ}^{(m_{χ})}.

The Z_{χ}, for χ ∈ W_{λ}, are simple, non-isomorphic A ∗ W -modules.

Proof. Note that (A ∗ W) ⊗_{A} Re^{λ} ≅ (A ∗ W) ⊗_{A∗W_{λ}} (A ∗ W_{λ}) ⊗_{A} Re^{λ} as an A ∗ W - module. Thus, the displayed equation is an immediate consequence of Proposition 2.4.

Let {h_{1} = ϵ, h_{2}, …, h_{s}} be a left transversal to W_{λ} in W and set Y_{χ} = Re^{λ} ⊗_{𝕜_{λ}} V_{χ}. As A-modules, Z_{χ} ≅ ⨁_{j=1}^{s} h_{j}Y_{χ}, and h_{i}Y_{χ} ≅ (Re^{h_{i}.λ})^{(m_{χ})} (use Lemma 2.3). Moreover, the A-modules Re^{h_{i}.λ} and Re^{h_{j}.λ} are simple and non-isomorphic for i ≠ j. Hence any simple A-submodule of Z_{χ} is contained in some h_{j}Y_{χ}.

We next show that Z_{χ} is a simple A ∗ W -module. Since Z_{χ} is a direct summand of (A ∗ W) ⊗_{A} Re^{λ}, it is semi-simple, by Lemma 2.1(iii). Thus, we may pick a simple A ∗ W -summand M such that the projection from M to h_{1}Y_{χ} is non-zero. It follows from the last paragraph that, as an A-module, M = ⨁_{i}(M ∩ h_{i}Y_{χ}). Thus M ∩ h_{1}Y_{χ} ≠ (0). But, Proposition 2.4 implies that Y_{χ} = h_{1}Y_{χ} is a simple A ∗ W_{λ}-submodule of Z_{χ}, whence h_{1}Y_{χ} ⊆ M. Therefore

M = (A ∗ W)M ⊇ (A ∗ W)Y_{χ} = ⨁_{i} h_{i}Y_{χ} = Z_{χ}.

Thus, Z_{χ} is indeed simple.

Finally, suppose that there exists an isomorphism θ : Z_{χ} → Z_{ψ} for some χ ≠ ψ ∈ W_{λ}. If we regard θ as a homomorphism of A-modules then, as in the second paragraph, Lemma 2.3(ii) implies that θ maps h_{i}Y_{χ} to h_{i}Y_{ψ}. In particular, for i = 1, we find that θ restricts to a non-zero A ∗ W_{λ}-module homomorphism θ′ : Y_{χ} → Y_{ψ}. Since Y_{χ} is a simple A ∗ W_{λ}-module, θ′ is an isomorphism and, by Proposition 2.4, χ = ψ.

3. The structure of O(V)e^{λ} as a D(V)^{W} -module

As in the last section, we fix an ℓ-dimensional 𝕜-vector space V and set R = O(V) and A = D(V), etc. In this section, we use the results of the last section, together with the Morita equivalence between A ∗ W and A^{W} to give a precise description of the structure of Re^{λ} as an A^{W} -module. In particular, in Theorem 3.4 we prove parts (i) and (ii) of Theorem A of the introduction.

We begin with some simple observations. Regard A as an (A∗W, A^{W})-bimodule in the natural way: (∑_{w_{i}∈W} a_{i}w_{i}) ∘ r ∘ b = ∑ a_{i}(w_{i}.r)b, for ∑ a_{i}w_{i} ∈ A ∗ W, r ∈ A and b ∈ A^{W}. Call this module A^{l}. We may also regard A as an (A^{W} , A ∗ W)- bimodule by a ∘ r ∘ ∑_{w_{i}∈W} w_{i}b_{i} = ∑ a(r^{w_{i}})b_{i}, for a ∈ A^{W}, r ∈ A and ∑ w_{i}b_{i} ∈ A ∗ W, in which case we write the module as A^{r}. Of course, it will usually be clear which bimodule structure is intended, in which case the superscript will be ignored.

The basic facts that we need about the Morita equivalence are collected in the next lemma.

Lemma 3.1. (i) A^{W} and A ∗ W are simple, Morita equivalent rings. The equivalence is obtained by the functors:

I(N) = A^{l} ⊗_{A^{W}} N  and  F (M) = A^{r} ⊗_{A∗W} (M),

for N ∈ A^{W} -mod, respectively M ∈ A ∗ W -mod.

(ii) If M ∈ A ∗ W -mod, then F (M) ≅ Hom_{A∗W} (A^{l}, M) ≅ M^{W}.

(iii) As A ∗ W -bimodules, A ⊗_{A^{W}} A ≅ A ∗ W.

Proof. (i) Note that Hom_{A∗W} (A^{r}, A ∗ W) ≅ A^{l}, as (A ∗ W , A^{W})-bimodules, and similarly for A^{l} (see [15, Proposition 7.8.5]). Thus, the result follows from [16, Theorem 2.5 and Corollary 2.6].

(ii) The first isomorphism is standard, while the second isomorphism, follows from the fact that each f ∈ Hom_{A∗W} (A, M) is determined by f(1) ∈ M^{W}.

(iii) This is just the assertion that IF (A ∗ W) ≅ A ∗ W.

Remark 3.2. The above results still apply if one replaces W by W_{λ}. In this situation we denote the equivalences by

F_{λ} = A ⊗_{A∗W_{λ}}  and  I_{λ} = A ⊗_{A^{W_{λ}}} .

Note that for all a ∈ A and j ∈ ℕ, there exists k ∈ ℕ such that m_{λ}^{k}a ⊂ Am_{λ}^{j}. Similarly, there exists i ∈ ℕ such that m_{λ}^{i} ⊆ n_{λ}S. It follows easily that, if M ∈ C_{λ} then I_{λ}(M) ∈ S_{λ}, using the notation of Remark 3.2. Conversely if N ∈ S_{λ} then F_{λ}(N) is obviously in C_{λ}. Combined with Proposition 2.2 this implies:

Lemma 3.3. The categories S_{λ} and C_{λ} are equivalent via the Morita equivalence of Lemma 3.1. Moreover C_{λ} ≈ W_{λ}-mod via the functor ΦI_{λ}.

Recall that V_{χ} is defined to be a simple W_{λ}-module of type χ. Define

V^{χ} = Hom_{W_{λ}}(V_{χ}^{∗}, Re^{λ}). (3.1)

Thus, V^{χ} ≅ (Re^{λ} ⊗_{𝕜_{λ}} V_{χ})^{W_{λ}}. Since the A^{W_{λ}}- and W_{λ}-module structures on Re^{λ} commute, the action of A^{W_{λ}} on Re^{λ} induces a left A^{W_{λ}}-module structure on V^{χ}. Note that V^{χ} ⊗ V_{χ}^{∗} is isomorphic with the isotypic component of type χ^{∗} in the W_{λ}-module Re^{λ}.

We can now complete the description of Re^{λ} as an A^{W} -module.

Theorem 3.4. (i) The (A^{W} , W_{λ})-module Re^{λ} decomposes as:

Re^{λ} ≅ ⨁_{χ∈W_{λ̂}} V^{χ} ⊗ V_{χ}^{∗}. (3.2)

The A^{W} -modules V^{χ}, for χ ∈ W_{λ}, are simple and pairwise non-isomorphic.

