NOTES ON EQUIVARIANT D-MODULES

T. LEVASSEUR

1. Generalities

All the varieties considered in these notes are quasi-projective algebraic varieties defined over ℂ.

Let X be a smooth algebraic variety. We denote by O_{X} the sheaf of regular functions on X and by D_{X} the sheaf of differential operators. The rings of global sections will be denoted by O(X) and D(X) respectively. We refer to [3] for the basic properties of D_{X}-modules. All the D_{X}-modules encountered in the sequel will be quasi-coherent. The category of quasi-coherent D_{X}-modules is denoted by Mod D_{X}.

Let F : Y → X be a morphism between 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 ↦ 1_{Y} ⊗_{F^{#}} v.

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; the O_{Z}-module M ⊗_{ℂ} N will be denoted by M ⊠ N.

In this section we recall the definitions, and well known properties, of equivariant D-modules. Our main references are [2, 3, 5, 6, 7, 8, 11, 12, 15] where the reader will find the proof of the results stated below.

Let G be a linear algebraic group and V be a smooth affine algebraic G-variety. Let e be the unit in G and set

μ : G × V → V, (g, v) ↦ g.v (the action of G on V)

μ_{G} : G × G → G, μ_{G}(g, h) = gh (the multiplication in G)

s : G → G, s(g) = g^{−1} (the inverse in G)

ε : {e} ↪ G (the inclusion)

p_{2} : G × V → V, p_{2}(g, v) = v

p_{23} : G × G × V → G × V, p_{23}(g, h, v) = (h, v)

ε_{V} : V → G × V, ε_{V} (v) = (e, v)

(1.1)

We then set: Δ = μ_{G}^{#}, S = s^{#}, ϵ = ε^{#}.

Let M be a rational G-module; the G-action on M is denoted by g.a, g ∈ G, a ∈ M. Recall that the G-module structure is equivalent to a left comodule structure λ_{M} : M → O(G) ⊠ M, such that

g.a = a_{(1)}(g^{−1})a_{(2)}

Date: October 1, 1997.

where we have used the (abbreviated) Sweedler notation λ_{M}(a) = a_{(1)}⊠a_{(2)}, a_{(1)} ∈ O(G), a_{(2)} ∈ M. This applies in particular to M = O(V) with the action (g.φ)(x) = φ(g^{−1}.x) for φ ∈ O(V), g ∈ G, x ∈ V. The corresponding coaction is denoted by λ_{V}. When V = G and G acts via left translations, we have λ_{G} = Δ.

Suppose furthermore that the G-module M has an O_{V} -module structure. Then we say that M is a G-equivariant O_{X}-module if the G and O(V) actions are compatible, i.e. g.(φa) = (g.φ)(g.a), g ∈ G, φ ∈ O(V), a ∈ M. This translates into λ_{M}(φa) = λ_{V} (φ)λ_{M}(a) for the coactions. We denote by 𝔐(O_{V} , G) the category of G-equivariant O_{V} -modules.

Recall that M ∈ 𝔐(O_{V} , G) if and only if there exists an isomorphism of O_{G×V} - modules

(1.2)  θ : p_{2}^{∗}M = O(G) ⊠ M \underrightarrow{∼} μ^{∗}M

such that

(1.3)  ε_{V}^{∗} (θ) = 1_{M}, (μ_{G} × 1_{V})^{∗}(θ) = (1_{G} × μ)^{∗}(θ) ∘ p_{23}^{∗}(θ).

When the coaction λ_{M} is given the isomorphism θ is:

θ(a⊠v) = aSv_{(1)} ⊗_{μ^{#}} v_{(2)}.

Endow O(G), O(G × V) = O(G) ⊠ O(V) and p_{2}^{∗}M = O(G) ⊠ M with the action induced by left translation on G. It is then easily seen that θ is G-linear when G acts on μ^{∗}M by

g.(b ⊗_{μ^{#}} m) = g.b ⊗_{μ^{#}} g.m

for all g ∈ G, b ∈ O(G)⊠O(V) and m ∈ M. For these actions, since (O(G)⊠M)^{G} = ℂ ⊠ M, we obtain the isomorphism

θ : ℂ ⊠ M \underrightarrow{∼}(μ^{∗}M)^{G}.

Recall that the G-action on V induces a rational G-module structure on the algebra D(V), given by

(g.D) · φ = g.(D · (g^{−1}.φ))

for all g ∈ G, D ∈ D(V), φ ∈ O(V). (We denote by D · φ the natural action of D(V) on O(V).) The corresponding coaction extends the coaction λ_{V} and we will still denote it by λ_{V} : D(V) → O(G) ⊠ D(V).

Let M ∈ Mod D_{V}. The module M is said to be a weakly G-equivariant D_{V} - module if M ∈ 𝔐(O_{V} , G) and

g.(D.v) = (g.D).(g.v)

for all g ∈ G, D ∈ D(V) and v ∈ M. (This is equivalent to saying that λ_{M}(D.v) = λ_{V} (D)λ_{M}(v).) We denote by 𝔐(D_{V} , G^{w}) the category of weakly G-equivariant D(V)-modules. Then, a module M ∈ Mod D_{V} is weakly G-equivariant if, and only if, there exists a map θ as in (1.2) & (1.3) which is O(G) ⊠ D(V) linear.

The differential of the G-action on V yields a Lie algebra map, τ_{V}, from 𝔤 = Lie(G) to the Lie algebra Θ_{V} of vector fields on V. It is defined by

(τ_{V} (ξ) · φ)(x) = \frac{d}{dt}_{|t=0}φ(exp(−tξ).x)

for all ξ ∈ 𝔤, φ ∈ O(V), x ∈ V. Notice that τ_{V} (ξ) identifies with a derivation of O(V) and that τ_{V} (Ad(g).ξ) = g.τ_{V} (ξ) for all g ∈ G. The differential of the G-action on D(V) is then given by

ξ.D = [τ_{V} (ξ), D], for all D ∈ D(V).

Let M ∈ 𝔐(O_{V} , G). The differential of the G-action on M gives a 𝔤-module structure on M:

ξ.v = \frac{d}{dt}_{|t=0}(exp(tξ).v), for all ξ ∈ 𝔤 and v ∈ M.

The module M is called a G-equivariant D_{V} -module if M ∈ 𝔐(D_{V} , G^{w}) and

(1.4)  τ_{V} (ξ).v = ξ.v, for all ξ ∈ 𝔤 and v ∈ M.

Then, a module M ∈ Mod D_{V} is G-equivariant if, and only if, there exists a map θ as in (1.2) & (1.3) which is D(G) ⊠ D(V) linear.

