Green Functions Via Hyperbolic Localization

Pramod N. Achar^{1}

Received: April 26, 2010

Communicated by Wolfgang Soergel

Abstract. Let G be a reductive algebraic group, with nilpotent cone N and flag variety ℬ. We construct an exact functor from perverse sheaves on N to locally constant sheaves on ℬ, and we use it to study Ext-groups and stalks of simple perverse sheaves on N in terms of the cohomology of ℬ.

2010 Mathematics Subject Classification: Primary 20G05; Secondary 14F05, 32S60. Keywords and Phrases: nilpotent cone; perverse sheaves; hyperbolic localization.

1. Introduction

Let G be a connected reductive algebraic group over an algebraically closed field 𝕜 of good characteristic. Let N denote the nilpotent cone in its Lie algebra 𝔤, and let W denote its Weyl group. An explicit description of the stalks of simple perverse sheaves on N has been given by Lusztig [L], building on earlier ideas of Shoji [S1, S2]. For most such perverse sheaves (those appearing in the Springer correspondence), this description involves the representation theory of W, and specifically its coinvariant algebra. The coinvariant algebra of W is also isomorphic to the cohomology ring H^{•}(ℬ) of the flag variety ℬ. However, that cohomology ring does not appear in the proofs in [L], which rely instead on orthogonality properties of character sheaves coming from the geometry of semisimple classes. The present paper is an attempt to understand Lusztig’s results directly in terms of the geometry of ℬ. Consider the diagram

(1.1)  \xymatrix{N & Ñ \ar[l]_{μ} \ar[r]^{π} & ℬ,}

where Ñ denotes the cotangent bundle of ℬ, π is the natural projection map, and μ is the Springer resolution. We study the functor

(1.2)  Φ = π_{!}μ^{∗} : D_{m}^{b}(N) → D_{m}^{b}(ℬ),

1The author was partly supported by National Security Agency grant H98230-09-1-0024.

where D_{m}^{b}(X) denotes the category of mixed complexes of ℚ̄_{ℓ}-sheaves (for some ℓ ≠ char 𝕜) that are constructible with respect to the G-orbits on X. The main results, proved in Sections 3–4, are summarized in the statement below. This statement involves the following categories: P_{G}(N) ⊂ D_{m}^{b}(N) is the abelian category of perverse sheaves; Spr ⊂ P_{G}(N) is the Serre subcategory containing the simple perverse sheaves appearing in the Springer correspondence; and Sh_{G}(ℬ) ⊂ D_{m}^{b}(ℬ) is the abelian category of locally constant sheaves.

Theorem 1.1. The functor Φ restricts to give an exact functor of abelian categories Φ|_{P_{G}(N)} : P_{G}(N) → Sh_{G}(ℬ). Moreover, for F, F′ ∈ Spr, the objects Φ(F) and Φ(F′) and the vector space \underline{Hom}^{i}(Φ(F), Φ(F′)) each carry a natural action of the Weyl group W, and Φ induces an isomorphism

(1.3)  \underline{Hom}_{D_{m}^{b}(N)}^{i}(F, F′) →^{∼} \underline{Hom}_{D_{m}^{b}(ℬ)}^{i} (Φ(F), Φ(F′))^{W} .

Here, the notation “\underline{Hom}” denotes a Hom-group equipped with an action of Frobenius; along the way to the theorem above, we show that \underline{Hom}_{D_{m}^{b}(N)}^{i}(F, F′) is pure. However, weights and purity are not used in any essential way; the main results are also valid in the unmixed setting. The W -action on Φ(F) induces one on the space of global sections Γ(Φ(F)), and the composition Γ ∘ Φ|_{Spr} : Spr → Rep(W) turns out to be an equivalence of categories that may be regarded as a categorical version of the Springer correspondence. On the other hand, the W -action on \underline{Hom}^{i}(Φ(F), Φ(F′)) is a generalization of the usual action of W on H^{•}(ℬ). Indeed, (1.3) can be used together with known formulas for the fake degrees of W to carry out explicit calculations of Ext-groups. As an application, in Sections 5–6, we use Theorem 1.1 to give new proofs of two results from [L]: a decomposition of D_{m}^{b}(N) into orthogonal subcategories, and the algorithmic description of stalks of perverse sheaves on N mentioned above.