(ii) If χ = triv is the trivial character, then V^{triv} = A^{W} · e^{λ} = R^{W_{λ}}e^{λ} is a simple A^{W} -module.

Remark 3.5. (1) By their construction, the V^{χ} are A^{W_{λ}}-modules. Therefore, the decomposition of part (i) of the theorem is also a decomposition of Re^{λ} into a direct sum of simple A^{W_{λ}}-modules.

(2) When λ = 0, one has W_{λ} = W and C_{0} ≈ W -mod. When λ is regular, that is, W_{λ} = {1}, Re^{λ} is a simple A^{W} -module (compare with [4, Lemmes 9 & 11]).

Proof. Note that one has a natural isomorphism of (A^{W_{λ}}, W_{λ})-modules

⨁_{χ∈W_{λ̂}} V^{χ} ⊗ V_{χ}^{∗} \underrightarrow{∼} Re^{λ}

given by f ⊗ v ↦ f(v), for f ∈ V^{χ} and v ∈ V_{χ}^{∗}. Thus (3.2) holds. In order to prove the rest of the theorem, we wish to reduce the result to that given by Theorem 2.5. As in §2, set Y_{χ} = Re^{λ} ⊗_{𝕜_{λ}} V_{χ} and Z_{χ} = (A ∗ W) ⊗_{A∗W_{λ}} Y_{χ}. Write U_{χ} = A⊗_{A∗W_{λ}}Y_{χ}, regarded as an A^{W} -module and note that U_{χ} = F (Z_{χ}) = F_{λ}(Y_{χ}). Clearly, A ≅ A ⊗_{A∗W} (A ∗ W), as an (A^{W} , A)-bimodule. Thus, by Theorem 2.5, one obtains the following isomorphism of A^{W} -modules:

Re^{λ} ≅ A ⊗_{A∗W} (A ∗ W) ⊗_{A} Re^{λ} ≅ ⨁_{χ} (A ⊗_{A∗W} Z_{χ})^{(m_{χ})} = ⨁_{χ} U_{χ}^{(m_{χ})}.

By Morita equivalence and Theorem 2.5, again, the A^{W} -modules U_{χ} are simple and pairwise non-isomorphic. Thus, in order to prove the theorem, it suffices to show that U_{χ} ≅ V^{χ} for each χ and that V^{triv} = R^{W_{λ}}e^{λ}.

Since U_{χ} = F_{λ}(Y_{χ}), it follows from Lemma 3.1(ii) that U_{χ} ≅ (Y_{χ})^{W_{λ}} as A^{W_{λ}}- modules and therefore as A^{W} -modules. Thus,

U_{χ} = (Re^{λ} ⊗_{𝕜_{λ}} V_{χ})^{W_{λ}} = Hom_{W_{λ}}(V_{triv}, Re^{λ} ⊗_{𝕜_{λ}} V_{χ}) = Hom_{W_{λ}}(V_{χ}^{∗}, Re^{λ}) = V^{χ}.

Finally, when χ = triv, we have V^{triv} = Hom_{W_{λ}}(V_{triv}, Re^{λ}) = R^{W_{λ}}e^{λ}. Since this A^{W} -module is simple and contains A^{W} · e^{λ}, part (ii) follows.

Corollary 3.6. Let λ, μ ∈ V^{∗}. Then the following are equivalent:

(i) Re^{λ} ≅ Re^{μ} as A^{W} -modules

(ii) Re^{λ} and Re^{μ} have a simple A^{W} -submodule in common

(iii) μ ∈ W.λ

Proof. (iii) ⇒ (i) Assume that μ = w.λ for some w ∈ W. We obviously have an isomorphism of vector spaces w̃ : Re^{λ} → Re^{μ}, given by fe^{λ} ↦ w.(fe^{λ}) = (w.f)e^{w.λ}. By definition w.(d · fe^{λ}) = (w.d) · (w.(fe^{λ})) for all d ∈ A. Hence the map w̃ is an isomorphism of A^{W} -modules.

(i) ⇒ (ii) is trivial.

(ii) ⇒ (iii) Generalizing the previous notation we set W = ⨆_{i} h_{i}(λ)W_{λ} with h_{1}(λ) = ϵ and write

(A ∗ W) ⊗_{A∗W_{λ}} Re^{λ} = ⨁_{χ∈W_{λ̂}}(A ∗ W) ⊗_{A∗W_{λ}} Z_{χ}(λ),

etc. Assume that the A^{W} -modules Re^{λ} and Re^{μ} have a simple A^{W} -module in common. Then there there is an A ∗ W -isomorphism

σ : (A ∗ W) ⊗_{A∗W_{λ}} Z_{χ}(λ) → (A ∗ W) ⊗_{A∗W_{μ}} Z_{ψ}(μ)

for some χ ∈ W_{λ} and ψ ∈ W_{μ}. As in the proof of Theorem 2.5, any simple Asubmodule of (A ∗ W) ⊗_{A∗W_{λ}} Z_{χ}(λ) is of the form Re^{h_{i}(λ).λ}, for some i. Thus, there exist h_{i}(λ), h_{j}(μ) such that Re^{h_{i}(λ).λ} ≅ Re^{h_{j}(μ).μ} as A-modules; this implies that h_{i}(λ).λ = h_{j}(μ).μ. Thus, μ ∈ W.λ.

4. The case of a group generated by pseudo-reflections

We continue with the notation of the last two sections. The A^{W} -module Re^{λ} clearly has finite length and so, by [1, Theorem 1.8.18] it is cyclic. In this section we will give a presentation Re^{λ} ≅ A^{W} /L in the case when W is generated by pseudoreflections. It is this presentation that will allow us to relate Re^{λ} to invariant holonomic systems, as described in the introduction to the paper.

We begin with some general results.

Lemma 4.1. Let C be a subalgebra of the 𝕜-algebra B. Let B/J = ⨁_{i=1}^{t} S_{i}^{n_{i}} be a cyclic, semi-simple B-module. Assume that the C-modules S_{i} are simple and non-isomorphic, and that End_{C}(S_{i}) = End_{B}(S_{i}) = 𝕜. Then B/J = (C + J)/J.

Proof. The hypotheses imply that End_{C}(B/J) = End_{B}(B/J) and so any decomposition of B/J as a C-module is also a B-module decomposition. But, as a C-module, B/J = N⊕(C + J)/J, for some module N. Therefore, this is a B-module decomposition with its generator [1+J] in the second factor. Thus B/J = (C +J)/J.

Lemma 4.2. Let E be a finite dimensional S ∗ W_{λ}-module such that m_{λ}^{j}E = (0) for some j ∈ ℕ. Then Hom_{A}(Re^{λ}, A ⊗_{S} E) ≅ E, as W_{λ}-modules.

Proof. Set E_{i} = m_{λ}^{i}E/m_{λ}^{i+1}E, and note that each E_{i} is a subfactor of E as an S ∗ W_{λ}-module and a summand of E as a W_{λ}-module. Since the elements of S act ad-nilpotently on A, clearly A ⊗_{S} E ∈ S_{λ}. Therefore, by Lemma 2.1, A ⊗_{S} E is semi-simple and hence A ⊗_{S} E = ⨁_{i} A ⊗_{S} E_{i}. Thus, it suffices to prove the result with E replaced by some E_{i}. In this case, A ⊗_{S} E_{i} ≅ A ⊗_{𝕜_{λ}} E_{i} and the lemma follows from Proposition 2.2 and the definition of Φ.

