EQUIVARIANT D-MODULES ATTACHED TO NILPOTENT ORBITS IN A SEMISIMPLE LIE ALGEBRA

T. LEVASSEUR

Abstract. Let 𝔤_{u} be a compact Lie algebra and 𝔤 be its complexification. Let ζ^{− \frac{1}{2}} be the inverse, on the set of regular elements of 𝔤_{u}, of a square root of the discriminant of 𝔤. Generalizing a result of W. Lichtenstein in the case 𝔤_{u} = 𝔰𝔲(n, ℂ) or 𝔰𝔬(n, ℝ), we prove that ∂(q).ζ^{− \frac{1}{2}} is non zero for all harmonic polynomials q ∈ S(𝔤)\{0}. This fact is deduced from results about equivariant D-modules supported on the nilpotent cone of 𝔤.

1. Introduction

Let G_{u} be a connected compact semisimple group with Lie algebra 𝔤_{u}. Denote by 𝔤 = 𝔤_{u} ⊗_{ℝ} ℂ the complexification of 𝔤_{u} and let G be the adjoint group of 𝔤. Choose a Cartan subalgebra 𝔥_{u} of 𝔤_{u} and set 𝔥 = 𝔥_{u} ⊗_{ℝ} ℂ. Fix a positive system of roots R^{+} for 𝔤 with respect to 𝔥_{u}.

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). For q ∈ S(𝔤), let ∂(q) ∈ D(𝔤) be the corresponding differential operator 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). Let S_{+}(𝔤^{∗})^{G} be the set of invariant elements without constant term and let H(𝔤^{∗}) be the space of harmonic polynomials on 𝔤. Then H(𝔤^{∗}) is a Gstable complement of S_{+}(𝔤^{∗})^{G}S(𝔤^{∗}) in S(𝔤^{∗}), see [9]. We adopt a similar notation for polynomial functions on 𝔤^{∗}, hence S(𝔤) = H(𝔤) ⊕ S_{+}(𝔤)^{G}S(𝔤).

Let ζ ∈ S(𝔤^{∗}) be the discriminant of 𝔤, see [5, §7] or [18, §4]. Hence ζ ∈ S(𝔤^{∗})^{G} corresponds, through the Chevalley isomorphism, to π^{2}, where π = ∏_{α∈R^{+}} α ∈ S(𝔥^{∗}). Define the set of generic elements by 𝔤′ = {x ∈ 𝔤 : ζ(x) ≠ 0} and set 𝔤′_{u} = 𝔤′ ∩ 𝔤_{u}. Then, one can define a function ζ^{\frac{1}{2}} , real analytic on 𝔤′_{u}, and one considers the “potential” ζ^{− \frac{1}{2}} on 𝔤′_{u}. Clearly, the algebra A(𝔤′_{u}) of analytic functions (with complex values) on 𝔤′_{u} has a natural D(𝔤)-module structure. W. Lichtenstein has proved the following result:

Theorem 1.1. ([14, Theorem 2]) Suppose that 𝔤_{u} = 𝔰𝔲(n, ℂ) or 𝔰𝔬(n, ℝ). Then the map

M : H(𝔤) → A(𝔤′_{u}), q ↦ ∂(q).ζ^{− \frac{1}{2}}

is injective.

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

Key words and phrases. semisimple Lie algebra, equivariant D-module, nilpotent orbit, Weyl group representation.

Actually, [14] states the result with a map M^{∗}(p) = ∂(p).ζ^{− \frac{1}{2}} , p ∈ H(𝔤^{∗}), after identifying 𝔤 with 𝔤^{∗} through the Killing form B. These elements are called multipole potentials. In [14, §3] it is shown that M(p) is the term of lowest degree in the asymptotic expansion, when λ → +∞, of the integral

∫_{O} e^{−iB(λx,y)}p(y)dμ(y),

where O ⊂ 𝔤_{u} is a regular orbit and dμ is a G_{u}-invariant measure on O.

One aim of this note is to establish Theorem 1.1 for an arbitrary compact Lie algebra 𝔤_{u}. We will show in §2 that this result is consequence of the decomposition of the invariant holonomic systems given in [12], which we now recall. Let W be the Weyl group associated to (𝔤, 𝔥). Denote by W the set of isomorphism classes of irreducible W -modules and, for χ ∈ W , let V_{χ} be a W -module in the class of χ. Set

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

Then, ℳ and 𝒩 are semisimple holonomic D(𝔤)-modules and the full subcategory of D(𝔤)-modules generated by 𝒩, or ℳ, is equivalent to the category of W -modules. Let 𝒩_{χ} and ℳ_{χ} be the simple submodules of 𝒩 and ℳ which correspond to V_{χ} through these equivalences (see §2 for details).

Then, in the general case, Theorem 1.1 follows from (and is explained by) the following result.

Theorem 1.2. Let triv ∈ W be the trivial character. Then:

(i) 𝒩_{triv} ≅ D(𝔤).ζ^{− \frac{1}{2}} ; (ii) The support of ℳ_{triv} is the nilpotent cone of 𝔤.

This theorem is presumably implicit in [6, 18]. Its significance for this paper is to show how the results of [12] can be applied to prove the generalization of Theorem 1.1, see Theorem 2.8.

In order to show that Theorem 1.2 (which deals with the regular nilpotent orbit) is a particular case of more general results, we present in §3 a construction, due to Hotta & Kashiwara [6] and Wallach [18], which associates in a natural way to any nilpotent orbit O ⊂ 𝔤 a D(𝔤)-module ℳ_{χ_{O}} (equivalently, a representation χ_{O} of W).

We end this note with some comments regarding the correspondence, via invariant holonomic systems, between nilpotent orbits and irreducible representations of the Weyl group [12, §7]. In particular, we show that the inverse image of ℳ_{χ_{O}} on O is isomorphic (as a D_{O}-module) to the standard module O_{O}; equivalently, χ_{O} corresponds to the pair (O, triv) where triv is the trivial representation of the component group A(O) of the orbit O.

2. The modules 𝒩_{triv} and ℳ_{triv}

We begin this section by recalling some facts from [12]. Set n = dim 𝔤, ℓ = dim 𝔥 and ν = #R^{+}, hence n = 2ν + ℓ. Let {e_{i}}_{1⩽i⩽n} be an orthonormal basis of 𝔤 with respect to B 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}. Recall that there exists an algebra homomorphism, defined by Harish-Chandra,

δ : D(𝔤)^{G} ⟶ D(𝔥)^{W}

which 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}, see [11]. This enables one to identify, through δ, modules over D(𝔤)^{G}/I with D(𝔥)^{W} -modules. Let R be the full subcategory of D(𝔤)-modules generated by the module 𝒩. Then, if M ∈ R, the D(𝔤)^{G}-module M^{G} can be considered as a D(𝔥)^{W} -module. Let

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

be the set of harmonic polynomials on 𝔥. The vector space H is graded by the H^{j} = H ∩ S^{j}(𝔥^{∗}) and, as a W -module, H identifies with the left regular representation.

The following theorem summarizes some of the results contained in [12]. (In the notation of [12], we are only concerned with the case λ = 0 and so the modules ℳ_{0} and 𝒩_{0} of [12] are denoted by ℳ, respectively 𝒩.)