Acknowledgments. The author is grateful to J.M. Douglass, A. Henderson, and D. Treumann for helpful conversations.

2. Preliminaries

2.1. General conventions. Throughout the paper, G and all related varieties will be assumed to be defined over the algebraic closure 𝕜 of a finite field 𝔽_{q} and equipped with an 𝔽_{q}-rational structure. For any G-variety X, D_{m}^{b}(X) will denote the category of mixed étale ℚ̄_{ℓ}-complexes that are constructible with respect to some G-stable stratification. Let 1_{X} denote the constant sheaf with value ℚ̄_{ℓ}, and let pt = Spec 𝕜. Let ω_{X} = a^{!}1_{pt} (where a : X → pt is the constant map) denote the dualizing complex, and let 𝔻 = RHom(·, ω_{X}) denote the Verdier duality functor. For F, F′ ∈ D_{m}^{b}(X), we let \underline{Hom}^{i}(F, F′) = H^{i}(a_{∗} RHom(F, F′)). This is a mixed ℚ̄_{ℓ}-sheaf on a point, i.e., a ℚ̄_{ℓ}-vector space equipped with an action of Frobenius. Forgetting that action yields the ℚ̄_{ℓ}-vector space of morphisms F → F′[i] over 𝕜. Nearly all results (those not explicitly involving purity) are also valid in the setting of unmixed sheaves over an arbitrary algebraically closed field 𝕜 of good characteristic, or for ℂ-sheaves in the classical topology when 𝕜 = ℂ.

2.2. Further notation. Let d = dim ℬ. We will frequently encounter shifts and Tate twists related to dim ℬ, so we adopt the following shorthand notation: if F is any sheaf, morphism, or functor, we put

F^{♭} = F [2d](d)  and  F^{♯} = F [−2d](−d).

Because ℬ is a smooth variety of dimension d, we have ω_{ℬ} ≃ 1_{ℬ}^{♭} , and because π is a smooth map of relative dimension d, we also have π^{!} ≃ (π^{∗})^{♭} and ω♯ ; Ñ ≃ 1♭ ; Ñ. Throughout the paper, W will denote the universal Weyl group of G, cf. [CG, §3.1]. This group does not depend on the choice of a maximal torus or a Borel subgroup, and we do not fix any such choice in this paper. Let Irr(W) denote the set of isomorphism classes of irreducible representations of W on ℚ̄_{ℓ}-vector spaces. For each χ ∈ Irr(W), choose a representative V_{χ}. We will sometimes regard V_{χ} as a pure object of weight 0 in D_{m}^{b}(pt), by letting the Frobenius act on it as the identity. It is known that

(2.1)  V_{χ} ≃ V_{χ}^{∗}

for all χ ∈ Irr(W). However, this isomorphism is not canonical. It is well known that W acts naturally on the cohomology ring H^{•}(ℬ), and that under this action, H^{•}(ℬ) can be identified with the coinvariant algebra of W. Let

κ : W → Aut(H^{•}(ℬ))

denote this action. Let Z denote the Steinberg variety Z = Ñ ×_{N} Ñ. Finally, let ι : ℬ → Ñ be the inclusion of the zero section, and let i_{0} : pt → N denote the inclusion of the point 0. We then have a cartesian square

(2.2)  \xymatrix{ℬ \ar[r]^{ι} \ar[d]_{a} & Ñ \ar[d]^{μ} \\ pt \ar[r]_{i_{0}} & N}

2.3. Springer correspondence. Let A = μ_{∗}1♭ ; Ñ = μ_{∗}1Ñ [2d](d). This is a semisimple perverse sheaf on N, known as the Springer sheaf. One approach to studying A, developed in detail in [CG], involves Borel–Moore homology, which is defined in terms of the hypercohomology of the dualizing complex. For our purposes, it is convenient to adopt a slightly nonstandard normalization and put

H̆^{i}(X) = H^{i}(X, ω_{X}[−4d](−2d)).