For the rest of this section we will be concerned with the case when the group W ⊂ GL(V) is generated by pseudo-reflections. Recall that this has the following consequences, see [3, Chapitre V §5]: S is a free S^{W} -module of rank |W | and the subgroup W_{λ} is also generated by pseudo-reflections. Also, there exists a W_{λ}-stable subspace 𝒦 ⊂ S such that S = 𝒦⊗_{𝕜}S^{W_{λ}}. As a W_{λ}-module, 𝒦 ≅ 𝕜[W_{λ}], the regular representation.

Corollary 4.3. Assume that W is generated by pseudo-reflections. Then, Re^{λ} ≅ A^{W_{λ}}/A^{W_{λ}}n_{λ}, as A^{W_{λ}}-modules.

Proof. We first note that ΦI_{λ}(Re^{λ}) = 𝕜[W_{λ}] under the left regular representation of W_{λ}. Indeed, by definition and Lemma 3.1(iii), I_{λ}(Re^{λ}) = (A ∗ W_{λ}) ⊗_{A} Re^{λ}, while, by the proof of Proposition 2.4, Φ(A ∗ W_{λ} ⊗_{A} Re^{λ}) = 𝕜[W_{λ}], as required.

Note that M = A^{W_{λ}}/A^{W_{λ}}n_{λ} ∈ C_{λ}. Therefore by the last paragraph and Lemma 3.3, it is enough to show that ΦI_{λ}(M) = 𝕜[W_{λ}]. However, as W_{λ}-modules,

ΦI_{λ}(M) = Hom_{A}(Re^{λ}, A ⊗_{A^{W_{λ}}} M) = Hom_{A}(Re^{λ}, A/An_{λ})
= Hom_{A}(Re^{λ}, A ⊗_{S} S/Sn_{λ}) = S/Sn_{λ},

where the final equality follows from Lemma 4.2. Since W_{λ} is generated by pseudoreflections,

S/Sn_{λ} = 𝒦 ⊗_{𝕜} (S^{W_{λ}}/n_{λ}) ≅ 𝕜[W_{λ}]

as W_{λ}-modules. Hence the result.

Theorem 4.4. Assume that W is generated by pseudo-reflections and set p_{λ} = S^{W} ∩ m_{λ}. Then Re^{λ} ≅ A^{W} /A^{W} p_{λ} as A^{W} -modules.

Proof. By Theorem 3.4, the hypotheses of Lemma 4.1 are satisfied by C = A^{W} ⊆ B = A^{W_{λ}} and B/J = Re^{λ}. Thus, Corollary 4.3 and Lemma 4.1 imply that Re^{λ} ≅ A^{W} /(A^{W} ∩ A^{W_{λ}}n_{λ}). Consequently, there exists a surjective morphism of A^{W} -modules ϕ : A^{W} /A^{W} p_{λ} ↠ Re^{λ}.

Using Lemma 3.1(iii), this induces a surjection of A ∗ W -modules

ϕ′ : A ⊗_{A^{W}} (A^{W} /A^{W} p_{λ}) ↠ A ⊗_{A^{W}} Re^{λ} ≅ (A ∗ W) ⊗_{A} Re^{λ}.

Thus, by Morita equivalence, it suffices to prove that ϕ′ is an isomorphism and, to do so, we need only to consider the A-module structure. By Lemma 2.3, (A ∗ W)⊗_{A} Re^{λ} has length |W |. On the other hand A ⊗_{A^{W}} (A^{W} /A^{W} p_{λ}) ≅ A ⊗_{S} (S/Sp_{λ}). Since W is generated by pseudo-reflections, S/Sp_{λ} is a 𝕜-vector space of dimension |W | and hence is an S-module of length |W |, with each simple factor isomorphic to 𝕜_{λ}. Thus, the A-module A ⊗_{A^{W}} (A^{W} /A^{W} p_{λ}) has length |W |, with each factor isomorphic to Re^{λ}. Therefore both ϕ′ and ϕ are isomorphisms.

5. On the equivalence C_{λ} ≈ W_{λ}-mod

By Lemma 3.3 the category C_{λ} is equivalent to W_{λ}-mod via the functor ΦI_{λ} = Hom_{A}(Re^{λ}, A⊗_{A^{W_{λ}}} ), with inverse functor F_{λ}Ψ. We study these functors in more depth in this section. Note that, by Remark 3.5, C_{λ} is also the full subcategory of A^{W} -modules generated by Re^{λ} and it makes little difference in this section whether we work with A^{W} -modules or A^{W_{λ}}-modules.

Note that ΦI_{λ} satisfies (and hence is defined by) ΦI_{λ}(V^{χ}) = V_{χ} for all χ. This permits one to give an alternative interpretation of this functor. Define functors from C_{λ} to W_{λ}-mod by

Sol(M) = Hom_{A^{W_{λ}}} (M, Re^{λ}) = Hom_{A^{W}} (M, Re^{λ}),

and

DR(M) = Hom_{A^{W_{λ}}} (Re^{λ}, M) = Hom_{A^{W}} (Re^{λ}, M),

By analogy with [2, VIII.13] we call these functors the solution functor, respectively the de Rham functor. Since Re^{λ} is an (A^{W_{λ}}, W_{λ})-module, their images do indeed lie in W_{λ}-mod.

Proposition 5.1. (i) DR = ΦI_{λ} provides an equivalence of categories from C_{λ} to W_{λ}-mod. Similarly, Sol gives an equivalence of categories between C_{λ} and (W_{λ}-mod)^{op}.

(ii) Sol(Re^{λ}) = DR(Re^{λ}) = 𝕜[W_{λ}] as (W_{λ} × W_{λ})-module.

Proof. (i) It follows from Theorem 3.4 that DR(V^{χ}) = V_{χ} and that Sol(V^{χ}) = (V_{χ})^{∗}. Thus, the result follows from the earlier observations.

(ii) By Theorem 3.4, there is a natural isomorphism:

End_{A^{W_{λ}}} (Re^{λ}) ≅ ⨁_{ψ,χ∈W_{λ̂}} Hom_{A^{W_{λ}}} (V^{ψ}, V^{χ}) ⊗ (V_{ψ} ⊗ V_{χ}^{∗}) = ⨁_{ψ∈W_{λ̂}} V_{ψ} ⊗ V_{ψ}^{∗} ≅ 𝕜[W_{λ}].

Since the A^{W_{λ}} and W_{λ} actions on Re^{λ} commute, this is an isomorphism of (W_{λ} × W_{λ})-modules.

Now, assume that W is generated by pseudo-reflections and adopt the notation of §4. By Theorem 4.4, Sol(Re^{λ}) = Hom_{A^{W}} (A^{W} /A^{W} p_{λ}, Re^{λ}) and so it can be identified with:

ℋ_{λ} = {f ∈ Re^{λ} : p_{λ} · f = 0} = {f ∈ Re^{λ} : p · f = p(λ)f, for all p ∈ S^{W} } .