Remarks. (1) Let M ∈ 𝔐(O_{V} , G). If v ∈ M^{G} we have λ_{M}(v) = 1_{G}⊠v and therefore θ(1_{G}⊠v) = 1_{G×V} ⊗_{μ^{#}} v.

(2) Set

(1.5)  𝒩 = D_{V} /D_{V} τ_{V} (𝔤).

Then 𝒩 ∈ 𝔐(D_{V} , G). Moreover, when G is connected every subquotient of 𝒩 is in 𝔐(D_{V} , G) (see [9, 14]).

(3) Suppose that M ∈ 𝔐(D_{V} , G). Then, when G is connected:

{M = D_{V} .v with v ∈ M^{G}} ⟺ {M is a quotient of 𝒩}

The definitions of 𝔐(O_{V} , G), 𝔐(D_{V} , G^{w}) and 𝔐(D_{V} , G) carry over to the case when the smooth variety V is not necessarily affine, see [3, 8, 15]. For instance, M ∈ Mod D_{V} is G-equivariant if there exists an isomorphism θ : O_{G} ⊠ M \underrightarrow{∼} μ^{∗}M of D_{G×V} -modules which satisfies the conditions of (1.3).

Let F : Y → X be a G-equivariant morphism between (not necessarily affine) smooth G-varieties. 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 in the derived category D^{b}(D_{Y}) of Mod D_{Y}, and the construction of F^{!} extends to D^{b}(D_{X}). Observe that D_{Y →X} is a weakly G-invariant right D_{X}- module for the G-action g.(φ ⊗_{F^{#}} D) = g.φ ⊗_{F^{#}} g.D, φ ∈ O(Y), D ∈ D(X).

Assume that M ∈ 𝔐(D_{X}, G) and let θ_{M} : p_{2,X}^{∗}M → μ_{X}^{∗}M be the associated isomorphism (with obvious notation). Set r = dim G. Since p_{2,X} and μ_{X} are smooth, p_{2,X}^{!}M and μ_{X}^{!}M have cohomology concentrated in degree −r, equal to p_{2,X}^{∗}M and μ_{X}^{∗}M respectively. Thus one can consider θ_{M} as an isomorphism of D_{G×X}-modules between p_{2,X}^{!}M and μ_{X}^{!}M. It follows from base change that we have an isomorphism in D^{b}(D_{G×Y})

θ_{F^{!}M} : p_{2,Y}^{!} F^{!}M \underrightarrow{∼} μ_{Y}^{!} F^{!}M.

Indeed, the isomorphism θ_{F^{!}M} is defined by the left vertical arrow which makes the following diagram commutative:

\begin{matrix}μ_{Y}^{!} F^{!}M & (1_{G} × F)^{!}μ_{X}^{!}M \\ ↑ & ↑^{(1_{G}×F)^{!}(θ_{M})} \\ p_{2,Y}^{!} F^{!}M & (1_{G} × F)^{!}p_{2,X}^{!}M \end{matrix}

Therefore, since p_{2,Y} and μ_{Y} are smooth, we obtain isomorphisms of D_{G×Y} -modules in cohomology:

θ_{F^{!}M} : p_{2,Y}^{∗} ℋ^{j}(F^{!}M) \underrightarrow{∼} μ_{Y}^{∗} ℋ^{j}(F^{!}M).

This implies that ℋ^{j}(F^{!}M) has a natural induced structure of G-equivariant D_{Y} - module. In particular, the G-action on ℋ^{j}(F^{!}M) is uniquely determined by the action of G on M. To compute this action one may proceed as follows.

Suppose that we are given a resolution (P^{•}, d) of D_{Y →X} by weakly G-equivariant right D_{X}-modules, see [6, Lemma 4.7] and [8, Proposition 2.1], such that each P^{k} is projective as a D_{X}-module and d is G-equivariant. Then, since (P^{•} ⊗_{D_{X}} M, d ⊗ 1_{M}) = F^{!}M[−d_{Y,X}], the G-action on ℋ^{j}(F^{!}M) is induced by the diagonal G-action on P^{•} ⊗_{D_{X}} M.

Moreover, since D_{G×Y →G×X} = D_{G} ⊠ D_{Y →X}, the objects (1_{G} × F)^{!}μ_{X}^{!}M and (1_{G} × F)^{!}p_{2,X}^{!}M are represented, up to a shift, by

(D_{G} ⊠ P^{•}) ⊗_{D_{G×X}} μ_{X}^{!}M and (D_{G} ⊠ P^{•}) ⊗_{D_{G×X}} p_{2,X}^{!}M.

Thus the isomorphism θ_{F^{!}M} is induced by

(1_{G} ⊠ 1_{P^{•}}) ⊗ θ_{M} : (D_{G} ⊠ P^{•}) ⊗ p_{2,X}^{∗}M ⟶ (D_{G} ⊠ P^{•}) ⊗ μ_{X}^{∗}M.

This applies for instance to the morphisms p_{2} and μ, and the isomorphism θ of (1.2) can be viewed as an isomorphism in 𝔐(D_{G×V} , G) between p_{2}^{!}M = p_{2}^{∗}M[r] and μ^{!}M = μ^{∗}M[r].

2. Reduction to a slice

In this section we apply the results of §1 to the case where the variety V is a finite dimensional rational G-module, which we denote by E. We keep the notation of (1.1). We set r = dim G, n = dim E and we fix x ∈ E. Denote by t_{x} : y ↦ y + x the translation by x and let μ_{x} = μ ∘ (1_{G} × t_{x}) : G × E → E, (g, v) ↦ g.(x + v).

Let F ⊆ E be a linear subspace of dimension m. We assume in this section that the following hypothesis holds:

(†)  μ_{x} : G × F → E is a smooth morphism and E = F ⊕ 𝔤.x

Set O = G.x, which is a quasi-affine subvariety of E. Then, by (†), x + F is a transverse slice to O at the point x, see [13, §5.1]. Let 𝔤^{x} be the stabilizer of x in 𝔤 and set s = dim 𝔤^{x}. Then, (†) implies that

dim O = r − s = n − m.