Theorem 2.1. (1) ([6, Theorem 5.3], [12, Theorem A]) ℳ and 𝒩 are semisimple (D(𝔤), W)-modules and, as such, have the following decompositions

ℳ ≅ ⨁_{χ∈W}ˆ ℳ_{χ} ⊗_{ℂ} V_{χ}, 𝒩 ≅ ⨁_{χ∈W}ˆ 𝒩_{χ} ⊗_{ℂ} V_{χ}

where the ℳ_{χ}, resp. 𝒩_{χ}, are pairwise non-isomorphic simple D(𝔤)-modules.

(2) ([12, Corollary 6.11]) The category R is equivalent to (W -mod)^{op}, through the functor M → Sol(M^{G}) = Hom_{D(𝔥)^{W}} (M^{G}, S(𝔥^{∗})). Moreover,

(i) when M is a quotient of 𝒩, Sol(M^{G}) identifies naturally with a W -submodule of H;

(ii) Sol(𝒩_{χ}^{G}) ≅ V_{χ}.

Corollary 2.2. Set L = D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤)^{G} + D(𝔤)(E_{𝔤} + ν). Then,

𝒩_{triv} ≅ D(𝔤)/L.

Proof. Clearly, N = D(𝔤)/L is a quotient of 𝒩. Observe that

N^{G} = D(𝔤)^{G}/(I + D(𝔤)^{G}S_{+}(𝔤)^{G} + D(𝔤)^{G}(E_{𝔤} + ν))
≅ D(𝔥)^{W} /(D(𝔥)^{W} S_{+}(𝔥)^{W} + D(𝔥)^{W} (δ(E_{𝔤}) + ν)).

It is easily checked that δ(E_{𝔤}) = E_{𝔥} − ν, (see, for example, [18, Proof of Theorem 6.1]). Therefore, as W -module,

Sol(N^{G}) ≅ {f ∈ H : E_{𝔥}.f = 0} = ℂ.

Thus Sol(N^{G}) = V_{triv} and the corollary follows from Theorem 2.1.

Corollary 2.3. The D(𝔤)-modules 𝒩_{triv} and D(𝔤).ζ^{− \frac{1}{2}} are isomorphic.

Proof. By [14, Theorem 1 and Corollary] ζ^{− \frac{1}{2}} is a G_{u}-invariant harmonic function on 𝔤′_{u}. Hence, τ(𝔤).ζ^{− \frac{1}{2}} = S_{+}(𝔤)^{G}.ζ^{− \frac{1}{2}} = 0. Observe that, since ζ is homogeneous of degree 2ν, E_{𝔤}.ζ^{− \frac{1}{2}} = −ν ζ^{− \frac{1}{2}} . The result then follows from Corollary 2.2.

Remark. Let sgn be the sign representation of W. Recall that the unique copy of V_{sgn} in H is H^{ν} = ℂπ. The modules 𝒩_{sgn} and ℳ_{sgn} are easy to describe:

𝒩_{sgn} = D(𝔤).1 = D(𝔤)/D(𝔤)𝔤, ℳ_{sgn} = D(𝔤).T_{0} = D(𝔤)/D(𝔤)𝔤^{∗},

where T_{0} is the Dirac distribution at 0 on a real form of 𝔤. This can be proved as follows. Set:

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

Then, the D(𝔤)^{G}-module N^{G} is isomorphic to the D(𝔥)^{W} -module

D(𝔥)^{W} /(D(𝔥)^{W} S_{+}(𝔥)^{W} + D(𝔥)^{W} (E_{𝔥} − ν))

and it follows that Sol(N^{G}) = ℂπ. This proves N ≅ 𝒩_{sgn}. Clearly, D(𝔤).1 = D(𝔤)/D(𝔤)𝔤 is a factor of N, hence N = D(𝔤).1. The description of ℳ_{sgn} can be deduced from that of 𝒩_{sgn} by Fourier transform (see below), or from a general argument (see §3).

Recall [12, §7] that the modules 𝒩_{χ} are the Fourier transforms of the modules ℳ_{χ}. The Fourier transformation is defined as follows: Use the G-invariant bilinear form B to define an isomorphism B : 𝔤 \underrightarrow{∼} 𝔤^{∗}. Since one has a G-module isomorphism D(𝔤) ≅ S(𝔤^{∗}) ⊗ S(𝔤), B induces an algebra automorphism F of D(𝔤) by F (f) = −B^{−1}(f) and F (x) = B(x) for f ∈ 𝔤^{∗} and x ∈ 𝔤. Moreover, 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 given by a ▪ m = F (a)m, for a ∈ D(𝔤) and m ∈ M. Then, as in [6, §6], 𝒩 = ℳ^{F}. Notice that, by definition, 𝒩_{χ} = ℳ_{χ}^{F}. For example, one deduces from Corollary 2.2 that

ℳ_{triv} ≅ D(𝔤)/(D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤^{∗})^{G} + D(𝔤)(E_{𝔤} + ν + ℓ)).

Let χ ∈ W . Set

b(χ) = inf {j : [S^{j}(𝔥^{∗}) : V_{χ}] ≠ 0} = inf {j : [H^{j} : V_{χ}] ≠ 0} .

Lemma 2.4. ([18, Lemma 6.5]) The highest eigenvalue of E_{𝔤} on ℳ_{χ}^{G} is ν−n−b(χ); this eigenvalue occurs with multiplicity [H^{b(χ)} : V_{χ}].

Proof. Recall [12] that as a D(𝔥)^{W} -module, 𝒩_{χ}^{G} ≅ V^{χ}, where

V^{χ} = Hom_{W} (V_{χ}, S(𝔥^{∗})).

Since E_{𝔥} acts semisimply on S(𝔥^{∗}), the same is true on V^{χ}. Moreover, the smallest eigenvalue of E_{𝔥} on V^{χ} is b(χ). Indeed: realize V^{χ} as a D(𝔥)^{W} -submodule of S(𝔥^{∗}), and note that ℂW.u ≅ V_{χ} for all u ∈ V^{χ}; this implies that the degree of each u is ⩾ b(χ). Observe also that the simple module V^{χ} occurs in S(𝔥^{∗}) with multiplicity dim V_{χ}, hence the multiplicity of the eigenvalue b(χ) is [H^{b(χ)} : V_{χ}].

Now, since δ(E_{𝔤}) = E_{𝔥} − ν, the smallest eigenvalue of E_{𝔤} on 𝒩_{χ}^{G} is b(χ) − ν. The lemma is then consequence of F (E_{𝔤}) = −E_{𝔤} − n and 𝒩_{χ} = ℳ_{χ}^{F}.

Let N(𝔤) be the set of nilpotent elements in 𝔤. Then, N(𝔤) is a finite union of G-orbits and its defining ideal is generated by S_{+}(𝔤^{∗})^{G}, see [9]. If O = G.x ⊂ 𝔤 is a nilpotent orbit, one sets [6, §7]

λ_{O} = \frac{1}{2} dim O − dim 𝔤 = −\frac{1}{2}(dim 𝔤 + dim 𝔤^{x}),

where 𝔤^{x} is the stabilizer of x. Note that if \overline{O_{1}} ⊂ \overline{O_{2}}, then λ_{O_{1}} ⩽ λ_{O_{2}}, with equality if and only if O_{1} = O_{2}.

The support of a coherent D(𝔤)-module M will be denoted by Supp M. (We refer to [4] for definitions related to D-modules.) The next proposition provides an analogue for G-equivariant D(𝔤)-modules of [1, Corollary 3.9] and [18, Lemma 6.2]. A version of this result is proved, in slightly more generality, in [13].

Proposition 2.5. Let M be a finitely generated G-equivariant D(𝔤)-module such that Supp M ⊂ N(𝔤).