By Proposition 5.1, ℋ_{λ} ≅ 𝕜[W_{λ}] as a W_{λ}-module. If M ∈ C_{λ} is a quotient of Re^{λ}, then Sol(M) is a submodule of Sol(Re^{λ}) and can therefore be identified with a W_{λ}-submodule of ℋ_{λ}.

We want to relate the subspace ℋ_{λ} with a subspace of W_{λ}-harmonic elements in R. Let V (λ) ⊂ Ker λ be the unique W_{λ}-stable complement of the subspace of invariants V^{W_{λ}} ⊆ V. We will identify V (λ)^{∗} with the W_{λ}-submodule (V/V^{W_{λ}})^{∗} of V^{∗}. Note that S(V (λ)^{∗})e^{λ} ≅ S(V (λ)^{∗}) as W_{λ}-modules. Define

ℋ(λ) = {f ∈ S(V (λ)^{∗}) : p · f = p(0)f, for all p ∈ S(V (λ))^{W_{λ}}} ⊆ S(V (λ)^{∗}).

Replacing V by V (λ) in the earlier argument, it follows that ℋ(λ) ≅ 𝕜[W_{λ}], as a W_{λ}-module.

Lemma 5.2. If W is generated by pseudo-reflections, then ℋ_{λ} = ℋ(λ)e^{λ}.

Proof. Since ℋ(λ)e^{λ} and ℋ_{λ} are subspaces of Re^{λ} of the same dimension, |W_{λ}|, it suffices to show that ℋ_{λ} ⊆ ℋ(λ)e^{λ}. Note that that q · (φe^{λ}) = (q · φ)e^{λ}, for all q ∈ S(V (λ)) and φ ∈ R. Similarly, r · (fe^{λ}) = r(λ)fe^{λ}, for all r ∈ S(V^{W_{λ}}) and f ∈ S(V (λ)^{∗}). Let f ∈ ℋ(λ) and p ∈ S^{W}. Write p = ∑_{j} r_{j}q_{j}, for r_{j} ∈ S(V^{W_{λ}}), and q_{j} ∈ S(V (λ))^{W_{λ}}. Observe that p(λ) = ∑_{j} r_{j}(λ)q_{j}(0).

Now, if p ∈ p_{λ}, equivalently p(λ) = 0, then these observations show that

p · (fe^{λ}) = ∑_{j} r_{j} · (q_{j} · (fe^{λ})) = ∑_{j} r_{j} · ((q_{j} · f)e^{λ})

= ∑_{j} r_{j} · (q_{j}(0)fe^{λ}) (since f ∈ ℋ(λ))

= ∑_{j} q_{j}(0)r_{j}(λ)fe^{λ} = p(λ)fe^{λ} = 0.

Thus fe^{λ} ∈ ℋ_{λ}.

This lemma gives us another way of defining the equivalence C_{λ} ≈ W_{λ}-mod:

Corollary 5.3. Assume that W is generated by pseudo-reflections If M is a quotient of Re^{λ}, define

Sol(M)e^{−λ} = {f ∈ ℋ(λ) : fe^{λ} ∈ Sol(M)}.

Then the functor M → Sol(M)e^{−λ} induces an equivalence of categories between C_{λ} and (W_{λ}-mod)^{op}.

6. Invariant holonomic systems

Let G be a connected reductive complex algebraic group with Lie algebra 𝔤. Fix a Cartan subalgebra 𝔥 of 𝔤 and denote by W the associated Weyl group. Let τ : 𝔤 → D(𝔤) be the differential of the adjoint action. Set ℓ = rk 𝔤 and n = dim 𝔤. Throughout the section, we fix λ ∈ 𝔥^{∗}. Following [10] we define the system of linear differential equations for invariant eigendistributions, also called the “invariant holonomic system on 𝔤”, by

𝒩_{λ} = D(𝔤)/(D(𝔤)τ(𝔤) +_{∑p∈S(𝔤)_{G}}D(𝔤)(p − p(λ))). (6.1)

It is not difficult to see that 𝒩_{λ} is an holonomic D(𝔤)-module, and therefore has finite length. The aim of this section is to give a complete description of the structure of 𝒩_{λ} (see Theorem 6.9), thereby proving Theorem B from the introduction. This generalizes the results of [10] which did this in the cases when λ = 0 and when λ is regular. The results of the previous sections apply here with V = 𝔥. We first need to relate 𝒩_{λ} to the D(𝔥)^{W} -module O(𝔥)e^{λ}.

In [6], Harish-Chandra defines a homomorphism δ : D(𝔤)^{G} → D(𝔥)^{W} such that the restriction of δ to O(𝔤)^{G} and to S(𝔤)^{G} are just the Chevalley isomorphisms δ : O(𝔤)^{G} → O(𝔥)^{W} and δ : S(𝔤)^{G} → S(𝔥)^{W}. Moreover, by [13] and [14], δ induces an isomorphism D(𝔥)^{W} ≅ D(𝔤)^{G}/I, where I = D(𝔤)^{G} ∩ D(𝔤)τ(𝔤). In particular, since τ(𝔤) commutes element-wise with D(𝔤)^{G}, we obtain a natural right D(𝔥)^{W} - module structure on 𝒩 = D(𝔤)/D(𝔤)τ(𝔤).

Recall the following definition from [20, §5]. Denote by Ad the adjoint action of G on 𝔤 or D(𝔤). A D(𝔤)-module M is said to be a compatible (D(𝔤), G)-module (or G-equivariant D_{𝔤}-module as in [2, 12.10]) if there is an action of G on M such that

(i) If p ∈ D(𝔤), g ∈ G, then g.(px) = (Ad g(p)) g.x for all x ∈ M; (ii) As a G-module, M is locally finite; (iii) The differential of the G-action on M is given via τ : 𝔤 → D(𝔤).

Lemma 6.1. Any subquotient of 𝒩 is a compatible (D(𝔤), G)-module.

Proof. Let P ⊆ L be left ideals of D(𝔤) such that D(𝔤)τ(𝔤) ⊆ P. Observe that, for all x ∈ L and ξ ∈ 𝔤, τ(ξ)x ≡ ad ∘τ(ξ)(x) = [τ(ξ), x] modulo D(𝔤)τ(𝔤). Recall that the differential of the G-action on D(𝔤) is given by ad ∘τ; it follows that L/P is a locally finite 𝔤-module via ad ∘τ. Therefore, since G is connected, L/P is a locally finite G-module. It is then clear that L/P is a compatible (D(𝔤), G)-module.

Set m_{λ} = Ker λ ⊂ S(𝔥) and p_{λ} = m_{λ} ∩ S(𝔥)^{W}. By Theorem 4.4, O(𝔥)e^{λ} is a (D(𝔥)^{W} , W_{λ})-module isomorphic, as a D(𝔥)^{W} -module, to D(𝔥)^{W} /D(𝔥)^{W} p_{λ}. Since δ : S(𝔤)^{G} → S(𝔥)^{W} is the Chevalley isomorphism,

δ(_{∑p∈S(𝔤)_{G}}S(𝔤)^{G}(p − p(λ))) = p_{λ}.

Thus, we have proved the first assertion of:

Lemma 6.2. (i) As left D(𝔤)-modules, 𝒩_{λ} ≅ 𝒩 ⊗_{D(𝔥)^{W}} O(𝔥)e^{λ}.