The Borel–Moore homology of the Steinberg variety H̆^{•}(Z) is equipped with a “convolution product,” making it into a graded algebra. Two key results are that there are natural algebra isomorphisms

(2.3)  H̆^{0}(Z) ≃ ℚ̄_{ℓ}[W]  and  H̆^{•}(Z) ≃ \underline{Hom}^{•}(A, A),

and that the latter is an isomorphism of graded algebras. In particular, we have a natural isomorphism ℚ̄_{ℓ}[W] ≃ \underline{End}(A), and so an action

σ : W → Aut(A).

Any action of W on A would allow us to decompose A into isotypic components, but since ℚ̄_{ℓ}[W] ≃ \underline{End}(A), we actually have

(2.4)  A ≃ ⨁_{χ∈Irr(W)} IC_{χ} ⊗ V_{χ},

where the IC_{χ} are various distinct simple perverse sheaves on N. This labeling of certain simple perverse sheaves by Irr(W) is what is usually known as the Springer correspondence. For χ ∈ Irr(W), let O_{χ} ⊂ N be the unique open G-orbit in the support of IC_{χ}, and let L_{χ} be the irreducible local system given by IC_{χ}|_{O_{χ}}[− dim O_{χ}](−\frac{1}{2} dim O_{χ}). Thus, IC_{χ} ≃ IC(O_{χ}, L_{χ}).

2.4. Modules for H̆^{•}(Z). The convolution product construction also makes the Borel–Moore homology of any subvariety of Ñ into a graded H̆^{•}(Z)-module. The convolution action of H̆^{0}(Z) on H̆^{•}(ℬ) ≃ H^{•}(ℬ)^{♯} is none other than the action κ. It is clear by base change in (2.2) that i_{0}^{∗}A ≃ H^{•}(ℬ)^{♭} ≃ H̆^{•}(ℬ)^{♭♭}. The functor i_{0}^{∗} therefore induces a map i_{0}^{∗} : \underline{End}(A) → End(H^{•}(ℬ)) that is a homomorphism of H̆^{•}(Z)-modules. In particular, it is W -equivariant, so we have

i_{0}^{∗}(σ(w)f) = κ(w)i_{0}^{∗}(f).

Finally, consider H̆^{•}(Ñ). This is a convolution algebra in its own right. Since H̆^{•}(Ñ) ≃ H^{•}(Ñ), it has another algebra structure given by cup product in ordinary cohomology, but it follows from [CG, Theorem 8.6.7] (cf. [DR, Theorem 2.3]) that these two algebra structures coincide. The following theorem of Douglass–Röhrle [DR] relates the W -action on H̆^{•}(Ñ) to that on H̆^{•}(Z).

Theorem 2.1 (Douglass–Röhrle). Let δ : Ñ → Z be the diagonal embedding. The induced map δ_{∗} : H̆^{•}(Ñ) → H̆^{•}(Z) in Borel–Moore homology satisfies

δ_{∗}(w ⋆ f) = w ⋆ δ_{∗}(f) ⋆ w^{−1}.

for any w ∈ W, where ⋆ denotes the convolution product.

2.5. Weakly 𝔾_{m}-equivariant objects. Let X be a variety endowed with an action of 𝔾_{m}. An object F ∈ D_{m}^{b}(X) is said to be weakly equivariant if it is in the image of the forgetful functor 𝕌 : D_{m,𝔾_{m}}^{b}(X) → D_{m}^{b}(X), where D_{m,𝔾_{m}}^{b}(X) denotes the 𝔾_{m}-equivariant derived category of X in the sense of Bernstein–Lunts [BL]. Weakly equivariant objects have the following useful property.

Lemma 2.2 (Springer [Sp, Proposition 1], cf. Braden [B, Lemma 6]). Let p : V → Z be a vector bundle, and suppose 𝔾_{m} acts linearly on the fibers of p with strictly positive weights (or strictly negative weights). Let i : Z → V be the inclusion of the zero section. For a weakly equivariant object S ∈ D_{m}^{b}(V), there are natural isomorphisms i^{!}S → p_{!}S and p_{∗}S → i^{∗}S. □

Remark 2.3. Any object obtained by pullback or push-forward of a weakly equivariant object along a 𝔾_{m}-equivariant map is automatically weakly equivariant, and the constant sheaf is always weakly equivariant. Therefore:

• If 𝔾_{m} acts on Ñ by scaling along the fibers, and on N by scaling, then the objects 1Ñ, A, μ^{∗}A, μ^{!}A, and all direct summands of the last three are weakly equivariant.