(i) Set λ_{M} = max{λ_{O} : O nilpotent orbit contained in Supp M}; then every eigenvalue of E_{𝔤} on M^{G} is ⩽ λ_{M}.

(ii) Assume that: Supp M = \overline{O}, O nilpotent orbit, and M = D(𝔤).v with 0 ≠ v ∈ M^{G} such that E_{𝔤}.v = λ_{O}v. Then, the λ_{O}-eigenspace of E_{𝔤} in M^{G} has dimension one.

Proof. Recall first that, since M is G-equivariant, the differential of the G-action on M coincides with the multiplication by the elements τ(ξ), ξ ∈ 𝔤 (see, for example, [17, Proposition 2.6] and Appendix A). In particular, τ(𝔤).v = 0 for all v ∈ M^{G}; it then follows from [16, §6], or [12, Lemma 6.1], that any subquotient of D(𝔤).v is G-equivariant.

(i) Let 0 ≠ v ∈ M^{G} be an eigenvector of E_{𝔤}. Set N = D(𝔤).v. By the previous remark and the inclusion Supp N ⊂ N(𝔤), the annihilator of v in D(𝔤) contains τ(𝔤) and a power of S_{+}(𝔤^{∗})^{G}. Then, [13, Remark 3.6] applied to N implies that the eigenvalue of v is ⩽ λ_{N} ⩽ λ_{M} (see [13, Remark following Lemma 3.4]).

(ii) Let v′ ∈ M^{G} be such that E_{𝔤}.v′ = λ_{O}v′. By [13, Lemma 3.4], there exists c ∈ ℂ such that M′ = D(𝔤).(v′ − cv) satisfies Supp M′ ⊊ \overline{O}. Since Supp M′ is a finite union of nilpotent orbits, (i) yields λ_{M′} < λ_{O}. Suppose that v′ − cv ≠ 0, so that v′ − cv is a non-zero eigenvector for λ_{O}. Then, by (i) again, λ_{O} ⩽ λ_{M′} and a contradiction.

As explained in [12, §7], ℳ_{χ} is a G-equivariant D(𝔤)-module, i.e. compatible in the sense of [12, §6], and Supp ℳ_{χ} = \overline{O(χ)} is the closure of some nilpotent orbit.

Corollary 2.6. One has: b(χ) + \frac{1}{2} dim O(χ) ⩾ ν.

Proof. By Lemma 2.4 and Proposition 2.5, ν − n − b(χ) ⩽ λ_{O(χ)} = \frac{1}{2} dim O(χ) − n, as desired.

Recall [9] that N(𝔤) is the closure of a nilpotent orbit, O^{reg}, of dimension 2ν. As observed in [6, §5] one has the following result, which proves Theorem 1.2(ii).

Proposition 2.7. Supp ℳ_{triv} = N(𝔤).

Proof. Since b(triv) = 0, Corollary 2.6 implies that dim O(triv) ⩾ 2ν, forcing dim O(triv) = 2ν and O(triv) = O^{reg}.

We can now complete the proof of Theorem 1.1 in the general case.

Theorem 2.8. The multipole mapping

M : H(𝔤^{∗}) → A(𝔤′_{u}), p ↦ ∂(F^{−1}(p)).ζ^{− \frac{1}{2}}

is injective.

Proof. Let L = D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤)^{G} + D(𝔤)(E_{𝔤} + ν) be as in Corollary 2.2. Assume that M(p) = 0 for some p ∈ H(𝔤^{∗}); by Corollary 2.3, this is equivalent to saying that ∂(F^{−1}(p)) ∈ L. Thus p = F (∂(F^{−1}(p))) ∈ F (L). Note that ℳ_{triv} = 𝒩_{triv}^{F^{−1}} ≅ D(𝔤)/F (L). Therefore, by definition of support, Supp ℳ_{triv} ⊂ p^{−1}({0}) ⊂ 𝔤 and Proposition 2.7 then implies that N(𝔤) ⊂ p^{−1}({0}). Since the defining ideal of the variety N(𝔤) is the prime ideal generated by S_{+}(𝔤^{∗})^{G} (see [9]) it follows that p ∈ S_{+}(𝔤^{∗})^{G}S(𝔤^{∗}). Hence p ∈ H(𝔤^{∗}) ∩ S_{+}(𝔤^{∗})^{G}S(𝔤^{∗}) = 0.

Remark. In the case when 𝔤_{u} = 𝔰𝔬(3, ℝ) the previous result is due to J. C. Maxwell, who used it in his work on electro-magnetism. We would like to thank V. Ginzburg for this historical remark.

3. Additional results

This section collects various results from [6, 18] which make more precise the correspondence (implied by Theorem 2.1) between nilpotent orbits and representations of the Weyl group.

We begin with a construction, using the results of [18], of some simple summands of ℳ. Let 𝔤_{0} be a real form of 𝔤 with adjoint group G_{0}. Let x ∈ 𝔤_{0} be a nilpotent element and set O_{x} = G.x. Define [18, §6] a distribution T_{x} on 𝔤_{0} by the formula

T_{x}(f) = ∫_{G_{0}.x} fdμ_{x}, f ∈ C_{c}^{∞}(𝔤_{0})

(dμ_{x} being the canonical G_{0}-invariant measure on G_{0}.x). Then, T_{x} is G-invariant and φ.T_{x} = 0 if φ ∈ O(𝔤) vanishes on O_{x}, thus Supp D(𝔤).T_{x} ⊂ \overline{O_{x}}. Notice that E_{𝔤}.T_{x} = λ_{O_{x}}T_{x}, see [18, Lemma 6.4]. The next result should be compared to [6, Proposition 8.4.1].

Proposition 3.1. There exists χ_{x} ∈ W such that D(𝔤).T_{x} ≅ ℳ_{χ_{x}}. Furthermore, the highest eigenvalue of E_{𝔤} on ℳ_{χ_{x}}^{G} is λ_{O_{x}}, and Supp D(𝔤).T_{x} = \overline{O_{x}}.

Proof. Set M = D(𝔤).T_{x} and observe that M is a quotient of ℳ. By [18, Theorem 6.3] and Theorem 2.1, M ≅ ℳ_{χ_{x}} for some χ_{x} ∈ W . Note also that, by [18, Theorem 6.3] again, b(χ_{x}) = \frac{1}{2}(dim 𝔤^{x} − ℓ). Thus the second assertion follows from Lemma 2.4. Let \overline{O} be the support of M. By Proposition 2.5, λ_{O_{x}} ⩽ λ_{O}; therefore \overline{O} ⊂ \overline{O_{x}} yields the third assertion.

Corollary 3.2. Let 𝔤_{0} be a (split) real form of 𝔤 and x ∈ 𝔤_{0} ∩ O^{reg}. Then,

ℳ_{triv} ≅ D(𝔤).T_{x}.

Proof. By Proposition 3.1, D(𝔤).T_{x} ≅ ℳ_{χ_{x}}. Since λ_{O^{reg}} = −ν − ℓ, we obtain from the same proposition and Lemma 2.4 that b(χ_{x}) = 0. Hence χ_{x} = triv.

Remark. Let x ∈ 𝔤_{0} be nilpotent. Denote by T̂_{x} the Fourier transform of the distribution T_{x}. Let 𝔥_{0} ⊂ 𝔤_{0} be a real form of 𝔥. Then, by [18, Theorem 6.7], the restriction of T̂_{x} to each connected component C of 𝔥_{0} ∩ 𝔤′ writes p_{C}/π for some polynomial p_{C} ∈ H^{b(χ_{x})}. In particular, when x is regular, b(χ_{x}) = 0 implies that T̂_{x}_{|C} = p_{C}/π for some constant p_{C}. (Recall that ζ^{− \frac{1}{2}} = ±1/\sqrt{π^{2}} on each connected component of 𝔥_{u} ∩ 𝔤′.)