(ii) Let {V^{χ} = Hom_{W_{λ}}(V_{χ}^{∗}, O(𝔥)e^{λ})} be the D(𝔥)^{W} -modules defined in (3.1). Then, there is a decomposition of (D(𝔤), W_{λ})-modules:

𝒩_{λ} = ⨁_{χ∈W_{λ̂}} 𝒩_{λ,χ} ⊗_{ℂ} V_{χ}^{∗}, where 𝒩_{λ,χ} = 𝒩 ⊗_{D(𝔥)^{W}} V^{χ}.

Here, D(𝔤) acts on 𝒩_{λ,χ} while W_{λ} acts on V_{χ}^{∗}. Finally, (𝒩_{λ,χ})^{G} = V^{χ}, for each χ.

Proof. (ii) The decomposition of 𝒩_{λ} follows from Theorem 3.4. The final assertion of the lemma is an immediate consequence of the fact that, as a left or right D(𝔤)^{G}- module, 𝒩 ≅ D(𝔤)^{G}/I ⊕ (⊕_{0≠ν∈G}ˆ 𝒩[ν]), where 𝒩[ν] denotes the sum of finite dimensional irreducible G-modules of type ν in 𝒩.

This result raises the question of whether the 𝒩_{λ,χ} are simple, pairwise nonisomorphic D(𝔤)-modules. That this is true will form the main result, Theorem 6.9, of this section. This will follow easily provided we can prove that each D(𝔤)- submodule of 𝒩_{λ} contains non-zero G-invariant elements.

In order to simplify the notation, we will write 𝒟 = D(𝔤), A = D(𝔥) and B = 𝒩 ⊗_{A^{W}} A, regarded as a (𝒟, A)-bimodule. Let d denote the discriminant of 𝔤. Thus, d ∈ O(𝔤)^{G} and under the Chevalley isomorphism O(𝔤)^{G} → O(𝔥)^{W}, d maps to d̃= π^{2}, where π is the product of the positive roots of 𝔥 in 𝔤. Since d ∈ O(𝔤)^{G}, it acts locally ad-nilpotently on 𝒟 and so 𝒞 = {d^{m} : m ∈ ℕ} is an Ore set in both 𝒟^{G} and 𝒟. Similarly, 𝒞̃ = {d̃^{m}} is an Ore set in both A^{W} and in A. Recall that 𝔤′ = {x ∈ 𝔤 : d(x) ≠ 0} is the set of generic elements. Thus, O(𝔤′) = O(𝔤)_{𝒞} and D(𝔤′) = 𝒟_{𝒞}.

The following identifications will prove useful:

Lemma 6.3. (i) As left 𝒟-modules, 𝒩_{λ} ≅ B ⊗_{A} A/Am_{λ}.

(ii) There is a natural isomorphism of localized modules:

𝒟_{C} ⊗_{𝒟} 𝒩_{λ} ≅ B ⊗_{A} (A/Am_{λ})𝒞̃

Proof. (i) As remarked in §2, O(𝔥)e^{λ} ≅ A/Am_{λ} as A^{W} -modules, via the natural A^{W} -module structure on A. Thus, part (i) follows from Lemma 6.2(i).

(ii) Since d acts locally ad-nilpotently on 𝒟, given any f ∈ 𝒟, there exists m ∈ ℕ and f_{1} ∈ 𝒟 such that fd^{m} = df_{1}. Hence, d^{−1}f = f_{1}d^{−m}, whenever the terms make sense. By the usual universality arguments, this implies that there is a natural isomorphism:

𝒟_{𝒞} ⊗_{𝒟} 𝒩 ≅ 𝒩 ⊗_{𝒟^{G}} 𝒟_{𝒞}^{G} (6.2)

Under the identification of 𝒟^{G}/I with A^{W}, d identifies with d̃ and so 𝒞 can be identified with 𝒞̃. Thus, combined with (6.2), part (i) implies that

𝒟_{𝒞} ⊗_{𝒟} 𝒩_{λ} ≅ 𝒩 ⊗_{A^{W}} AW ; 𝒞̃ ⊗_{A^{W}} A/Am_{λ} ≅ B ⊗_{A} (A/Am_{λ})𝒞̃

as desired.

One consequence of Lemma 6.3 is that localizing with respect to 𝒞 or 𝒞̃ is always going to have the same result and so, from now on we will identify d̃ with d and hence identify 𝒞̃ with 𝒞.

We will need the following general result:

Lemma 6.4. Let C = 𝕜[y_{1}, …, y_{ℓ}] be a commutative polynomial ring over an algebraically closed field 𝕜 of characteristic zero and set A_{ℓ} = D(C) (the ℓ-th Weyl algebra). Let D be a left Noetherian ring and L be a (D, A_{ℓ})-bimodule, finitely generated as a left D-module. Then L is a flat right C-module.

Proof. Use the first two paragraphs of the proof of [14, Theorem 2.4].

Proposition 6.5. Let m be a maximal ideal of S(𝔥) and set P = A/Am. Then, Tor_{1}^{A}(B, P_{𝒞} /P) = 0.

Proof. To begin with, let α ∈ 𝔥^{∗} be one positive root. Thus, we may choose a basis x_{1} = α, x_{2}, …, x_{ℓ} of 𝔥^{∗} and regard O(𝔥) as the polynomial ring in the {x_{i}}. Write ∂_{i} = \frac{∂}{∂x_{i}}; thus the \{∂_{i}\} are generators of S(𝔥) and m = ∑_{i} S(𝔥)(∂_{i} − λ_{i}), for some scalars λ_{i}. Set 𝒮 = 𝒮_{α} = {x_{1}^{m} : m ∈ ℕ}.

We first consider the module P_{𝒮}/P = A_{𝒮}/(A + ∑_{i} A_{𝒮}(∂_{i} − λ_{i})). Certainly, this is generated by the images [x_{1}^{−m}] ∈ P_{𝒮}/P of the x_{1}^{−m}, for m ∈ ℕ. However,

(∂_{1} − λ_{1})[x_{1}^{−m}] = [x_{1}^{−m}(∂_{1} − λ_{1}) + (−m)x_{1}^{−m−1}] = [(−m)x_{1}^{−m−1}],

and so, in fact, P_{𝒮} is generated by [x_{1}^{−1}]. This element is clearly killed by x_{1} and by (∂_{i} − λ_{i}) for 2 ≤ i ≤ ℓ. Hence, either P_{𝒮}/P = 0 or P_{𝒮}/P is isomorphic to the simple A-module

P_{𝒮}/P ≅ A/(Ax_{1} + ∑_{i=2}^{ℓ}A(∂_{i} − λ_{i})). (6.3)

Now, consider P_{𝒞} /P. By [1, Theorem 1.5.5] this is a finitely generated A-module, generated by the elements [d^{−q}], for q ∈ ℕ. Hence, there exists m ∈ ℕ, such that P_{𝒞} /P = A[d^{−m}]. Write d^{−m} = ∏_{i∈Q} α_{i}^{−2m}, for distinct positive roots α_{i}. By the

Chinese Remainder Theorem, d^{−m} = ∑_{i} g_{i}α_{i}^{−j(i)} for some g_{i} ∈ O(𝔥). Thus, as in the last paragraph,

A[d^{−m}] = A∑_{i} g_{i}[α_{i}^{−j(i)}] ⊆ _{∑i}A[α_{i}^{−j(i)}] = _{∑i}A[α_{i}^{−1}].