• If 𝔾_{m} acts on N by an action that factors through G̃, where G̃ is a group isogenous to G with simply-connected derived subgroup, then every semisimple perverse sheaf on N is weakly equivariant, since any local system on any nilpotent orbit is G̃-equivariant.

Remark 2.4. Suppose ϕ is a morphism of functors that is an isomorphism on weakly equivariant objects. If the domain category of ϕ is generated as a triangulated category by weakly equivariant objects, then a standard dévissage argument shows that ϕ actually induces isomorphisms for all objects; in other words, ϕ is an isomorphism of functors outright. This observation will be used when we apply Lemma 2.2 and other results to D_{m}^{b}(N), which is generated by the objects of P_{G}(N).

3. Exactness of Φ

In this section, we use hyperbolic localization to prove exactness results for the functor Φ = π_{!}μ^{∗} : D_{m}^{b}(N) → D_{m}^{b}(ℬ) of (1.2), as well as for the dual functor

Φ′ = (π_{∗}μ^{!})^{♯} : D_{m}^{b}(N) → D_{m}^{b}(ℬ).

To study these functors, we will make use of the additional functor

Ψ = μ_{∗}π^{!} ≃ (μ_{!}π^{∗})^{♭} : D_{m}^{b}(ℬ) → D_{m}^{b}(N).

It is clear that Ψ is left-adjoint to Φ′ and right-adjoint to Φ. In addition, we have

Φ ≃ ι^{!}μ^{∗}, Φ′ ≃ (ι^{∗}μ^{!})^{♯}, Ψ(1_{ℬ} ) ≃ A,

with the first two assertions relying on Remark 2.4. Moreover, since all objects of Sh_{G}(ℬ) are direct sums of copies of 1_{ℬ} , it follows that Ψ restricts to an exact functor of abelian categories Ψ : Sh_{G}(ℬ) → P_{G}(N). The main result of this section is the following.

Theorem 3.1. The functors Φ, Φ′ : D_{m}^{b}(N) → D_{m}^{b}(ℬ) restrict to give isomorphic exact functors of abelian categories Φ ≃ Φ′ : P_{G}(N) → Sh_{G}(ℬ).

We first require the following preliminary result.

Lemma 3.2. Let e : pt → ℬ be the inclusion of a point. There is a natural isomorphism of functors e^{∗}π_{∗}μ^{!♯} →^{∼} e^{∗}π_{!}μ^{∗} : D_{m}^{b}(N) → D_{m}^{b}(pt).

Proof. For this proof, we fix a choice of Borel subgroup B ⊂ G and maximal torus T ⊂ B. Recall that these choices yield a canonical identification W = N_{G}(T)/T. Let 𝔤 = 𝔲̄ ⊕ 𝔱 ⊕ 𝔲 be the corresponding triangular decomposition of 𝔤. That is, 𝔱 = Lie(T), 𝔲 is the nilpotent radical of Lie(B), and 𝔲̄ is the nilpotent radical of the Lie algebra of the opposite Borel subgroup. Choose a regular dominant cocharacter λ : 𝔾_{m} → T, and let 𝔾_{m} act on 𝔤 by composing λ with the adjoint action of T on 𝔤. Clearly, the triangular decomposition of 𝔤 is stable under this action. Moreover, 𝔾_{m} acts on 𝔲 with positive weights and on 𝔲̄ with negative weights, and it acts trivially on 𝔱. From these observations, it is easy to see that the point 0 ∈ N is the unique fixed point for the action of 𝔾_{m} on N, and that

𝔲 = \{x ∈ N | lim_{\begin{matrix}z∈𝔾_{m} \\ z→0 \end{matrix}} zẋ = 0\}  and  𝔲̄ = \{x ∈ N | lim_{\begin{matrix}z∈𝔾_{m} \\ z→∞ \end{matrix}} zẋ = 0\}.