We now recall how one can attach, in a natural way, to any nilpotent orbit O a representation χ_{O} of W. This result is due to Hotta & Kashiwara [6, Corollary 7.1.5] but we include a proof for completeness. If O ⊂ 𝔤 is a nilpotent orbit, we denote by J(O) ⊂ O(𝔤) the prime ideal defining \overline{O}.

Theorem 3.3. Let O be a nilpotent orbit in 𝔤.

(1) There exists χ_{O} ∈ W such that

ℳ_{χ_{O}} ≅ D(𝔤)/(D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤^{∗})^{G} + D(𝔤)J(O)^{k} + D(𝔤)(E_{𝔤} − λ_{O})) for all k ∈ ℕ^{∗}.

(2) Let 𝔤_{0} be a real form of 𝔤 and x ∈ 𝔤_{0} be a nilpotent element. Set O_{x} = G.x. Then, χ_{x} = χ_{O_{x}}, i.e. D(𝔤).T_{x} ≅ ℳ_{χ_{Ox}}.

Proof. Denote by M_{k} the D(𝔤)-module on the right hand side of the isomorphism in (1). Observe that M_{k} is a quotient of M_{k+1} and

M_{1} = D(𝔤)/(D(𝔤)τ(𝔤) + D(𝔤)J(O) + D(𝔤)(E_{𝔤} − λ_{O})) (3.1)

Assume that M_{k} ≠ 0 and denote by v its canonical generator (the class of 1). Notice that E_{𝔤}.v = λ_{O}v. It is clear that M_{k} is a quotient of ℳ with support contained in \overline{O}. Therefore, by Theorem 2.1, M_{k} ≅ ⨁_{χ∈W}ˆ ℳ_{χ}^{(m_{χ})} for some integers m_{χ}. Recall that Supp ℳ_{χ} = \overline{O(χ)} and that the eigenvalues of E_{𝔤} on ℳ_{χ}^{G} are ⩽ λ_{O(χ)}, see Proposition 2.5(i). Suppose that m_{χ} ≠ 0. Then M_{k} surjects onto ℳ_{χ}; therefore ℳ_{χ} = D(𝔤).v̄ with E_{𝔤}.v̄ = λ_{O}v̄. Hence λ_{O} ⩽ λ_{O(χ)} and, since \overline{O(χ)} ⊂ \overline{O}, we obtain that O(χ) = O. It follows that Supp M_{k} = \overline{O}. But, by Proposition 2.5(ii), the λ_{O}-eigenspace of E_{𝔤} in M_{k}^{G} has dimension 1, thus M_{k} ≅ ℳ_{χ} for some χ = χ_{O}.

To complete the proof of the theorem it suffices to show that, if 𝔤_{0} is a real form of 𝔤 such that O ∩ 𝔤_{0} ≠ ∅, then D(𝔤).T_{x} ≅ M_{1} for all x ∈ O ∩ 𝔤_{0}. Observe that

τ(𝔤).T_{x} = J(O).T_{x} = (E_{𝔤} − λ_{O}).T_{x} = 0.

Therefore, D(𝔤).T_{x} is a quotient of M_{1}. This proves that M_{1} ≠ 0 and, by the first part of the proof, D(𝔤).T_{x} ≅ ℳ_{χ} ≅ M_{k} for all k ∈ ℕ^{∗}.

Remark. Set L(O) = D(𝔤)τ(𝔤)+D(𝔤)J(O)+D(𝔤)(E_{𝔤}−λ_{O}). From (3.1) we get that ℳ_{χ_{O}} = D(𝔤)/L(O); thus, 𝒩_{χ_{O}} = ℳ_{χ_{O}}^{F} = D(𝔤)/F^{−1}(L(O)) with F^{−1}(L(O)) = D(𝔤)τ(𝔤)+D(𝔤)F^{−1}(J(O))+D(𝔤)(F^{−1}(E_{𝔤})−λ_{O}). By definition of F, the defining ideal of the closure of the (nilpotent) orbit B(O) ⊂ 𝔤^{∗} is I(O) = F^{−1}(J(O)). Hence,

𝒩_{χ_{O}} = D(𝔤)/(D(𝔤)τ(𝔤) + D(𝔤)I(O) + D(𝔤)(E_{𝔤} + \frac{1}{2} dim O)).

Setting K(O) = δ((D(𝔤)I(O))^{G}) and d_{O} = \frac{1}{2} codim_{N(𝔤)} \overline{O}, we obtain that

𝒩_{χ_{O}}^{G} ≅ D(𝔥)^{W} /(K(O) + D(𝔥)^{W} (E_{𝔥} − d_{O}))

as a D(𝔥)^{W} -module. Therefore, the representation χ_{O} is defined by

V_{χ_{O}} ≅ Sol(𝒩_{χ_{O}}^{G})
= {φ ∈ S^{d_{O}}(𝔥^{∗}) : ∀D ∈ K(O), D.φ = 0}
= {φ ∈ H^{d_{O}} : ∀D ∈ K(O), D.φ = 0}.

One can also observe that (in particular cases) the description of the left ideal L(O) can be simpler than the one given above. For example, when O = {0}, i.e. χ_{O} = sgn, the previous definition gives L({0}) = D(𝔤)𝔤^{∗} but we have seen in §2 that L({0}) = D(𝔤)τ(𝔤) + D(𝔤)S_{+}(𝔤^{∗})^{G} + D(𝔤)(E_{𝔤} + n).

We end this section by showing how the module ℳ_{χ_{O}} is related to the trivial representation of the component group of O via the “Springer correspondence”.

We first need to recall a few facts from [12, §7]. Let χ ∈ W and consider the module ℳ_{χ}. Set O = O(χ) and denote by ı : O ↪ 𝔤 the natural inclusion. Fix x ∈ O and set A(O) = G^{x}/G_{0}^{x}, where G_{0}^{x} is the identity component of the stabilizer G^{x} of x in G. We denote by m_{x} ⊂ O(𝔤) the maximal ideal associated to x. Then (see [4, 10.4 & 10.6]) ℳ_{χ}, or equivalently χ, is uniquely determined by the coherent connection ℒ_{χ} = ı^{!}ℳ_{χ}. This connection is, in turn, defined by an irreducible representation ψ ∈ A(O) on the geometric fibre ℒ_{χ}(x) = (ℒ_{χ})_{x}/m_{x}(ℒ_{χ})_{x} [7, Proposition 4.11.1]. We express these facts by setting χ = σ(O, ψ). Thus, we have a correspondence (O, ψ) → σ(O, ψ) from a subset of {nilpotent orbits O} × {ψ ∈ A(O) } onto W . This correspondence is related to the Springer correspondence, see [6, 18]. The results of Appendix A show how the representation ψ ∈ A(O) can be computed. We summarize below some of these results.