Therefore, for some subset Q′ of Q (in fact, one can prove that Q′ = Q),

P_{𝒞} /P = A[d^{−m}] = ⨁_{i∈Q′} A[α_{i}^{−1}] = ⨁_{i∈Q′} P_{𝒮_{α_{i}}} /P.

Thus, in order to prove the proposition, it suffices to prove that

Tor_{1}^{A}(B, P_{𝒮_{α_{i}}} /P ) = 0 for each i.

We may assume that α_{i} = α_{1} and revert to the notation of the first two paragraphs. By (6.3), we may write P_{𝒮}/P = A/ ∑_{i=1}^{ℓ} Ay_{i}, where y_{1} = x_{1} and y_{i} = ∂_{i} − λ_{i} for i > 1. Moreover, the ring C = ℂ[y_{1}, …, y_{ℓ}] is a polynomial ring in the {y_{j}} and A ≅ D(C). Then, by Lemma 6.4, B is a flat right C-module. Thus,

Tor_{1}^{A}(B, A/_{∑i}Ay_{i}) ≅ Tor_{1}^{A}(B, A ⊗_{C} C/_{∑i}Cy_{i}) ≅ Tor_{1}^{C}(B, C/_{∑i}Cy_{i}) = 0,

where the second isomorphism comes from [17, Theorem 11.53]. This completes the proof of the proposition.

The significance of this result is that it implies:

Corollary 6.6. The module 𝒩_{λ} is torsion-free as a left O(𝔤)-module.

Proof. By Proposition 6.5, Tor_{1}^{A}(B, P_{𝒞} /P) = 0, where P = A/Am_{λ}. It follows then from Lemma 6.3 that the natural map 𝒩_{λ} → (𝒩_{λ})_{𝒞} is injective.

By [14, Lemma 3.1], the characteristic variety Ch (𝒩_{λ})_{𝒞} of the D(𝔤′)-module (𝒩_{λ})_{𝒞} is contained in C(𝔤) ∩ (𝔤′ × N), where C(𝔤) denotes the commuting variety {(x, y) ∈ 𝔤 × 𝔤 : [x, y] = 0} and N is the cone of nilpotent elements of 𝔤. Now, for any element x ∈ 𝔤′, the commutant 𝔤^{x} of x is a Cartan subalgebra of 𝔤, and so 𝔤^{x} ∩ N = 0. Hence, Ch (𝒩_{λ})_{𝒞} ⊆ (𝔤′ × {0}).

This implies that (𝒩_{λ})_{𝒞} is an integrable connection on 𝔤′, see [2, VI.1.7 and VII.10.4]. In particular, (𝒩_{λ})_{𝒞} is a finitely generated, locally free O(𝔤′)-module. Consequently, 𝒩_{λ} ↪ (𝒩_{λ})_{𝒞} is a torsion-free O(𝔤)-module.

The fact that 𝒩_{λ} has no 𝒞-torsion also follows from the proof of [10, Theorem 6.1]. The corollary provides a nice analogue of a deep result of Harish-Chandra [7, 8] that proves that 𝒩_{λ} has no factor that is O(𝔤)-torsion.

Let κ be a G-invariant nondegenerate symmetric bilinear form on 𝔤. If u ∈ O(𝔤), denote by grad(u) ∈ Der O(𝔤) the vector field defined by κ(grad(u)_{x}, y) = \frac{d}{dt}_{|t=0}u(x + ty), for x, y ∈ 𝔤.

Lemma 6.7. Let {u_{1}, …, u_{ℓ}} be algebraically independent generators of O(𝔤)^{G} and set θ_{i} = grad(u_{i}). Then, each θ_{i} is G-invariant and

Der O(𝔤′) = O(𝔤′) τ(𝔤) ⊕ (⊕_{i=1}^{ℓ}O(𝔤′) θ_{i}).

Consequently, 𝒩_{𝒞} = D(𝔤′)/D(𝔤′)τ(𝔤) is a free left O(𝔤′)-module with basis

{Θ^{i} = θ_{1}^{i_{1}} ⋯ θ_{ℓ}^{i_{ℓ}} : i = (i_{1}, …, i_{ℓ}) ∈ ℕ^{ℓ}} .

Proof. It follows from the definition that, if u ∈ O(𝔤)^{G}, then the vector field grad(u) is G-invariant. Furthermore, it is easily seen that grad(u)_{x} ∈ 𝔤^{x}, for all x ∈ 𝔤. Let (𝒪_{x}, m_{x}) be the local ring of 𝔤 at x. Identify the tangent space T_{x}𝔤 ≡ 𝔤 with Der 𝒪_{x}/m_{x} Der 𝒪_{x}, and T_{x}(G.x) with [x, 𝔤] = {τ(ξ)_{x} : ξ ∈ 𝔤}.

Define a submodule of Der O(𝔤) by setting L = ∑_{i=1}^{ℓ} O(𝔤) θ_{i}. We first want to show that L = ⊕_{i}O(𝔤) θ_{i} is a free O(𝔤)-module of rank ℓ, and that L∩O(𝔤)τ(𝔤) = 0. Thus, suppose that ∑_{i=1}^{ℓ} a_{i}θ_{i} ∈ O(𝔤)τ(𝔤), for some a_{i} ∈ O(𝔤). Let x ∈ 𝔤′. By the identifications of the last paragraph, ∑_{i} a_{i}(x) grad(u_{i})_{x} ∈ τ(𝔤)_{x} = [x, 𝔤]. Therefore, ∑_{i} a_{i}(x) grad(u_{i})_{x} ∈ 𝔤^{x} ∩ [x, 𝔤] = 0. Recall [12, Theorem 0.1] that the {grad(u_{i})_{x}} are linearly independent. Hence, for all i and x ∈ 𝔤′, a_{i}(x) = 0. Since 𝔤′ is dense in 𝔤, it follows that a_{i} = 0, for all i. Thus, L is free and Der O(𝔤) ⊃ L ⊕ O(𝔤)τ(𝔤), as desired.

Let x ∈ 𝔤′. Since 𝔤 = 𝔤^{x}⊕[x, 𝔤], we know that Der 𝒪_{x}/m_{x} Der 𝒪_{x} is spanned by the {grad(u_{i})_{x}} and τ(𝔤)_{x}. Thus, by Nakayama’s Lemma, Der 𝒪_{x} = L_{x} ⊕ 𝒪_{x}τ(𝔤). This proves the second assertion of the lemma, from which the final assertion follows easily.

Recall the following well known result:

Lemma 6.8. Let I be a non-zero G-stable ideal in O(𝔤). Then, I^{G} ≠ 0.

Proof. Since I contains a product of prime ideals, we may assume that I is prime. If d ∈ I there is nothing to prove. If d ∉ I, there exists a generic point in the variety of zeroes of I and, by [12, Theorem 0.5], I is generated by I ∩ O(𝔤)^{G}.

We can now prove the main result of this section:

Theorem 6.9. Fix λ ∈ 𝔥^{∗} and let

𝒩_{λ} = D(𝔤)/(D(𝔤)τ(𝔤) +_{∑p∈S(𝔤)_{G}}D(𝔤)(p − p(λ))).