Proposition 2.1. There exists an affine open neighborhood U of 0 in F such that, if ψ is the restriction of μ_{x} to G × F,

(1) ψ is smooth on Y := G × U, Ω = ψ(Y) = G.(x + U) is a G-stable open subset of E;

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

Proof. Let T = (x + F) ×_{E} O be the intersection of x + F and O; thus T is a subscheme of E. From (†) and [4, IV.17.13.8] it follows that this intersection is transverse of dimension 0 at the point x. Hence [ibid], there exists an affine open subset 0 ∈ U_{0} ⊆ F such that the intersection (x + U_{0}) ∩ O is transverse at x. In particular dim ((x + U_{0}) ∩ O) = 0, therefore (x + U_{0}) ∩ O is a finite set [4, O.14.1.9], say {x = x_{0}, x_{1}, …, x_{t}}. For each i = 1, …, t, pick an affine open subset 0 ∈ U(x_{i}) ⊂ F such that x_{i} ∉ x + U(x_{i}). Set U_{1} = U_{0} ∩ U(x_{1}) ∩ ⋯ ∩ U(x_{t}). Then, U_{1} is an affine open neighborhood of 0 and (x + U_{1}) ∩ O = {x}.

Since O is open in its closure, we can find a G-stable open subset V ⊂ E such that V ∩ \overline{O} = O. Define an open neighborhood of 0 in F by:

U_{2} = p_{2}ψ^{−1}(V) = {u ∈ F : ∃ g ∈ G, g.(x + u) ∈ V }

Let v ∈ U_{2}; since ψ is G-equivariant, we have ψ(G × {v}) ⊂ V. Hence, G × U_{2} ⊂ ψ^{−1}(V). Observe that, if 0 ∈ U ⊂ U_{2} is any open subset, ψ(G × U) is open and ψ(G × U) ∩ \overline{O} = O.

Now, let U_{1} be as in the first paragraph and choose an affine open subset 0 ∈ U ⊆ U_{1} ∩ U_{2}. Then U satisfies the required properties. □

We will keep the notation of Proposition 2.1 for the rest of this section. In particular, ψ will be the smooth morphism

ψ : Y = G × U ⟶ E, ψ(g, u) = g.(x + u).

Set X = x + U and notice that t_{x} induces an isomorphism (of varieties) from U onto X, with inverse t_{−x}. Define:

(2.1)  β : U ↪ E, β_{x} : X ↪ E, ı : {x} ↪ E (the natural inclusions)

Observe that β_{x} = t_{x} ∘ β ∘ t_{−x} and ψ = μ_{x} ∘ (1_{G} × β).

Let M ∈ 𝔐(D_{E}, G). We assume that M is a coherent D_{E}-module, i.e. M is a finitely generated G-equivariant D(E)-module. Recall from §1 that we have an isomorphism of G-equivariant D_{G×E}-modules,

θ : p_{2}^{!}M = p_{2}^{∗}M[r] \underrightarrow{∼} μ^{∗}M[r] = μ^{!}M.

By using the translation t_{x} we can construct the D_{E}-module t_{x}^{∗}M, which identifies with t_{x}^{!}M. Observe that t_{x} ∈ Aut(E) induces an automorphism of O(E), (t_{x}.φ)(y) = φ(y −x) for φ ∈ O(E), y ∈ E, and therefore yields an automorphism of D(E), (t_{x}.D)(φ) = t_{x}.D(t_{−x}.φ) for D ∈ D(E). Then, the D(E)-module t_{x}^{∗}M can be identified with the vector space M endowed with the action D ▪ u = (t_{x}.D)u. As in §1 we have maps t_{x}^{#} : M → t_{x}^{∗}M and t_{−x}^{#} : M → t_{−x}^{∗}M. We set:

t_{−x}.v = t_{x}^{#}(v), t_{x}.v = t_{−x}^{#}(v).

We will adopt a similar notation for the inverse images by t_{x}, or t_{−x}, for modules over X, or U.

Lemma 2.2. There exists an isomorphism of G-equivariant D_{G×E}-modules:

θ_{x} : p_{2}^{!}t_{x}^{∗}M \underrightarrow{∼} μ_{x}^{!}M

Proof. Since p_{2} and μ are smooth, p_{2}^{!}t_{x}^{!}M = p_{2}^{∗}t_{x}^{∗}M[r] and μ_{x}^{!}M = μ^{∗}M[r]. Notice that μ_{x}^{!}M = (1_{G} × t_{x})^{!}μ^{!}M = (1_{G} × t_{x})^{∗}μ^{!}M and p_{2}^{!}t_{x}^{∗}M = (O(G) ⊠ t_{x}^{∗}M)[r] = (1_{G} × t_{x})^{∗}p_{2}^{!}M. Now, the G-equivariant isomorphism 1_{G} × t_{x} : G × E → G × E yields the isomorphism, in 𝔐(D_{G×E}, G),

θ_{x} = (1_{G} × t_{x})^{∗}(θ) : (1_{G} × t_{x})^{∗}p_{2}^{!}M = p_{2}^{!}t_{x}^{∗}M \underrightarrow{∼}(1_{G} × t_{x})^{∗}μ^{!}M = μ_{x}^{!}M

as desired. □

Remark. The isomorphism θ_{x} yields the isomorphism

θ_{x} : ℋ^{−r}(p_{2}^{!}t_{x}^{∗}M) = O(G) ⊠ t_{x}^{∗}M \underrightarrow{∼} μ_{x}^{∗}M = ℋ^{−r}(μ_{x}^{!}M),

which is given by θ_{x}(a⊠t_{−x}.v) = aSv_{(1)} ⊗_{μ_{x}^{#}} v_{(2)} for a ∈ O(G) and v ∈ M.

Since ψ is smooth and G-equivariant, ψ^{∗}M is coherent [3, VI.4.8] and ψ^{!}M = ψ^{∗}M[r + m − n] ∈ 𝔐(D_{G×U}, G). Thus,

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

The G-action on ψ^{∗}M = O(G × U) ⊗_{ψ^{#}} M is given by g.(b ⊗_{ψ^{#}} v) = g.b ⊗_{ψ^{#}} g.v, b ∈ O(G × U), v ∈ M.

Lemma 2.3. There exists an isomorphism of G-equivariant D_{Y} -modules

ψ^{!}M ≅ (O(G) ⊠ β^{!}t_{x}^{∗}M)[r]

and β^{!}t_{x}^{!}M = (β^{∗}t_{x}^{∗}M)[m − n]. Hence

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

Proof. 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 morphism 1_{G} × β, we obtain

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

Then, (1_{G} × β)^{!}p_{2}^{!}t_{x}^{∗}M = (1_{G} × β)^{!}(O(G) ⊠ t_{x}^{∗}M)[r] = (O(G) ⊠ β^{!}t_{x}^{∗}M)[r] yields ψ^{!}M ≅ (O(G)⊠β^{!}t_{x}^{∗}M)[r]. From this isomorphism one deduces that ℋ^{j−r}(ψ^{!}M) ≅ O_{G} ⊠ ℋ^{j}(β^{!}t_{x}^{∗}M). Then, by (2.2), ℋ^{j}(β^{!}t_{x}^{∗}M) = 0 unless j = n − m, and

ℋ^{n−m}(β^{!}t_{x}^{∗}M) = ℋ^{n−m}((O_{U} ⊗_{β^{#}}^{L} t_{x}^{∗}M)[m − n]) = O_{U} ⊗_{β^{#}} t_{x}^{∗}M.