Consider the following diagram of inclusion maps:

\xymatrix{pt \ar[r]^{i} \ar[d]_{ī} & 𝔲 \ar[d]^{g} \\ 𝔲̄ \ar[r]_{ḡ} & N}

This is a setting in which we may apply the formalism of hyperbolic localization, following [B]. The main theorem of [B] states that there is a natural morphism of functors ī^{∗}ḡ^{!} → i^{!}g^{∗} that is an isomorphism on weakly equivariant objects. By Remark 2.4, this is an isomorphism of functors in our situation. Next, consider the constant maps p : 𝔲 → pt and p̄: 𝔲̄ → pt. Using Lemma 2.2, we obtain a natural isomorphism p̄_{∗}ḡ^{!} →^{∼} p_{!}g^{∗}. Finally, let e : pt → ℬ (resp. ē: pt → ℬ) be the inclusion of the point corresponding to the Borel subgroup B (resp. the opposite Borel subgroup to B). Forming pullbacks over π, we obtain the diagrams

\xymatrix{ & 𝔲 \ar[d]_{ẽ} \ar[r]^{p} & pt \ar[d]^{e} &  & 𝔲̄ \ar[d]_{˜ē} \ar[r]^{p̄} & pt \ar[d]^{ē} \\ N & Ñ \ar[l]_{μ} \ar[r]^{π} & ℬ & N & Ñ \ar[l]_{μ} \ar[r]^{π} & ℬ}

It is clear that μẽ = g and μ˜ē = ḡ. By base change, we have p_{!}ẽ^{∗} ≃ e^{∗}π_{!} and p̄_{∗}˜ē^{!} ≃ ē^{!}π_{∗}. Combining with our earlier observations, we obtain an isomorphism e^{!}π_{∗}μ^{!} →^{∼} ē^{∗}π_{!}μ^{∗}. It is clear that on D_{m}^{b}(ℬ), we have ē^{∗} ≃ e^{∗} and e^{!} ≃ e^{∗♯}, so we now have the isomorphism e^{∗}π_{∗}μ^{!♯} →^{∼} e^{∗}π_{!}μ^{∗}, as desired. □

Proof of Theorem 3.1. We begin by showing that Φ|_{P_{G}(N)} is exact. Let F ∈ P_{G}(N). We wish to show that H^{j}(Φ(F)) = 0 for j ≠ 0. First, observe that for n < 0, we have

\underline{Hom}(Φ(F), 1_{ℬ} [n]) ≃ \underline{Hom}(F, Ψ(1_{ℬ} )[n]) ≃ \underline{Hom}(F, A[n]) = 0.

This shows that H^{j}(Φ(F)) = 0 for j > 0.

By duality, we have H^{j}(π_{∗}μ^{!}F) ≃ H^{j}(𝔻π_{!}μ^{∗}𝔻F) ≃ (𝔻H^{−2d−j}(π_{!}μ^{∗}𝔻F))[−2d]. Thus, H^{j}(π_{∗}μ^{!}F) = 0 if j < −2d. For the inclusion of a point e : pt → ℬ, we know that e^{∗} is an exact functor on Sh_{G}(ℬ), so it follows that H^{j}(e^{∗}π_{∗}μ^{!}F^{♯}) = 0 if j < 0. By Lemma 3.2, this implies that H^{j}(e^{∗}π_{!}μ^{∗}F) = 0 for j < 0 as well. Now, e^{∗} is also faithful on the abelian category Sh_{G}(ℬ) (though not, of course, on D_{m}^{b}(ℬ)), so that vanishing implies that H^{j}(π_{!}μ^{∗}F) = 0 for j < 0, as desired. Thus, Φ|_{P_{G}(N)} is exact. Finally, note that the exact functor e^{∗} : Sh_{G}(ℬ) → Sh(pt) is an equivalence of categories. The isomorphism Φ|_{P_{G}(N)} ≃ Φ′|_{P_{G}(N)} then follows from Lemma 3.2. □

Corollary 3.3. The functor Φ ≃ Φ′ : P_{G}(N) → Sh_{G}(ℬ) preserves purity. That is, it takes pure objects of weight w to pure objects of weight w.