When O = {0}, A(O) is trivial and we have seen in §2 that sgn = σ({0}, triv). In the rest of this section we assume that x ≠ 0. Then, there exists y ∈ N(𝔤), z ∈ 𝔤 such that [z, x] = 2x, [z, y] = −2y, [x, y] = z. Let G^{ϕ} = {g ∈ G : g.x = x, g.y = y} be the centralizer of ℂx + ℂy + ℂz ≅ 𝔰𝔩(2, ℂ) and let G_{0}^{ϕ} be its identity component. By [2, Proposition 2.4], A(O) = G^{ϕ}/G_{0}^{ϕ} and G^{ϕ} is reductive. Note that we have the G^{ϕ}-stable decomposition 𝔤 = 𝔤^{y}⊕[𝔤, x]. Let {y_{1}, …, y_{m}} and {y_{m+1}, …, y_{n}} be coordinate functions on 𝔤^{y} and [𝔤, x] respectively. Define G^{ϕ}-stable ideals of O(𝔤) by J_{x} = (y_{m+1} − y_{m+1}(x), …, y_{n} − y_{n}(x)) and n_{x} = (y_{1} − y_{1}(x), …, y_{m} − y_{m}(x)). Set:

T = \{\overline{v} ∈ ℳ_{χ}/J_{x}ℳ_{χ} : v ∈ ℳ_{χ} and n_{x}.\overline{v} = 0\}

Observe that G^{ϕ} acts naturally on T by g.\overline{v} = \overline{g.v}.

Theorem 3.4. (1) The natural action of G^{ϕ} on T yields the irreducible representation ψ : A(O) → GL(T).

(2) Write ℳ_{χ} = D(𝔤).v with 0 ≠ v ∈ ℳ_{χ}^{G}. Assume that E_{𝔤}.v = λ_{O}v. Then,

T = ℂ \overline{v} and χ = χ_{O} = σ(O, triv).

Proof. (1) follows from Theorem A.4.

(2) By Corollary A.5 and the proof of [13, Lemma 3.4(i)] we obtain that n_{x}.\overline{v} = 0 and T = ℂ \overline{v}. Since τ(𝔤).v = 0, E_{𝔤}.v = λ_{O}v and Supp ℳ_{χ} = \overline{O}, it follows from Theorem 3.3 that ℳ_{χ} = ℳ_{χ_{O}}. Thus χ = χ_{O} and χ_{O} = σ(O, triv) is consequence of (1) and v ∈ ℳ_{χ}^{G}.

Remark. As pointed out by a referee and M. Van den Bergh, Theorem 3.4(2) can be proved in the following alternative way. Let Y be a non empty affine open subset of O and let ı : Y ↪ 𝔤 be the inclusion. We aim to show that ℳ_{χ_{O}} is isomorphic to the unique irreducible submodule of ı_{+}(O_{Y}), see [4, VII.10.5, VII.10.6]. Set X = 𝔤 \ (\overline{Y} \ Y). Then ı′ : Y ↪ X is a closed embedding, ȷ : X ↪ 𝔤 is open and ı_{+}(O_{Y}) = ȷ_{+}(ı′_{+}(O_{Y})). Recall [4, VI.7.1, VI.7.8] that ı′_{+}(O_{Y}) contains ω_{Y/X} := ω_{Y} ⊗_{O_{X}} ω_{X}^{−1}. The orbit O carries a G-invariant symplectic form β_{O}, thus

β_{Y}^{\frac{1}{2} dim O} is a non zero section of ω_{Y}. On the other hand, since 𝔤 is unimodular, ω_{𝔤} is generated by a G-invariant form dx. Hence,

γ = β_{Y}^{\frac{1}{2} dim O} ⊗ dx_{|X}^{−1}

gives a non zero section of ω_{Y/X}. It is not difficult to see that γ ∈ Γ(X, ı′_{+}(O_{Y})) extends to a section of ı_{+}(O_{Y}) and satisfies τ(𝔤).γ = (E_{𝔤} − λ_{O}).γ = J(O).γ = 0. It follows then from Theorem 3.3 that ℳ_{χ_{O}} ≅ D(𝔤).γ is the socle of the D(𝔤)- module ı_{+}(O_{Y}).

Appendix A.

The material of this section is taken from [10]. A lot of results stated here are probably well known but do not appear in the literature in the form we need. The proofs are very often straightforward (but lengthy) verifications and we will sometimes only give a sketch of them.

Let G be a linear algebraic group and 𝔤 = Lie(G). We first recall a few definitions about G-equivariant D-modules, see [3, 4, 7, 8, 17]. All the varieties considered are quasi-projective algebraic varieties defined over ℂ. If X is a smooth algebraic variety, O_{X} is the sheaf of regular functions on X and D_{X} the sheaf of differential operators. We refer to [4] for the basic properties of D_{X}-modules; all the D_{X}- modules encountered in the sequel will be quasi-coherent and the category of quasicoherent D_{X}-modules is denoted by Mod D_{X}. Suppose that Z = X × Y is the product of two varieties. Let M be an O_{X}-module and N be an O_{Y} -module, we will write M ⊠ N for the O_{Z}-module M ⊗_{ℂ} N.

Let F : Y → X be a morphism between smooth varieties. The comorphism of F is denoted by F^{#} and the inverse image of an O_{X}-module M by F^{∗}M = O_{Y} ⊗_{F^{#}} M. When X and Y are affine, there exists a natural map F^{#} : M → F^{∗}M, v ↦ F^{#}(v) = 1_{Y} ⊗_{F^{#}} v. Recall that if M ∈ Mod D_{X} one defines the inverse image of M by setting

F^{!}M =(D_{Y →X} ⊗_{D_{X}}^{L} M)[d_{Y,X}]

where D_{Y →X} = F^{∗}D_{X} = O_{Y} ⊗_{F^{#}} D_{X} and d_{Y,X} = dim Y − dim X. The inverse image is an object of the derived category D^{b}(D_{Y}) of Mod D_{Y}, and the construction of F^{!} extends to D^{b}(D_{X}). When F is smooth and M ∈ Mod D_{X}, F^{!}M is a complex concentrated in degree −d_{Y,X} and ℋ^{−d_{Y,X}} (F^{!}M) coincides, as an O_{Y} -module, with F^{∗}M.

Assume that X is a smooth G-variety, define: μ : G × X → X, (g, v) ↦ g.v (the action of G on X), μ_{G} : G × G → G, μ_{G}(g, h) = gh (the multiplication in G), ε_{X} : X ↪ G × X, ε_{X}(v) = (e, v), p_{2} : G × X → X, p_{2}(g, v) = v, p_{23} : G × G × X → G × X, p_{23}(g, h, v) = (h, v).

Let M ∈ Mod D_{X}; then M is a G-equivariant D_{X}-module if there exists an isomorphism of D_{G×X}-modules

θ : p_{2}^{∗}M = O_{G} ⊠ M \underrightarrow{∼} μ^{∗}M (A.1)

such that ε_{X}^{∗}(θ) = 1_{M}, (μ_{G} × 1_{X})^{∗}(θ) = (1_{G} × μ)^{∗}(θ) ∘ p_{23}^{∗}(θ). One can then construct the category of G-equivariant D_{X}-modules, which we denote by 𝔐(D_{X}, G). Notice that if F : Y → X is a G-equivariant morphism between smooth G-varieties and M ∈ 𝔐(D_{X}, G), the D_{Y} -modules ℋ^{j}(F^{!}M) have a natural structure of Gequivariant D_{Y} -modules (see [17, §2]). Remark. Suppose that X is affine. The differential of the G-action on X yields a Lie algebra map, τ_{X}, from 𝔤 to the Lie algebra of vector fields on X, defined by (τ_{X}(ξ).φ)(x) = \frac{d}{dt}_{|t=0}φ(exp(−tξ).x), ξ ∈ 𝔤, φ ∈ O(X), x ∈ X. Then, the D(X)- module M is G-equivariant if, and only if, M has a rational G-module structure such that g.(D.v) = (g.D).(g.v) and τ_{X}(ξ).v = ξ.v, for all g ∈ G, D ∈ D(X), ξ ∈ 𝔤, v ∈ M, where ξ.v = \frac{d}{dt}_{|t=0}(exp(tξ).v). (See [17, Proposition 2.6] for a proof.)