This completes the proof of the lemma. □

In order to simplify the notation we set

t_{x}^{∗}M_{|U} = ℋ^{dim O}(β^{!}t_{x}^{!}M) = O_{U} ⊗_{β^{#}} t_{x}^{∗}M.

Recall that the natural inclusion β_{x} : X ↪ E is equal to t_{x} ∘ β ∘ t_{−x}. Therefore, β_{x}^{!}M = t_{−x}^{!}β^{!}t_{x}^{!}M has non-zero cohomology only in degree dim O = n − m, where ℋ^{dim O}(β_{x}^{!}M) = O_{X} ⊗_{β^{#}} M. We set

M_{|X} = ℋ^{dim O}(β_{x}^{!}M) = O_{X} ⊗_{β_{x}^{#}} M.

Thus we have:

(2.3)  β_{x}^{!}M = M_{|X}[m − n]

Notice that it follows easily from β_{x} ∘ t_{x} = t_{x} ∘ β (on U) that

(2.4)  t_{x}^{∗}(M_{|X}) = t_{x}^{∗}M_{|U}.

Let S be a complementary subspace to F, E = F ⊕S. Let {v_{1}, …, v_{m}} be a basis of F and {v_{m+1}, …, v_{n}} be a basis of S. 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(F) = S(E^{∗})/J, O(x + F) = S(E^{∗})/J_{x}

where J = (y_{m+1}, …, y_{n})S(E^{∗}) and J_{x} = t_{x}.J = (f_{m+1}, …, f_{n})S(E^{∗}). Let ȷ : U ↪ F and ȷ_{x} : X ↪ x + F be the natural inclusions.

Lemma 2.4. We have:

(1) M_{|X} = O_{X} ⊗_{ȷ_{x}^{#}} (M/J_{x}M);

(2) t_{x}^{∗}M_{|U} = O_{U} ⊗_{ȷ^{#}} (t_{x}^{∗}M/Jt_{x}^{∗}M) = O_{U} ⊗_{ȷ^{#}} t_{x}^{∗}(M/J_{x}M).

Proof. 1. Let β̃_{x} : x + F ↪ E 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+F →E} ⊗_{D_{E}}^{L} M)[m − n]. Set V_{x} = ⊕_{i=m+1}^{n}ℂdf_{i} and 𝒞_{x}^{−p} = ⋀^{p} V_{x} ⊠ D_{E}. By [3, VI.7.4], β̃_{x}^{!}M = (𝒞_{x}^{•} ⊗_{D_{E}} M, ∂_{x})[m − n], with 𝒞_{x}^{p} ⊗_{D_{E}} 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_{E}} M, ∂_{x}) = M/J_{x}M = O_{x+F} ⊗β̃_{x}^{#} M. 2. follows from 1. and (2.4). □

Let Supp M be the support of the D_{E}-module M. Since M is coherent and G-equivariant, Supp M is a closed G-stable subvariety of E and we have Supp M = ⋃_{v∈M} Supp O_{E}.v. From now on we assume that

Supp M ⊆ \overline{O}.

Recall that the local cohomology group H_{[0]}^{m}(O_{U}) is a D_{U}-module isomorphic to D_{U}/ (D_{U}y_{1} + ⋯ + D_{U}y_{m}). It is easily seen that t_{−x}^{∗}H_{[0]}^{m}(O_{U}) = H_{[x]}^{m}(O_{X}) ≅ D_{X}/ (D_{X}f_{1} + ⋯ + D_{X}f_{m}).

Proposition 2.5. Let M be as above. Then, there exists k ∈ ℕ such that (1) M_{|X} ≅ H_{[x]}^{m}(O_{X})^{⊕k}, t_{x}^{∗}M_{|U} ≅ H_{[0]}^{m}(O_{U})^{⊕k};

(2) ψ^{!}M ≅ (O_{G} ⊠ H_{[0]}^{m}(O_{U})^{⊕k})[s];
(3) if Supp M = \overline{O}, then k ⩾ 1.

Proof. 1. Clearly, the support of M_{|X} = O_{X} ⊗_{β_{x}^{#}} M is 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 [3, VI.7.11], M_{|X} = H_{[x]}^{m}(O_{X})^{⊕k} for some k. The proof is similar for t_{x}^{∗}M_{|U}.

2. follows from 1. and Lemma 2.3.

3. The hypothesis implies that there exists v ∈ M such that O ∩ Supp O_{E}.v ≠ ∅. Then, since ψ is flat and Ω ⊃ O, we have 1 ⊗_{β^{#}} v ∈ ψ^{∗}M \ {0}. Thus ψ^{∗}M ≠ 0, i.e. k ⩾ 1. □

Denote by m_{x} = (f_{1}, …, f_{n})O(E) and n_{x} = (f_{1}, …, f_{m})O(F) the maximal ideals asociated to x ∈ E and x ∈ (x+F) (respectively). Set ℂ_{x} = O(E)/m_{x}O(E).

Theorem 2.6. Let ı : {x} ↪ E be the inclusion. Set

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

Then, (1) T ≅ {v ∈ H_{[x]}^{m}(O_{X})^{⊕k} : n_{x}v = 0} is a ℂ-vector space of dimension k;

(2) ı^{!}M has cohomology concentrated in degree dim O and

ℋ^{dim O}(ı^{!}M) = Tor_{m}^{O(E)}(ℂ_{x}, M) = ω^{−1} ⊗_{ℂ_{x}} T.