Then, as (D(𝔤), W_{λ})-modules:

𝒩_{λ} = ⨁_{χ∈W_{λ̂}} 𝒩_{λ,χ} ⊗_{ℂ} V_{χ}^{∗},

where each 𝒩_{λ,χ} = 𝒩 ⊗_{D(𝔤)^{G}} V^{χ} is a simple D(𝔤)-module. Moreover, if χ ≠ ψ, then 𝒩_{λ,χ} ≅ 𝒩̸_{λ,ψ}.

Proof. The main aim of the proof is to show that

(6.4) If L is a non-zero submodule of 𝒩_{λ}, then L^{G} ≠ 0.

Indeed, suppose that (6.4) holds. By Lemma 6.2, it remains to prove that the 𝒩_{λ,χ} are simple and non-isomorphic. Suppose that 𝒩_{λ,χ} has a non-zero submodule L. Since L ⊆ 𝒩_{λ,χ} ⊆ 𝒩_{λ}, (6.4) implies that L^{G} ≠ 0. By Theorem 3.4, L^{G} is then a non-zero submodule of the simple D(𝔥)^{W} -module (𝒩_{λ,χ})^{G} = V^{χ}. Thus, L^{G} = V^{χ} and hence L ⊇ D(𝔤)V^{χ} = 𝒩_{λ,χ}. Thus, 𝒩_{λ,χ} is simple. Lemma 6.1 implies that (𝒩_{λ,χ})^{G} = {θ ∈ 𝒩_{λ,χ} : τ(𝔤) θ = 0}. Hence, if 𝒩_{λ,χ} ≅ 𝒩_{λ,ψ}, then (𝒩_{λ,χ})^{G} ≅ (𝒩_{λ,ψ})^{G}, as D(𝔤)^{G}-modules. Thus, χ = ψ, by Theorem 3.4(i) and Lemma 6.2(ii).

Thus, it remains to prove (6.4). By Corollary 6.6, L ⊆ L_{𝒞}. Moreover, since d ∈ O(𝔤)^{G}, the action of G on 𝒟 extends to an action on 𝒟_{𝒞} and hence to L_{𝒞}. If 0 ≠ f ∈ L_{𝒞}^{G}, then some d^{m}f ∈ L^{G} and hence L^{G} ≠ 0 (Corollary 6.6). Thus, it suffices to prove:

(6.5) If L is a non-zero submodule of 𝒩_{λ}, then L_{𝒞}^{G} ≠ 0.

We next need to study 𝒩_{𝒞} = 𝒟_{𝒞} /𝒟_{𝒞} τ(𝔤), for which we use the notation of Lemma 6.7. Fix a lexicographic ordering ⪯ on ℕ^{ℓ}. Then, by Lemma 6.7, any element α ∈ 𝒩_{𝒞} can be uniquely written α = a_{j}Θ^{j} + ∑_{i≺j} a_{i}Θ^{i}, for some a_{i} ∈ O(𝔤′). For i ∈ ℕ^{ℓ}, set

F_{i}𝒩_{𝒞} = ∑_{j⪯i} O(𝔤′)Θ^{j} ⊂ 𝒩_{𝒞} .

Since the {θ_{i}} are G-invariant, each F_{i}𝒩_{𝒞} is an (O(𝔤′), G)-module. It follows from Lemma 6.7 that the {F_{i}𝒩_{𝒞} } form a filtration on 𝒩_{𝒞}. Let P be a 𝒟_{𝒞} -submodule of 𝒩_{𝒞} and set F_{i}P = P ∩ F_{i}𝒩_{𝒞}. By Lemma 6.1, the {F_{i}P } provide a filtration of P by (O(𝔤′), G)-modules that are locally finite as G-modules. For j ∈ ℕ^{ℓ}, set gr_{j} P = F_{j}P/(∑_{i≺j} F_{i}P ). Thus, gr_{j} P ⊆ gr_{j} 𝒩_{𝒞} ≅ O(𝔤′)Θ^{j}. Observe that, since G is reductive, F_{q}P ≅ gr_{q} P ⊕ ∑_{i≺q} F_{i}P as G-modules. Thus, (F_{q}P)^{G} ≠ 0 whenever (gr_{q} P)^{G} ≠ 0. Suppose that gr_{q} P ≠ 0. Then gr_{q} P = I Θ^{q} for some nonzero, G-stable ideal I of O(𝔤′). Lemma 6.8 implies that I^{G} ≠ 0, and so Lemma 6.7 implies that (gr_{q} P)^{G} = I^{G} Θ^{q} ≠ 0. To summarize, we have proved:

gr_{q} P ≠ 0 ⟹ (F_{q}P)^{G} ≠ 0 (6.6)

Let Q ⊊ P be submodules of 𝒩_{𝒞} and assume that P/Q is a torsion-free O(𝔤′)- module. Pick q ∈ ℕ^{ℓ} minimal such that F_{q}Q ⊊ F_{q}P. We claim that F_{q}Q = 0. If not, we may pick b = βΘ^{q} + b′ ∈ F_{q}Q and f = αΘ^{q} + f′ ∈ F_{q}P \ F_{q}Q, with 0 ≠ β, α ∈ O(𝔤′) and b′, f′ ∈ ∑_{i≺q} F_{i}𝒩′. Then, βf − αb = βf′ − αb′ ∈ F_{j}𝒩′ ∩ P = F_{j}P for some j ≺ q. Thus, βf ∈ αb + F_{j}Q ⊆ Q. Since P/Q is torsion-free, this forces β = 0 and a contradiction. Thus, F_{q}Q = 0.

Finally, write 𝒩′_{λ} = (𝒩_{λ})_{𝒞} = 𝒟_{𝒞} /Q, where Q ⊇ 𝒟_{𝒞} τ(𝔤) denotes the annihilator of the natural generator of 𝒩′_{λ}. Let L_{𝒞} = P/Q ⊆ 𝒩′_{λ} be a non-zero submodule of 𝒩′_{λ}. By Corollary 6.6, L_{𝒞} ⊆ 𝒩′_{λ} is a torsion-free O(𝔤′)-module and so the comments of the last paragraph apply. If q is chosen as there, then F_{q}Q = 0 and gr_{q} P ≠ 0. Thus, (6.6) implies that L_{𝒞}^{G} ⊇ (F_{q}P)^{G} ≠ 0. By (6.5), this completes the proof of the theorem.

Let R_{λ} denote the full subcategory of D(𝔤)-mod generated by 𝒩_{λ} or, equivalently, by the modules 𝒩_{λ,χ}, χ ∈ W_{λ}. We end this section by giving two alternative descriptions of this category.

Corollary 6.10. Set p_{λ} = ∑_{p∈S(𝔤)^{G}} S(𝔤)^{G}(p − p(λ)). The category R_{λ} is the subcategory of D(𝔤)-modules M such that:

1. M is a compatible (D(𝔤), G)-module, finitely generated as a D(𝔤)-module;

2. M = D(𝔤)M^{G};

3. For all x ∈ M, there exists k ∈ ℕ such that p_{λ}^{k}x = 0.

Proof. It is clear that any object in R_{λ} satisfies these conditions. Conversely, let M satisfy (1), (2) and (3). Since M is finitely generated we may write M = D(𝔤)E for some finite dimensional subspace E ⊂ M^{G}. Then, X = D(𝔤)^{G}E can be viewed (through δ) as a finitely generated D(𝔥)^{W} -module. It follows from the hypotheses that X ∈ C_{λ}. Hence, 𝒩_{λ}⊗_{D(𝔥)^{W}} X ∈ R_{λ} and there is a natural surjection 𝒩_{λ} ⊗_{D(𝔥)^{W}} X ↠ M. This proves that M ∈ R_{λ}.