Now, let G, 𝔤 be as in §1 and consider the adjoint action of G on X = 𝔤. Let x ∈ N(𝔤) and set O = G.x. Let M be a G-equivariant coherent D_{𝔤}-module such that Supp M = \overline{O}. If x = 0, it follows from Kashiwara’s equivalence [4, VI.7.11] that M is isomorphic to H_{[0]}^{n}(O_{𝔤})^{⊕k} for some k ∈ ℕ^{∗}, where H_{[0]}^{n}(O_{𝔤}) ≅ D_{𝔤}/D_{𝔤}𝔤^{∗}. Therefore, we will suppose from now on that x ≠ 0. Then we can find an Striplet {x, y, z} containing x, i.e. [x, y] = z, [z, x] = 2x, [z, y] = −2y and 𝔰 = ℂx + ℂy + ℂz ≅ 𝔰𝔩(2, ℂ). Recall that 𝔤^{y} = {ξ ∈ 𝔤 : [ξ, y] = 0} and set n = dim 𝔤, m = dim 𝔤^{y}, thus dim O = n − m. In this situation it is well known [15, III.5.1, III.7.4] that x + 𝔤^{y} is a transverse slice to O at the point x. Let μ_{x} : G × 𝔤^{y} → 𝔤 be given by μ_{x}(g, u) = g.(x + u). The following lemma is classical (see, for example, [13, Proposition 2.3]).

Lemma A.1. There exists an affine open neighborhood U of 0 in 𝔤^{y} such that:

(1) ψ = μ_{x}_{|Y} is smooth on Y = G × U, Ω = ψ(Y) = G.(x + U) is a G-stable open subset of 𝔤;

(2) Ω ∩ \overline{O} = O and O ∩ \{x + U\} = \{x\}.

Observe that, since ψ is smooth on Y, we have

ℋ^{j}(ψ^{!}M) = \begin{cases}0 & if j ≠ −m, \\ ψ^{∗}M & if j = −m. \end{cases}

Furthermore, it follows from Ω ⊃ O and the flatness of ψ that ψ^{∗}M ≠ 0.

Denote by t_{x} : 𝔤 → 𝔤 the translation by x, t_{x}(v) = x + v, and set X = t_{x}(U) = x + U. Let β : U ↪ 𝔤, β_{x} : X ↪ 𝔤, ȷ_{x} : X ↪ x + 𝔤^{y} be the canonical inclusions (note that β_{x} ∘ t_{x} = t_{x} ∘ β).

Recall that [𝔤, x] is a complementary subspace to 𝔤^{y}. Let {v_{1}, …, v_{m}} be a basis of 𝔤^{y} and {v_{m+1}, …, v_{n}} be a basis of [𝔤, x]. Denote by {y_{j} = v_{j}^{∗}}_{j} the dual basis and set f_{j} = t_{x}.y_{j} = y_{j} − y_{j}(x), 1 ⩽ j ⩽ n. Then, O(x + 𝔤^{y}) = O(𝔤)/J_{x} where J_{x} = (f_{m+1}, …, f_{n})O(𝔤). Let m_{x} = (f_{1}, …, f_{n})O(𝔤) and n_{x} = (f_{1}, …, f_{m})O(𝔤^{y}) be the maximal ideals asociated to x ∈ 𝔤 and x ∈ (x + 𝔤^{y}) (respectively).

Theorem A.2. (1) There exists an isomorphism of G-equivariant D_{Y} -modules ψ^{!}M ≅ (O(G) ⊠ β^{!}t_{x}^{∗}M)[n] (where G acts by left translation on O(G)). Moreover,

ℋ^{j}(β^{!}t_{x}^{!}M) = \begin{cases}0 & if j ≠ dim O, \\ β^{∗}t_{x}^{∗}M & if j = dim O. \end{cases}

(2) Set M_{|X} = β_{x}^{∗}M. Then, β_{x}^{!}M = M_{|X}[− dim O] and M_{|X} = β_{x}^{∗}(M/J_{x}M) is a D_{X}-module isomorphic to H_{[x]}^{m}(O_{X})^{⊕k} for some k ⩾ 1.

(3) One has β^{∗}t_{x}^{∗}M ≅ H_{[0]}^{m}(O_{U})^{⊕k} and ψ^{!}M ≅ (O_{G} ⊠ H_{[0]}^{m}(O_{U})^{⊕k})[m].

Proof. (1) Notice first that, since t_{x} is an isomorphism, we can identify t_{x}^{!}M with t_{x}^{∗}M. From ψ = μ_{x} ∘ (1_{G} × β) we deduce that ψ^{!}M = (1_{G} × β)^{!}μ_{x}^{!}M. Therefore, using the G-equivariant morphisms 1_{G} × β and 1_{G} × t_{x}, we obtain

(1_{G} × β)^{!}(1_{G} × t_{x})^{!}(θ) : (1_{G} × β)^{!}p_{2}^{!}t_{x}^{∗}M \underrightarrow{∼} ψ^{!}M

(θ being as in (A.1)). Then, (1_{G} × β)^{!}p_{2}^{!}t_{x}^{∗}M = (1_{G} × β)^{!}(O_{G} ⊠ t_{x}^{∗}M)[n] = (O_{G} ⊠ β^{!}t_{x}^{∗}M)[n] yields ψ^{!}M ≅ (O_{G}⊠β^{!}t_{x}^{∗}M)[n]. From this isomorphism one deduces that ℋ^{j−n}(ψ^{!}M) ≅ O_{G} ⊠ ℋ^{j}(β^{!}t_{x}^{∗}M). Therefore, ℋ^{j}(β^{!}t_{x}^{∗}M) = 0 unless j = dim O, and

ℋ^{dim O}(β^{!}t_{x}^{∗}M) = ℋ^{dim O}((O_{U} ⊗_{β^{#}}^{L} t_{x}^{∗}M)[dim O]) = O_{U} ⊗_{β^{#}} t_{x}^{∗}M,

as required.

(2) Let β̃_{x} : x + 𝔤^{y} ↪ 𝔤 be the inclusion. Thus β_{x} = β̃_{x} ∘ ȷ_{x}. Since ȷ_{x} is an open immersion, ℋ^{j}(β_{x}^{!}M) = O_{X} ⊗_{ȷ_{x}^{#}} ℋ^{j}(β̃_{x}^{!}M) for all j. Recall that β̃_{x}^{!}M = (D_{(x+𝔤^{y})→𝔤} ⊗_{D_{𝔤}}^{L} M)[m − n]. Set V_{x} = ⊕_{i=m+1}^{n}ℂdf_{i} and 𝒞_{x}^{−p} = ⋀^{p} V_{x} ⊠ D_{𝔤}. By [4, VI.7.4], β̃_{x}^{!}M = (𝒞_{x}^{•} ⊗_{D_{𝔤}} M, ∂_{x})[m − n], with 𝒞_{x}^{p} ⊗_{D_{𝔤}} M = ⋀^{−p} V_{x} ⊠ M and

∂_{x}(df_{j_{1}} ∧ ⋯ ∧ df_{j_{p}} ⊠v) = ∑_{a=1}^{p}(−1)^{a+1}df_{j_{1}} ∧ ⋯ ∧ d̂f_{j_{a}} ∧ ⋯ ∧ df_{j_{p}} ⊠f_{j_{a}}v.