Proof. Recall [3, VI.4.2] that ℋ^{j}(ı^{!}M) = Tor_{−j+n}^{O(E)}(ℂ_{x}, M). Let γ : {x} ↪ X be the inclusion. Then, ı = β_{x} ∘ γ and ı^{!}M = γ^{!}β_{x}^{!}M. Recall from (2.3) that β_{x}^{!}M = M_{|X}[m − n]. By Lemma 2.4 and Proposition 2.5, we know that M_{|X} = O_{X} ⊗_{ȷ_{x}^{#}} (M/J_{x}M) ≅ H_{[x]}^{m}(O_{X})^{⊕k} is supported on {x}. Thus, T_{|X} = {u ∈ M_{|X} : n_{x}u = 0} is a ℂ-vector space of dimension k. Furthermore, by [3, VI.7.4],

ℋ^{j}(γ^{!}M_{|X}) = \begin{cases}0 & if j ≠ 0, \\ ω_{\{x\}/X}^{−1} ⊗_{ℂ_{x}} T_{|X} & if j = 0. \end{cases}

Since ω_{{x}/X}^{−1} = df_{1} ∧ ⋯ ∧ df_{m}, we obtain that ı^{!}M has cohomology concentrated in degree n − m with ℋ^{n−m}(ı^{!}M) = ω^{−1} ⊗_{ℂ_{x}} T_{|X}. To finish the proof it suffices to apply the following standard result to the module N = M/J_{x}M: Let N be any O_{x+F} -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′, ȷ_{x}^{#}(u) = 1_{X} ⊗_{ȷ_{x}^{#}} u. □

One can factorize the inclusion ı : {x} ↪ E as follows: ı : {x} ↪^{ı_{1}} O ↪^{ı_{2}} E. We now compute ı_{2}^{!}M, the “restriction to O”.

Proposition 2.7. ı_{2}^{!}M has cohomology concentrated in degree 0 and, as an O_{O}- module, ℋ^{0}(ı_{2}^{!}M) = Tor_{m}^{O_{E}}(O_{O}, M).

Proof. Notice first the following commutative diagram of G-equivariant morphisms

Y = G × U ⟶^{ψ} E
ȷ_{2}↑  ↑^{ı_{2}}
G × {0} ⟶^{π} O

where π(g, 0) = g.x.

Recall that, by Proposition 2.5, ψ^{!}M ≅ ψ^{∗}M[s] with ψ^{∗}M = O_{G} ⊠ H_{[0]}^{m}(O_{U})^{⊕k}. Thus ψ^{∗}M = ℋ^{−s}(ψ^{!}M) is supported on G × {0}. Since ȷ_{2} : G × {0} ↪ Y is a closed embedding, [3, 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 −s.

On the other hand,

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

Since π is 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+s}(ı_{2}^{!}M).

Now, since π is faithfully flat and π^{!}ı_{2}^{!}M = ȷ_{2}^{!}ψ^{!}M, the previous paragraph implies that ı_{2}^{!}M has cohomology concentrated in degree 0. By definition, ı_{2}^{!}M = (D_{O} ⊗_{D_{E}}^{L} M)[−m]. Hence, see [3, VI.4.2], ℋ^{0}(ı_{2}^{!}M) = Tor_{m}^{O_{E}}(O_{O}, M). □

We set:

(2.5)  M_{|O} = ℋ^{0}(ı_{2}^{!}M)

Denote by G^{x} the stabilizer of x in G and let G_{0}^{x} be its connected component. Define the component group of O by

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

Let π : G ↠ O, g ↦ g.x, be the natural morphism and L be a rational representation of G^{x}. Define an O_{O}-module ℒ by setting, for any open subset W ⊆ O,

Γ(W, ℒ) = {f : π^{−1}(W) → L : f(gh) = h^{−1}.f(g) for all g ∈ π^{−1}(W), h ∈ G^{x}}.

Then ℒ ∈ 𝔐(O_{O}, G) [7, Theorem 4.8.1]. When L is a representation of the (finite) group A(O), i.e. when G_{0}^{x} acts trivially on L, ℒ ∈ 𝔐(D_{O}, G) and, conversely, any G-equivariant D_{O}-module is of this form, see [7, Proposition 4.11.1] and [8, §4]. An object of 𝔐(D_{O}, G) will be called a connection on O. The representation of A(O) associated to a connection ℒ is the “geometric fibre at the point x”, ℒ(x) = ℂ_{x} ⊗_{O_{O}} ℒ = ℒ_{x}/m_{x}ℒ_{x}, where the A(O)-action is coming from the natural action of G^{x}.

Proposition 2.8. The D_{O}-module M_{|O} is a connection and its geometric fibre at the point x is

M_{|O}(x) = ω^{−1} ⊗_{ℂ} T.

Proof. Since ı_{2} : O ↪ E is G-equivariant, the results of §1 ensure that ı_{2}^{!}M = M_{|O} (see Proposition 2.7) is in 𝔐(D_{O}, G). Therefore, by the remarks above, M_{|O} is a connection. 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 n−m, where ℋ^{n−m}(ı^{!}M) = ℂ_{x} ⊗_{O_{O}} M_{|O} = M_{|O}(x). The proposition then follows from Theorem 2.6. □

Remark. We will see (in a particular case) in the next section how to compute the A(O)-action on ω^{−1} ⊗_{ℂ} T.

Recall that t_{x}^{∗}M_{|U} = O_{U} ⊗_{β^{#}} t_{x}^{∗}M, M_{|X} = O_{X} ⊗_{β_{x}^{#}} M. Thus we have maps β^{#} : t_{x}^{∗}M → t_{x}^{∗}M_{|U}, β^{#}(t) = 1_{U} ⊗_{β^{#}} t, and β_{x}^{#} : M → M_{|X}, β_{x}^{#}(v) = 1_{X} ⊗_{β_{x}^{#}} v. We set:

(2.6)  ρ(v) = β^{#}(t_{−x}.v) = 1_{U} ⊗_{β^{#}} t_{−x}.v

Recall also from Lemma 2.4 that t_{x}^{∗}M_{|U} = O_{U} ⊗_{ȷ^{#}} (t_{x}^{∗}M/Jt_{x}^{∗}M) and M_{|X} = O_{X} ⊗_{ȷ_{x}} (M/J_{x}M). Let ϖ : t_{x}^{∗}M ↠ t_{x}^{∗}M/Jt_{x}^{∗}M and ϖ_{x}M ↠ M/J_{x}M be the canonical projections. It is easy to see that, since t_{x}^{∗}M/Jt_{x}^{∗}M = t_{x}^{∗}(M/JM), one has t_{−x}.(ϖ_{x}(v)) = ϖ(t_{−x}.v). One obtains from the definitions that, for all v ∈ M,

1_{X} ⊗_{ȷ_{x}^{#}} ϖ_{x}(v) = 1_{X} ⊗_{β_{x}^{#}} v, ρ(v) = 1_{U} ⊗_{β^{#}} t_{−x}.v = 1_{U} ⊗_{ȷ^{#}} ϖ(t_{−x}.v).