It is implicit in Theorem 6.9 that the categories R_{λ} and W_{λ}-mod are equivalent. We make this explicit in the next result, in which we freely use the notation of Section 5 for the case V = 𝔥. We state the result using the functor Sol, although one could as easily use DR.

Corollary 6.11. The category R_{λ} is equivalent to (W_{λ}-mod)^{op}, through the functor M → Sol(M^{G}) = Hom_{D(𝔥)^{W}} (M^{G}, O(𝔥)e^{λ}). Moreover, if M ∈ R_{λ},

(i) Sol(M^{G}) ≅ Hom_{D(𝔤)}(M, 𝒩_{λ}) as W_{λ}-modules; (ii) When M is a quotient of 𝒩_{λ}, Sol(M^{G}) identifies naturally with a W_{λ}-sub-

module of ℋ(λ)e^{λ}; (iii) End_{D(𝔤)} 𝒩_{λ} ≅ ℂ[W_{λ}] as (W_{λ} × W_{λ})-modules.

Proof. By Theorem 6.9, the functor M → M^{G} provides an equivalence of categories R_{λ} ≈ C_{λ}. Thus, by Proposition 5.1, the map M ↦ Sol(M^{G}) gives an equivalence R_{λ} ≈ (W_{λ}-mod)^{op}. The proof now follows easily from the results of Section 5 and the details are left to the interested reader.

7. Fourier transforms and the Springer correspondence

As has been remarked in the introduction, the module 𝒩_{λ} was defined and intensively studied in [10]. In that paper, 𝒩_{λ} is constructed as a “Fourier transform” of a second module

ℳ_{λ} = D(𝔤)/(D(𝔤)τ(𝔤) +_{∑q∈S(𝔤_{∗})_{G}}D(𝔤)(q − q(λ))). (7.1)

(see [10, (6.1.2)]). In this section, we apply our results to give a detailed description of ℳ_{λ} and use this to obtain an analogue of Springer correspondence for nonnilpotent orbits.

The Fourier transformation is defined as follows: Use the G-invariant bilinear form κ to define an isomorphism κ : 𝔤 \underrightarrow{∼} 𝔤^{∗}. Since one has a G-module isomorphism D(𝔤) ≅ S(𝔤^{∗}) ⊗ S(𝔤), κ induces an algebra automorphism F of D(𝔤) by F (f) = −κ^{−1}(f) and F (x) = κ(x) for f ∈ 𝔤^{∗} and x ∈ 𝔤. It is easily checked that F is a G-automorphism of D(𝔤) such that F (τ(x)) = τ(x) for all x ∈ 𝔤. Given a D(𝔤)- module M, define the Fourier transform M^{F} of M to be the abelian group M with multiplication defined by a ∘ m = F (a)m, for a ∈ D(𝔤) and m ∈ M. Then, as in [10, Section 6], 𝒩_{λ} is the Fourier transform 𝒩_{λ} = ℳ_{λ}^{F}.

Using the automorphism F, Theorem 6.9 immediately gives the following generalization of [10, Theorem 5.3].

Corollary 7.1. For each χ ∈ W_{λ}, set ℳ_{λ,χ} = 𝒩_{λ,χ}^{F^{−1}}. As (D(𝔤), W_{λ})-modules:

ℳ_{λ} = ⨁_{χ∈W_{λ̂}} ℳ_{λ,χ} ⊗_{ℂ} V_{χ}^{∗},

where each ℳ_{λ,χ} is a simple D(𝔤)-module. If χ ≠ ψ, then ℳ_{λ,χ} ≅ ℳ̸_{λ,ψ}. □

This decomposition of ℳ_{λ} is also related to the G-orbit structure of 𝔤, for which we need some definitions. Let π : 𝔤 ↠ 𝔤/G \underrightarrow{∼} 𝔥/W be the natural projection from 𝔤 to the space of W -orbits in 𝔥. Set N(λ) = π^{−1}(W.κ^{−1}(λ)) ⊂ 𝔤. By [12, Theorem 3], N(λ) is a finite union of G-orbits:

N(λ) = \overline{O_{λ}^{r}} = O_{λ}^{r} ∪ ⋯ ∪ O_{λ}^{s},

where dim O_{λ}^{r} = n − ℓ and O_{λ}^{s} = G.κ^{−1}(λ) is the unique closed orbit in N(λ). In the sequel, we adopt the notation of [2] for concepts related to algebraic D-modules. In particular, for any D(𝔤)-module M, one defines the closed subset Supp M ⊆ 𝔤 to be the support of the O(𝔤)-module M.

Corollary 7.2. The support of ℳ_{λ,χ} is the closure of an orbit O(χ) ⊆ N(λ).

Proof. Note first that each ℳ_{λ,χ} is a compatible (D(𝔤), G)-module (use Lemma 6.1 and the equivariance of F). It therefore follows from (7.1) that Supp ℳ_{χ} is a closed G-stable subset of N(λ). But, since ℳ_{λ,χ} is simple, its support is irreducible, (see the proof of [1, Lemmas 3.3.16 and 3.3.17]). Therefore, Supp ℳ_{λ,χ} must be the closure of a single orbit contained in N(λ).

As in [11], one can relate the simple modules ℳ_{λ,χ} to connections on the orbits contained in N(λ), and interpret the equivalence R_{λ} ≈ W_{λ}-mod through a “Springer correspondence”.

Let M = ℳ_{λ,χ} be a simple quotient of ℳ_{λ} and, by Corollary 7.2, write Supp M = \overline{O}, for the appropriate orbit O = O(χ). If ı : O ↪ 𝔤 is the natural embedding, then [2, Proposition 10.4] implies that ı^{!}M is a connection, E, on a dense affine Zariski open subset of O. Note that, since O is a G-space, ı^{!}M is a connection on O. Moreover, ı^{!}M is regular holonomic, and is uniquely determined by a representation ψ ∈ A(O) , [18, Lemma 2.1], where A(O) = G^{x}/(G^{x})^{0}, x ∈ O, is the component group of O. Then, by [2, Theorem 10.6], M is of the form ı_{!+}E (this is a regular holonomic module, minimal extension of E in the algebraic category). Thus, we have proved:

Corollary 7.3. The module ℳ_{λ,χ} is uniquely determined by the associated datum (O, ψ), where O an orbit in N(λ) and ψ ∈ A(O) . Consequently, any representation χ ∈ W_{λ} is also uniquely determined by this datum; written χ = σ(O, ψ).

When λ = 0, the correspondence (O, ψ) → σ(O, ψ) is closely related to the Springer correspondence (see [5], [10, Section 8] and [19, Theorem 6.10]).

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Département de Mathématiques, Université de Poitiers, 86022 Poitiers, France.

E-mail address: levasseu@mathlabo.univ-poitiers.fr

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.

E-mail address: jts@math.lsa.umich.edu