Proof. Let F ∈ P_{G}(N) be pure of weight w. By the well-known rules [BBD, 5.1.14] for behavior of weights under various sheaf functors, we see that Φ = π_{!}μ^{∗} takes F to an object with weights ≤ w, whereas Φ′ = π_{∗}μ^{!} takes it to one with weights ≥ w. □

4. Action of the Weyl Group

In Section 2, we considered the action σ of W on \underline{End}(A), and the action κ on H^{•}(ℬ). In this section, we discuss several additional actions, and prove a W -equivariance result for Ψ. There are two natural commuting actions λ, ρ : W → Aut(\underline{Hom}^{i}(A, A)), given by

λ(w)(f) = σ(w) ∘ f  and  ρ(w)(f) = f ∘ σ(w^{−1}).

The exactness result of Section 3 allows us to construct a new action as follows.

Proposition 4.1. For any F ∈ P_{G}(N), the sheaf Φ(F) carries a natural action of W. If F is simple, then we have

Φ(F) ≃ \begin{cases}0 & if F ∉ Spr, \\ 1_{ℬ} ⊗ V_{χ}^{∗} & if F ≃ IC_{χ}. \end{cases}

Proof. The following general principle is easy to see: if 𝒜 is a semisimple 𝕜linear abelian category containing a unique simple object S up to isomorphism, and End(S) ≃ 𝕜, then any object A is canonically isomorphic to S⊗Hom(S, A). Applying this to Sh_{G}(ℬ), we have Φ(F) ≃ 1_{ℬ} ⊗ \underline{Hom}(1_{ℬ} , Φ(F)). By adjunction, \underline{Hom}(1_{ℬ} , Φ(F)) ≃ \underline{Hom}(Ψ(1_{ℬ} ), F) ≃ \underline{Hom}(A, F). The W -action on A induces one on \underline{Hom}(A, F) for any F, and therefore on Φ(F) ≃ 1_{ℬ} ⊗\underline{Hom}(A, F). For simple F, it is clear from (2.4) that Hom(A, F) = 0 if F ∉ Spr, and that Hom(A, IC_{χ}) ≃ V_{χ}^{∗}. □

The action described in this proposition will be denoted ν : W → Aut(Φ(F)). This action gives rise to a W -action on the vector space Γ(Φ(F)) for any F ∈ P_{G}(N). In other words, the functor Γ ∘ Φ may be regarded as taking values in Rep(W). Since P_{G}(N) and Rep(W) are both semisimple categories, the following result is immediate from Proposition 4.1.

Theorem 4.2. For a simple perverse sheaf F ∈ P_{G}(N), the functor Γ ∘ Φ : P_{G}(N) → Rep(W) is given by

(Γ ∘ Φ)(F) ≃ \begin{cases}0 & if F ∉ Spr, \\ V_{χ}^{∗} & if F ≃ IC_{χ}. \end{cases}

In particular, Γ ∘ Φ|_{Spr} : Spr → Rep(W) is an equivalence of categories. □

Corollary 4.3. The category Spr ⊂ P_{G}(N) is stable under Verdier duality 𝔻. In fact, for each simple perverse sheaf IC_{χ} ∈ Spr, we have 𝔻IC_{χ} ≃ IC_{χ}.

Proof. It follows from Theorem 3.1 and the fact that ℬ is a projective variety that Γ ∘ Φ commutes with 𝔻. For a simple perverse sheaf F ∈ P_{G}(N), we see that (Γ ∘ Φ)(F) ≠ 0 if and only if (Γ ∘ Φ)(𝔻F) ≠ 0, so 𝔻 preserves Spr. Moreover, for F ≃ IC_{χ}, we have (Γ ∘ Φ)(𝔻IC_{χ}) ≃ 𝔻(V_{χ}^{∗}) ≃ V_{χ}. The result follows using the noncanonical isomorphism (2.1). □

Note that when F = A, ν is obtained via the adjunction isomorphism

(4.1)  θ : \underline{Hom}(1_{ℬ} , Φ(A)) →^{∼} \underline{Hom}(A, A)

from the action on Hom(A, A) that we have called ρ. The other action λ on Hom(A, A) also induces an action on Φ(A), which we denote

λ̂ : W → Aut(Φ(A)).