It is also easily seen that the map t_{x}^{#} : M_{|X} → t_{x}^{∗}(M_{X}) = t_{x}^{∗}M_{|U} is a bijection given by

t_{x}^{#}(a ⊗_{ȷ_{x}^{#}} ϖ_{x}(v)) = t_{−x}.a ⊗_{ȷ^{#}} ϖ(t_{−x}.v) = (t_{−x}.a)ρ(v)

for all a ∈ O(X) and v ∈ M.

Recall from Lemma 2.2, and the remark thereafter, that θ_{x} yields an isomorphism O(G) ⊠ μ_{x}^{∗}M \underrightarrow{∼} μ_{x}^{∗}M, θ_{x}(a⊠t_{−x}.v) = aSv_{(1)} ⊗_{μ_{x}^{#}} v_{(2)}. It follows that θ̄_{x} = (1_{G} × β)^{∗}(θ_{x}) is the isomorphism:

θ̄_{x} : O(G) ⊠ t_{x}^{∗}M_{|U} \underrightarrow{∼} ψ^{∗}M, θ̄_{x}(a⊠ρ(v)) = aSv_{(1)} ⊗_{ψ^{#}} v_{(2)}.

Lemma 2.9. Let α : \{e\} × U ↪ Y be the inclusion and φ : \{e\} × U \underrightarrow{∼} X be the restriction of ψ. Then, ℋ^{dim O}(α^{!}ψ^{!}M) = α^{∗}ψ^{∗}M = φ^{∗}M_{|X} and θ̄_{x} induces an isomorphism

α^{∗}(θ̄_{x}) : ℂ ⊠ t_{x}^{∗}M_{|U} \underrightarrow{∼} φ^{∗}M_{|X}, 1⊠(t_{−x}.a)ρ(v) ↦ φ^{#}(a ⊗_{ȷ_{x}^{#}} ϖ_{x}(v)),

a ∈ O(X), v ∈ M.

Proof. Recall that

α^{!}ψ^{!}M = (D_{{e}×U→𝒴} ⊗_{D_{𝒴}}^{L} ψ^{!}M)[−r] = (D_{{e}×U→𝒴} ⊗_{D_{𝒴}}^{L} ψ^{∗}M)[m − n].

It follows that ℋ^{n−m}(α^{!}ψ^{!}M) = α^{∗}ψ^{∗}M. On the other hand, ψ ∘ α = β_{x} ∘ φ yields α^{∗}ψ^{∗}M = φ^{∗}β_{x}^{∗}M = φ^{∗}M_{|X}.

Let v ∈ M, then we have:

α^{∗}(θ̄_{x})(1⊠ρ(v)) = 1_{{e}×U} ⊗_{α^{#}} Sv_{(1)} ⊗_{ψ^{#}} v_{(2)}
= (Sv_{(1)})(e) ⊗_{(ψ∘α)^{#}} v_{(2)}
= 1 ⊗_{(ψ∘α)^{#}} v
= φ^{#}(1_{X} ⊗_{β_{x}^{#}} v)
= φ^{#}(1_{X} ⊗_{ȷ_{x}^{#}} ϖ_{x}(v)),

as required. □

From now on, in order to simplify the notation, we will identify the D_{Y} -modules ψ^{∗}M and O(G) ⊠ t_{x}^{∗}M_{|U} through the isomorphism θ̄_{x}. By Lemma 2.9, the D_{{e}×U}- module φ^{∗}M_{|X} then identifies with ℂ ⊠ t_{x}^{∗}M_{|U} via φ^{#}; this imply that we will identify the elements 1⊠ρ(v) and φ^{#}(1_{X} ⊗_{ȷ_{x}} ϖ_{x}(v)).

Let o : {0} ↪ U be the inclusion and recall that γ : {x} ↪ X. We have a commutative diagram:

Y  ⟶^{ψ} E
α↑  ↑^{β_{x}}
{e} × U ⟶^{φ} X
ε×o↑  ↑^{γ}
{e} × {0} ⟶_{φ̄} {x}

It follows that φ̄ gives an isomrphism:

ı^{!}M = γ^{!}β_{x}^{!}M \underrightarrow{∼}(ε × o)^{!}α^{!}ψ^{!}M

Since ℂ ⊠ t_{x}^{∗}M_{|U} is supported on {e} × {0} (see Proposition 2.5), (ε × o)^{!}α^{!}ψ^{!}M has cohomology concentrated in degree dim O and φ̄^{#} yields the isomorphism

φ̄^{#} : ℋ^{dim O}(ı^{!}M) \underrightarrow{∼} ℂ ⊠ ℋ^{0}(o^{!}t_{x}^{∗}M_{|U}).

Let n_{0} = (y_{1}, …, y_{m})O(F) be the defining ideal of the point 0 ∈ F. Since f_{i} = y_{i} − y_{i}(x), we have

ω_{{0}/U}^{−1} = dy_{1} ∧ ⋯ ∧ dy_{m} = ω^{−1} = df_{1} ∧ ⋯ ∧ df_{m}.

Theorem 2.10. Set T_{0} = {u ∈ t_{x}^{∗}M/Jt_{x}^{∗}M : n_{0}u = 0}. Then,

(1) T_{0} ≅ {u ∈ H_{[0]}^{m}(O_{U})^{⊕k} : n_{0}u = 0} is a vector space of dimension k and

ℋ^{0}(o^{!}t_{x}^{∗}M_{|U}) = ω^{−1} ⊗_{ℂ} T_{0};

(2) the isomorphism φ̄^{#} : ℋ^{dim O}(ı^{!}M) \underrightarrow{∼} ℂ ⊠ ℋ^{0}(o^{!}t_{x}^{∗}M_{|U}) coincides with

φ̄^{#} : ω^{−1} ⊗_{ℂ} T ⟶ ℂ ⊠ (ω^{−1} ⊗_{ℂ} T_{0}), ω^{−1} ⊗ ϖ_{x}(v) ↦ 1⊠(ω^{−1} ⊗ ϖ(t_{−x}.v)).

Proof. Since O_{U} ⊗_{ȷ^{#}} (t_{x}^{∗}M/Jt_{x}^{∗}M) ≅ H_{[0]}^{m}(O_{U})^{⊕k} (see Lemma 2.4 and Proposition 2.5), the proof of 1. is the same as the proof of Theorem 2.6. Observe in particular that

{u ∈ t_{x}^{∗}M_{|U} : n_{0}u = 0} ≡ {u ∈ t_{x}^{∗}M/Jt_{x}^{∗}M : n_{0}u = 0}.