It follows that ℋ^{n−m}(β̃_{x}^{!}M) = ℋ^{0}(𝒞_{x}^{•} ⊗_{D_{𝔤}} M, ∂_{x}) = M/J_{x}M = O_{x+𝔤^{y}} ⊗β̃_{x}^{#} M. Hence, M_{|X} identifies with the D_{X}-module ℋ^{dim O}(β_{x}^{!}M).

Now, the support of M_{|X} = O_{X} ⊗_{β_{x}^{#}} M is clearly contained in X ∩ Supp M ⊂ X ∩\overline{O}. But, X ∩\overline{O} ⊂ X ∩Ω∩\overline{O} = X ∩O = \{x\}. Thus M_{|X} is a D_{X}-module whose support is contained in {x}; therefore, by Kashiwara’s equivalence [4, VI.7.11], M_{|X} ≅ H_{[x]}^{m}(O_{X})^{⊕k} for some k. Notice that it follows easily from β_{x} ∘ t_{x} = t_{x} ∘ β (on U) that t_{x}^{∗}(M_{|X}) = β^{∗}t_{x}^{∗}M. Since ψ^{∗}M ≠ 0, (1) implies that k ≠ 0.

(3) is an easy consequence of (1) and (2).

In the sequel, thanks to Theorem A.2(3), we will currently identify ψ^{!}M with (O_{G} ⊠ H_{[0]}^{m}(O_{U})^{⊕k})[m].

We factorize the inclusion ı : {x} ↪ 𝔤 as the composition {x} ↪^{ı_{1}} O ↪^{ı_{2}} 𝔤 and we set M_{|O} = ℋ^{0}(ı_{2}^{!}M).

Lemma A.3. The cohomology of ı_{2}^{!}M is concentrated in degree 0.

Proof. Notice first that ψ ∘ ȷ_{2} = ı_{2} ∘ π, where π : G × {0} → O, π(g, 0) = g.x, and ȷ_{2} : G × {0} ↪ Y is the natural inclusion. From ψ^{∗}M = O_{G} ⊠ H_{[0]}^{m}(O_{U})^{⊕k} we get that ψ^{∗}M = ℋ^{−m}(ψ^{!}M) is supported on G × {0}. Since ȷ_{2} is a closed embedding, [4, VI.7.4] gives that ȷ_{2}^{!}ψ^{∗}M has cohomology concentrated in degree 0; equivalently, ȷ_{2}^{!}ψ^{!}M has cohomology concentrated in degree −m. On the other hand,

ℋ^{j}(π^{!}ı_{2}^{!}M) = ℋ^{j}((D_{G×{0}→O} ⊗_{D_{O}}^{L} ı_{2}^{!}M)[m]) = ℋ^{j+m}(D_{G×{0}→O} ⊗_{D_{O}}^{L} ı_{2}^{!}M).

From π smooth and D_{G×{0}→O} = π^{∗}D_{O} it follows that, as an O_{G×{0}}-module, ℋ^{j}(π^{!}ı_{2}^{!}M) = O_{G×{0}} ⊗_{π^{#}} ℋ^{j+m}(ı_{2}^{!}M). Now, since π is faithfully flat and π^{!}ı_{2}^{!}M = ȷ_{2}^{!}ψ^{!}M, we deduce that ı_{2}^{!}M has cohomology concentrated in degree 0.

Since ı_{2} : O ↪ 𝔤 is G-equivariant, ı_{2}^{!}M = M_{|O} (Lemma A.3) is in 𝔐(D_{O}, G). Therefore, see [7, Proposition 4.11.1] and [8, §4], M_{|O} is the (integrable) connection on O associated to the representation of A(O) on the geometric fibre M_{|O}(x) = (M_{|O})_{x}/m_{x}(M_{|O})_{x}. In particular, M_{|O} is flat as an O_{O}-module and it follows that

ı^{!}M = ı_{1}^{!}ı_{2}^{!}M = ı_{1}^{!}M_{|O} = (D_{{x}→O} ⊗_{D_{O}}^{L} M_{|O})[m − n]

has cohomology concentrated in degree dim O with ℋ^{dim O}(ı^{!}M) = M_{|O}(x).

Let G^{ϕ} = {g ∈ G : g.a = a for all a ∈ 𝔰} be the centralizer of the Lie subalgebra 𝔰 and denote by G_{0}^{ϕ} its identity component. Recall that G^{ϕ} is reductive and that the map G^{ϕ} ↪ G^{x} induces the identification A(O) = G^{ϕ}/G_{0}^{ϕ} [2, Proposition 2.4]. Observe that the subspaces

⨁_{i=1}^{m} ℂy_{i}, ⨁_{i=m+1}^{n} ℂy_{i}, ⨁_{i=1}^{m} ℂf_{i}, ⨁_{i=m+1}^{n} ℂf_{i}

are G^{ϕ}-stable. Set

ω^{−1} = df_{1} ∧ ⋯ ∧ df_{m} and T = {ū ∈ M/J_{x}M : n_{x}.ū = 0}.

It is easily shown that ω^{−1} = dy_{1} ∧ ⋯ ∧ dy_{m} is G^{ϕ}-invariant. Furthermore, G^{ϕ} acts naturally on M/J_{x}M and T. Therefore, ℂ ω^{−1} ⊗_{ℂ} T carries a representation of G^{ϕ} isomorphic to T. A consequence of the next result is that G_{0}^{ϕ} acts trivially on T (it is not difficult to prove this fact directly).

Theorem A.4. The D_{O}-module M_{|O} is the connection on O defined by representation ℋ^{dim O}(ı^{!}M) = ω^{−1} ⊗_{ℂ} T of the group A(O). Furthermore, T ≅ {v ∈ H_{[x]}^{m}(O_{X})^{⊕k} : n_{x}v = 0} is a ℂ-vector space of dimension k.

Proof. Write ı = β̃_{x} ∘ γ, where γ : {x} ↪ x + 𝔤^{y} and β̃_{x} : x + 𝔤^{y} ↪ 𝔤. Since the maps γ and β̃_{x} are G^{ϕ}-equivariant, we have a natural G^{ϕ}-action on M_{|O}(x) = ℋ^{dim O}(ı^{!}M) which yields the representation of the group A(O) that we want to compute.

Set V_{x} = ⊕_{i=m+1}^{n}ℂdf_{i} = ⊕_{i=m+1}^{n}ℂdy_{i}; then V_{x} is a G^{ϕ}-stable subspace of 𝔤^{∗}. We have seen in the proof of Theorem A.2 that (setting ∂_{II} = ∂_{x}),

β̃_{x}^{!}M = (𝒞_{II}^{•} = 𝒞_{x}^{•} ⊗_{D_{𝔤}} M, ∂_{II})[m − n].