Since ρ and λ commute, ν and λ̂ commute as well. By an abuse of notation, we will also write ν and λ̂ for the corresponding actions on the vector space \underline{Hom}^{i}(1_{ℬ} , Φ(A)). By definition, we have

(4.2)  θ(λ̂(v)ν(w)f) = λ(v)ρ(w)θ(f) = σ(v) ∘ θ(f) ∘ σ(w^{−1}).

Lemma 4.4. The map Ψ : \underline{Hom}^{i}(1_{ℬ} , 1_{ℬ} ) → \underline{Hom}^{i}(A, A) has the property that Ψ(κ(w)f) = σ(w) ∘ Ψ(f) ∘ σ(w^{−1}).

Proof. This is essentially a restatement of Theorem 2.1 due to Douglass–Röhrle. Note first that the functor π^{!} induces a W -equivariant isomorphism H^{•}(ℬ) →^{∼} H^{•}(Ñ), so it suffices to study the W -equivariance of μ_{∗} : D_{m}^{b}(Ñ) → D_{m}^{b}(N). Let q : Z → N be the natural projection map. We then have a commutative diagram:

\xymatrix{ & Ñ \ar[dl]_{δ} \ar[dr]^{μ} \\ Z \ar[rr]_{q} &  & N}

Recall [CG, Section 8.6] that the second isomorphism in (2.3) arises from the fact that q_{∗}ω_{Z}^{♯♯} ≃ RHom(A, A). There is also a natural adjunction map δ_{∗}ωÑ → ω_{Z}. Consider the composition

\xymatrix{μ_{∗}ω♯♯ ; Ñ \ar[r] & q_{∗}ω_{Z}^{♯♯} \ar[r]^{∼} & RHom(A, A).}

Applying RΓ to the first map yields the induced map δ_{∗} : H̆^{•}(Ñ) → H̆^{•}(Z) in Borel–Moore homology. On the other hand, we can identify ω♯♯ ; Ñ ≃ 1Ñ ≃ RHom(1Ñ , 1Ñ), and then the composition above becomes the canonical morphism

μ_{∗} RHom(1Ñ , 1Ñ) → RHom(A, A),

and applying RΓ gives us the map \underline{Hom}^{•}(1Ñ , 1Ñ) → \underline{Hom}^{•}(A, A) induced by μ. To summarize, we have the following commutative diagram:

\xymatrix{H̆^{•}(Ñ) \ar[r]^{∼} \ar[d]_{δ_{∗}} & \underline{Hom}^{•}(Ñ , Ñ) \ar[d]^{μ} \\ H̆^{•}(Z) \ar[r]_{q} & \underline{Hom}^{•}(A, A)}

Since the top isomorphism is W -equivariant, and the bottom one is an algebra isomorphism sending w ∈ ℚ̄_{ℓ}[W] ≃ H̆^{0}(Z) to σ(w) ∈ \underline{End}(A), the result follows from Theorem 2.1. □

Proposition 4.5. There is a natural isomorphism

α : H^{•}(ℬ) ⊗ \underline{Hom}(1_{ℬ} , Φ(A)) →^{∼} \underline{Hom}^{•}(1_{ℬ} , Φ(A)).

Its composition with the adjunction θ, denoted

Θ = θ ∘ α : H^{•}(ℬ) ⊗ \underline{Hom}(1_{ℬ} , Φ(A)) →^{∼} \underline{Hom}^{•}(A, A),

is W -equivariant in the following way: for u, v, w ∈ W, we have

Θ(κ(u)f ⊗ λ̂(v)ν(w)g) = λ(v)ρ(w)Θ(κ(w^{−1}u)f ⊗ g).

Proof. Recall from the proof of Proposition 4.1 that Φ(A) ≃ 1_{ℬ} ⊗ \underline{Hom}(1_{ℬ} , Φ(A)). It follows that

\underline{Hom}^{i}(1_{ℬ} , Φ(A)) ≃ \underline{Hom}^{i}(1_{ℬ} , 1_{ℬ} ⊗ \underline{Hom}(1_{ℬ} , Φ(A)))

≃ \underline{Hom}^{i}(1_{ℬ} , 1_{ℬ} ) ⊗ \underline{Hom}(1_{ℬ} , Φ(A)).