The assertion 2. then follows from 1. and the identification of φ^{#}(1_{X} ⊗_{ȷ^{#}} ϖ_{x}(v)) with 1⊠ρ(v) = 1⊠(1_{U} ⊗_{ȷ^{#}} ϖ(t_{−x}.v)) □

Remark. We notice for further use the following consequence of Theorem 2.10. Let v ∈ M^{G}. Then, λ_{M}(v) = 1_{G}⊠v and therefore

ψ^{#}(v) = 1_{Y} ⊗_{ψ^{#}} v = θ̄_{x}(1_{G}⊠ρ(v)).

Thus we may identify ψ^{#}(v) with 1_{G}⊠ρ(v). Assume moreover that n_{0}ρ(v) = 0, then ρ(v) = 1_{U} ⊗_{ȷ^{#}} ϖ(t_{−x}.v) ∈ ℋ^{0}(o^{!}t_{x}^{∗}M_{|U}) can be identified with ϖ(t_{−x}.v) ∈ T_{0}.

3. The case of the adjoint representation

In this section we consider the case where G is the adjoint group of a semisimple Lie algebra 𝔤 of dimension n. We are going to apply the results of §2 to the case of the adjoint action of G on E = 𝔤. Moreover, we will assume that the element x ∈ 𝔤 is nilpotent, hence O = G.x is a nilpotent orbit. We fix a coherent equivariant D_{𝔤}-module M ∈ 𝔐(D_{𝔤}, G) such that Supp M = \overline{O}.

Suppose that x = 0. Then, by Kashiwara’s equivalence [3, VI.7.11] one has M ≅ H_{{0}}^{n}(O_{𝔤})^{⊕k} for some k ⩾ 1. In this case A(O) = {e} and the connection M_{|O} is the vector space ℂ^{k}.

Therefore, we now will suppose that x ≠ 0. Then, we can find an S-triplet {x, y, z} containing x, i.e. [x, y] = z, [z, x] = 2x, [z, y] = −2y and 𝔰 = ℂx + ℂy + ℂz ≅ 𝔰𝔩(2, ℂ). We take F = 𝔤^{y} = {ξ ∈ 𝔤 : [ξ, y] = 0}. Thus,

n = r = dim 𝔤, m = s = dim 𝔤^{y}.

In this situation it is well known [13, III.5.1, III.7.4] that x + F = x + 𝔤^{y} is a transverse slice to O at the point x. Thus the condition (†) of §2 is satisfied. We adopt the notation of the previous section; in particular, we have a smooth morphism ψ : Y = G × U → 𝔤 as in Proposition 2.1. We can summarize the results about the equivariant D_{Y} -module ψ^{∗}M in the following theorem, see Proposition 2.5, Theorem 2.6 and Proposition 2.8. Recall that we have the natural embedding

ı : {x} ↪^{ı_{1}} O ↪^{ı_{2}} 𝔤.

Theorem 3.1. (1) The D_{G×U}-module ψ^{∗}M is isomorphic to O_{G} ⊠H_{[0]}^{m}(O_{U})^{⊕k} for some k ⩾ 1.

(2) The D_{O}-module M_{|O} = ℋ^{0}(ı_{2}^{!}M) is the connection defined by the k-dimen-sional representation ℋ^{dim O}(ı^{!}M) = ω^{−1} ⊗_{ℂ} T of the finite group A(O), where ω^{−1} = dy_{1} ∧ ⋯ ∧ dy_{m} and T = {ϖ_{x}(v) ∈ M/J_{x}M : n_{x}ϖ_{x}(v) = 0}.

We now want to be more explicit on the action of A(O) on M_{|O}(x) = ω^{−1} ⊗ T. We first recall the following “Levi decomposition” of the stabilizer G^{x}.

Lemma 3.2. ([1, Proposition 2.4]) 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. Then, G^{ϕ} is reductive and there exists a semidirect product decomposition G^{x} = U^{x}.G^{ϕ}, where U^{x} is a normal unipotent subgroup. Furthermore, the map G^{ϕ} ↪ G^{x} induces the identification A(O) = G^{ϕ}/G_{0}^{ϕ}.

Recall that we have chosen a decomposition 𝔤 = 𝔤^{y}⊕S. Since F = 𝔤^{y} is G^{ϕ}-stable and G^{ϕ} is reductive, we may choose S to be G^{ϕ}-stable (e.g. S = [𝔤, x] = T_{x}(G.x)). Hence, with the notation of §2, 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. Observe that ℂ ω^{−1} = ℂdy_{1} ∧ ⋯ ∧ dy_{m} carries a representation of G^{ϕ}. Furthermore, G^{ϕ} acts naturally on M/J_{x}M and T. We will need the following well known result.

Lemma 3.3. Let g ∈ G^{ϕ} and ξ ∈ 𝔤^{ϕ} = Lie(G^{ϕ}). Then, (1) det Ad_{𝔤^{y}}(g) = det Ad_{𝔤/𝔤^{y}}(g) = 1;

(2) g.ω^{−1} = ω^{−1};
(3) tr ad_{𝔤^{y}}(ξ) = tr ad_{𝔤/𝔤^{y}}(ξ) = 0.

Proof. Since 𝔤 is semisimple, det Ad_{𝔤}(g) = 1 for all g ∈ G. Recall that the Killing form B induces a symplectic form B_{y} on 𝔤/𝔤^{y} by the formula B_{y}(ξ̄, η̄) = B(y, [ξ, η]). It is easily seen that B_{y} is G^{y}-invariant. Hence, if g ∈ G^{ϕ}, Ad_{𝔤/𝔤^{y}}(g) belongs to the symplectic group Sp(𝔤/𝔤^{y}, B_{y}). This implies det Ad_{𝔤/𝔤^{y}}(g) = 1. Now, the G^{ϕ}-stable decomposition 𝔤 = 𝔤^{y}⊕S with S ≅ 𝔤/𝔤^{y} (as a G^{ϕ}-module) yields det Ad_{𝔤^{y}}(g) = 1. This proves 1., and 2. follows from g.ω^{−1} = det Ad_{𝔤^{y}}(g^{−1}) ω^{−1}. The proof of 3. is similar. □

Theorem 3.4. The representation of A(O) on the fibre M_{|O}(x) = ω^{−1} ⊗_{ℂ} T is induced by the natural G^{ϕ}-action on T.