Let G^{ϕ} act diagonally on 𝒞_{II}^{p} = 𝒞_{x}^{p+m−n} ⊗_{D_{𝔤}} M = ⋀^{−p+dim O} V_{x} ⊠ M. Then, ∂_{II} is G^{ϕ}-equivariant and the G^{ϕ}-action on the cohomology group ℋ^{j}(β̃_{x}^{!}M) ∈ 𝔐(D_{x+𝔤^{y}}, G^{ϕ}) is induced by the diagonal action of G^{ϕ} on 𝒞_{II}^{•}. Notice, in particular, that ℋ^{dim O}(β̃_{x}^{!}M) = Tor_{0}^{O_{𝔤}}(O_{x+𝔤^{y}}, M) = M/J_{x}M is endowed with the natural action of G^{ϕ}. By Theorem A.2(1), ℋ^{j}(β_{x}^{!}M) = O_{X} ⊗_{O_{x+𝔤^{y}}} ℋ^{j}(β̃_{x}^{!}M) = 0 when j ≠ dim O. Thus,

Supp ℋ^{j}(β̃_{x}^{!}M) ⊂ (x + 𝔤^{y}) \ X ⊂ (x + 𝔤^{y}) \ {x} if j ≠ dim O.

Now, ı^{!}M = γ^{!}β̃_{x}^{!}M = (D_{{x}→x+𝔤^{y}} ⊗_{D_{x+𝔤^{y}}}^{L} β̃_{x}^{!}M)[−m] can be computed as follows. Notice that n_{x}/n_{x}^{2} = ⊕_{j=1}^{m}ℂdf_{j} and consider the complex (𝒞_{I}^{•}, ∂_{I}) where 𝒞_{I}^{p} = ⋀^{−p}(n_{x}/n_{x}^{2}) ⊠ D_{x+𝔤^{y}} and

∂_{I}(df_{j_{1}} ∧ ⋯ ∧ df_{j_{p}} ⊠D) = ∑_{s=1}^{p}(−1)^{s+1}df_{j_{1}} ∧ ⋯ ∧ d̂f_{j_{s}} ∧ ⋯ ∧ df_{j_{p}} ⊠f_{j_{s}}D.

Observe that G^{ϕ} acts diagonally on 𝒞_{I}^{p} and that ∂_{I} is G^{ϕ}-equivariant. Let (𝒞_{tot}^{•}, ∂_{tot}) be the total complex associated to the double complex 𝒞^{•} = 𝒞_{I}^{•} ⊗_{D_{x+𝔤^{y}}} 𝒞_{II}^{•}. Then,

ı^{!}M = γ^{!}β̃_{x}^{!}M = (𝒞_{tot}^{•}, ∂_{tot})[−m]

and therefore ℋ^{j}(ı^{!}M) = ℋ^{j−m}(𝒞_{tot}^{•}). This group is computed by the spectral sequence E_{2}^{pq} = ℋ_{I}^{p}(ℋ_{II}^{q}(𝒞^{•})) ⟹ ℋ^{p+q}(𝒞_{tot}^{•}). But, E_{2}^{pq} = Tor_{−p}^{O_{x+𝔤^{y}}} (ℂ_{x}, ℋ^{q}(𝒞_{II})) as O_{x+𝔤^{y}}-module, where ℂ_{x} = O(𝔤)/m_{x}O(𝔤), and we have noticed that the support of ℋ^{q}(𝒞_{II}) = ℋ^{q}(β̃_{x}^{!}M) is contained in (x + 𝔤^{y}) \ {x} when q ≠ dim O. Therefore E_{2}^{pq} = 0 for all q ≠ dim O and E_{2}^{p dim O} = Tor_{−p}^{O_{x+𝔤^{y}}} (ℂ_{x}, ℋ^{dim O}(β̃_{x}^{!}M)). Hence, the spectral sequence E_{2}^{pq} collapses to E_{2}^{p dim O} = ℋ^{p+dim O}(𝒞_{tot}^{•}). In particular, we obtain

ℋ^{dim O}(ı^{!}M) = ℋ^{dim O−m}(𝒞_{tot}^{•}) = Tor_{m}^{O_{x+𝔤^{y}}} (ℂ_{x}, ℋ^{dim O}(β̃_{x}^{!}M))
= Tor_{m}^{O_{x+𝔤^{y}}} (ℂ_{x}, M/J_{x}M) = ω^{−1} ⊗_{ℂ} T.

Furthermore, the group G^{ϕ} acts diagonally on the complexes 𝒞^{•}, 𝒞_{tot} and it follows from the previous computation that the action of A(O) on ℋ^{dim O}(ı^{!}M) is coming from the induced action of G^{ϕ} on E_{2}^{−m dim O} = ω^{−1} ⊗_{ℂ} T.

To finish the proof it suffices, since M_{|X} = O_{X} ⊗_{ȷ_{x}^{#}} M/J_{x}M ≅ H_{[x]}^{m}(O_{X})^{⊕k} by Theorem A.2, to apply the following standard result to the module N = M/J_{x}M: Let N be any O_{x+𝔤^{y}}-module; then, if N′ = O_{X} ⊗_{ȷ_{x}^{#}} N, one has

\{u ∈ N : n_{x}.u = 0\} \underrightarrow{∼} \{u′ ∈ N′ : n_{x}.u′ = 0\}

through the natural map ȷ_{x}^{#} : N → N′.

Recall that we identify ψ^{∗}M with O_{G} ⊠ H_{[0]}^{m}(O_{U})^{⊕k} ∈ 𝔐(D_{Y} , G), where G acts

by left translation on O_{G}; thus (ψ^{∗}M)^{G} = H_{[0]}^{m}(O_{U})^{⊕k}. By Theorem A.2 we can

identify M_{|X} with H_{[x]}^{m}(O_{X})^{⊕k} and, by Theorem A.4, the A(O)-module M_{|O}(x) identifies with T.

Let v ∈ M^{G} and denote by v̄ the class of v in M/J_{x}M. Then, ψ^{#}(v) ∈ (ψ^{∗}M)^{G} and we write ψ^{#}(v) = 1_{G}⊠ρ(v) with ρ(v) ∈ H_{[0]}^{m}(O_{U})^{⊕k}. Denote by φ : {e} × U \underrightarrow{∼} X the restriction of ψ. Following the previous identifications, it is not difficult to see that φ yields an isomorphism ℂ ⊠ H_{[0]}^{m}(O_{U})^{⊕k} \underrightarrow{∼} M_{|X} = O_{X} ⊗_{ȷ_{x}^{#}} M/J_{x}M such that 1⊠ρ(v) ↔ ȷ_{x}^{#}(v̄). For simplicity we will still denote the element ȷ_{x}^{#}(v̄) by v̄. Set n_{0} = (y_{1}, …, y_{m})O(𝔤^{y}) and T_{0} = {f ∈ H_{[0]}^{m}(O_{U})^{⊕k} : n_{0}.f = 0}. Under the previous identifications, we have ℂ ⊠ T_{0} \underrightarrow{∼} T.

Corollary A.5. Assume that M = D_{𝔤}.v with v ∈ M^{G}. Then ψ^{∗}M is generated by 1_{G}⊠ρ(v). Moreover, if k = 1 and ρ(v) ∈ T_{0}, then M_{|O}(x) = ℂv̄ is the trivial representation of A(O) and the connection M_{|O} is the standard D_{O}-module O_{O}.

Proof. Since ψ is smooth, it is easy to see [13, Lemma 3.2] that ψ^{∗}M = D_{Y} ψ^{#}(v). Now, assume that k = 1 and n_{0}.ρ(v) = 0. Since dim T_{0} = 1 and ρ(v) ≠ 0, we obtain that T_{0} = ℂρ(v) and, by the previous identifications, T = ℂv̄. Since v ∈ M^{G}, the group G^{ϕ} acts trivially on M_{|O}(x) = T. The corollary then follows from Theorem A.4.

Acknowledgement. We wish to thank M. Raïs for bringing the reference [14] to our attention, and M. Van den Bergh for a useful correspondence.

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

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