Proof. Write ı = β̃_{x}∘γ̃, where γ̃ : {x} ↪ x+F and β̃_{x} : x+F ↪ 𝔤. The maps γ̃ and β̃_{x} being 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 Lemma 2.4 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+F} , G^{ϕ}) is induced by the diagonal action of G^{ϕ} on 𝒞_{II}^{•} (see §1). Notice, in particular, that

ℋ^{dim O}(β̃_{x}^{!}M) = Tor_{0}^{O_{𝔤}}(O_{x+F} , M) = M/J_{x}M

is endowed with the natural action of G^{ϕ}. By (2.3), ℋ^{j}(β_{x}^{!}M) = O_{X} ⊗_{O_{x+F}} ℋ^{j}(β̃_{x}^{!}M) = 0 when j ≠ dim O. Thus,

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

Now,

ı^{!}M = γ̃^{!}β̃_{x}^{!}M = (D_{{x}→x+F} ⊗_{D_{x+F}}^{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+F} 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+F}} 𝒞_{II}^{•}. Then,

ı^{!}M = γ̃^{!}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+F}} (ℂ_{x}, ℋ^{q}(𝒞_{II})) as O_{x+F} -module, and we have noticed that the support of ℋ^{q}(𝒞_{II}) = ℋ^{q}(β̃_{x}^{!}M) is contained in (x + F) \ {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+F}} (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+F}} (ℂ_{x}, ℋ^{dim O}(β̃_{x}^{!}M))
= Tor_{m}^{O_{x+F}} (ℂ_{x}, M/J_{x}M) = ω^{−1} ⊗_{ℂ} T

(as expected). 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. Then, by Lemma 3.3,

g.(ω^{−1} ⊗ ϖ_{x}(v)) = g.ω^{−1} ⊗ g.ϖ_{x}(v) = ω^{−1} ⊗ ϖ_{x}(g.v)

for all ϖ_{x}(v) ∈ T. Hence the result. □

Remark. A consequence of Theorem 3.4 is that the identity component G_{0}^{ϕ} acts trivially on ω^{−1} ⊗_{ℂ} T. It is not difficult to prove this fact directly. Denote by τ_{ϕ} : 𝔤^{ϕ} → Der O(x + F) the differential of the (adjoint) action of G^{ϕ} on x + F (thus τ_{ϕ} = τ_{x+F} in the notation of §1). Let ξ ∈ 𝔤^{ϕ}. Since O(x + F) = O(𝔤)/J_{x}O(𝔤) ≡ ℂ[y_{1}, …, y_{m}], we may write τ_{ϕ}(ξ) = ∑_{j=1}^{m} ξ_{j}∂_{j}, where ∂_{j} = \frac{∂}{∂y_{j}}, 1 ⩽ j ⩽ m. Lemma 3.3 and a straightforward computation yield

τ_{ϕ}(ξ) = ∑_{j=1}^{m} ∂_{j}ξ_{j} + tr ad_{𝔤^{y}}(ξ) = ∑_{j=1}^{m} ∂_{j}ξ_{j}.

Notice that ξ_{j}(x) = y_{j}([x, ξ]) = 0, hence ξ_{j} ∈ n_{x}. Recall that M/J_{x}M = ℋ^{dim O}(β̃_{x}^{!}M) ∈ 𝔐(D_{x+F} , G^{ϕ}). Then, for all ϖ_{x}(v) ∈ T ⊂ M/J_{x}M,

ξ.(ω^{−1} ⊗ ϖ_{x}(v)) = \frac{d}{dt}_{|t=0}(e^{tξ}.ω^{−1} ⊗ e^{tξ}.ϖ_{x}(v))

= ω^{−1} ⊗ \frac{d}{dt}_{|t=0}(e^{tξ}.ϖ_{x}(v)) (by Lemma 3.3)

= ω^{−1} ⊗ τ_{ϕ}(ξ)ϖ_{x}(v) (since M/J_{x}M ∈ 𝔐(D_{x+F} , G^{ϕ}))

= ∑_{j=1}^{m}ω^{−1} ⊗ ∂_{j}ξ_{j}ϖ_{x}(v)

= 0.

Thus 𝔤^{ϕ} = Lie(G_{0}^{ϕ}) acts trivially on ω^{−1} ⊗_{ℂ} T and the result follows.

We end these notes by the following particular case of Theorem 3.4. Recall that n_{0} = (y_{1}, …, y_{m})O(F) and ρ(v) = 1_{U} ⊗_{ȷ^{#}} ϖ(t_{−x}.v).

Corollary 3.5. Let M = D_{𝔤}v ∈ 𝔐(D_{𝔤}, G) with v ∈ M^{G}. Then, t_{x}^{∗}M_{|U} = D_{U}ρ(v) and ψ^{∗}M = D_{Y} (1_{G}⊠ρ(v)). Furthermore, if k = 1 and n_{0}ρ(v) = 0, then we have (i) T = ℂϖ_{x}(v); (ii) the representation of A(O) on M_{|O}(x) is the trivial representation and the

connection M_{|O} is isomorphic to the standard D_{O}-module O_{O}.

Proof. Since ψ is smooth, it is easy to see [10, Lemma 3.2] that ψ^{∗}M = D_{Y} ψ^{#}(v). As explained in §2 (cf. Lemma 2.9, Theorem 2.10 and Remark at the end of §2) we may identify the D_{Y} -module ψ^{∗}M with O(G) ⊠ t_{x}^{∗}M_{|U}, and, since v ∈ M^{G}, ψ^{#}(v) identifies with 1_{G}⊠ρ(v). Thus,

ψ^{∗}M = O_{G} ⊠ t_{x}^{∗}M_{|U} = D_{G×U}(1_{G}⊠ρ(v)) = O_{G} ⊠ D_{U}ρ(v),

proving the first assertions of the corollary.

Now, assume that k = 1 and n_{0}ρ(v) = 0. Then, t_{x}^{∗}M_{|U} ≅ H_{[0]}^{m}(O_{U}) and ρ(v) identifies with ϖ(t_{−x}.v) inside T_{0} = {u ∈ t_{x}^{∗}M_{|U} : n_{0}u = 0} (loc. cit.). Since dim T_{0} = 1 and ρ(v) ≠ 0, we obtain T_{0} = ℂρ(v). It follows then from Theorem 2.10 that ω^{−1} ⊗_{ℂ} T = ℂ(ω^{−1} ⊗ ϖ_{x}(v)). By Theorem 3.4, since v ∈ M^{G}, the group G^{ϕ} acts trivially on M_{|O}(x) = ω^{−1} ⊗_{ℂ} T. The isomorphism M_{|O} ≅ O_{O} then follows from Proposition 2.8. □

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

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