Since H^{i}(ℬ) ≃ \underline{Hom}^{i}(1_{ℬ} , 1_{ℬ} ), we obtain the desired isomorphism α. Note that α is given by composition: that is, if f ∈ \underline{Hom}^{i}(1_{ℬ} , 1_{ℬ} ) and g ∈ \underline{Hom}(1_{ℬ} , Φ(A)), then

α(f ⊗ g) = g ∘ f.

The adjunction isomorphism θ behaves on compositions according to the rule θ(g ∘ f) = θ(g) ∘ Ψ(f). Using (4.2) and Lemma 4.4, we find

Θ(κ(u)f ⊗ λ̂(v)ν(w)g) = θ(λ̂(v)ν(w)g) ∘ Ψ(κ(w)f)
= σ(v) ∘ θ(g)σ(w^{−1}) ∘ σ(u) ∘ Ψ(f) ∘ σ(u^{−1})

= σ(v) ∘ θ(g) ∘ σ(w^{−1}u) ∘ Ψ(f) ∘ σ(u^{−1}w) ∘ σ(w^{−1})

= σ(v) ∘ θ(g) ∘ Ψ(κ(w^{−1}u)f) ∘ σ(w^{−1})
= λ(v)ρ(w)θ(g ∘ κ(w^{−1}u)f).

The result follows. □

Theorem 4.6. For χ, ψ ∈ Irr(W), there is a natural isomorphism

(4.3)  \underline{Hom}^{i}(Φ(IC_{χ}), Φ(IC_{ψ})) ≃ V_{χ} ⊗ H^{i}(ℬ) ⊗ V_{ψ}^{∗},

and thus \underline{Hom}^{i}(Φ(IC_{χ}), Φ(IC_{ψ})) is endowed with a natural action of W. Moreover, Φ induces isomorphisms

\underline{Hom}^{i}(IC_{χ}, IC_{ψ}) ≃ \underline{Hom}^{i}(Φ(IC_{χ}), Φ(IC_{ψ}))^{W} ≃ (V_{χ} ⊗ H^{i}(ℬ) ⊗ V_{ψ}^{∗})^{W} .

Proof. The isomorphism (4.3) is immediate from Proposition 4.1. Next, using (2.4), we can decompose \underline{Hom}^{i}(A, A) as

Hom^{i}(A, A) ≃ ⨁_{χ,ψ} Hom^{i}(IC_{χ}, IC_{ψ}) ⊗ V_{χ}^{∗} ⊗ V_{ψ}.

Thus, in terms of the action of W × W on Hom^{i}(A, A) by λ ⊠ ρ, we can find Hom^{i}(IC_{χ}, IC_{ψ}) by picking out the χ^{∗} ⊠ ψ-isotypic component:

Hom^{i}(IC_{χ}, IC_{ψ}) ≃ Hom_{W ×W} (V_{χ}^{∗} ⊠ V_{ψ}, Hom^{i}(A, A)).

Using Proposition 4.5, this is isomorphic to

\underline{Hom}_{W ×W} (V_{χ}^{∗} ⊠ V_{ψ}, H^{i}(1_{ℬ} ) ⊗ \underline{Hom}(1_{ℬ} , Φ(A))),

where W × W acts on H^{i}(1_{ℬ} ) ⊗ \underline{Hom}(1_{ℬ} , Φ(A)) by λ̂ ⊠ (κ ⊗ ν). That is, for v, w ∈ W and f ⊗ g ∈ H^{i}(1_{ℬ} ) ⊗ \underline{Hom}(1_{ℬ} , Φ(A)), we put

(v, w) · (f ⊗ g) = κ(w)f ⊗ λ̂(v)ν(w)g.

Using the adjunction (4.1) and the isomorphism (2.3), we see that \underline{Hom}(1_{ℬ} , Φ(A)) decomposes under λ̂ ⊠ ν as \underline{Hom}(1_{ℬ} , Φ(A)) ≃ ⨁_{ϕ} V_{ϕ} ⊗ V_{ϕ}^{∗}. Picking off the χ^{∗}-isotypic component for the first factor of W, we find that

\underline{Hom}^{i}(IC_{χ}, IC_{ψ}) ≃ Hom_{W} (V_{ψ}, H^{i}(ℬ) ⊗ V_{χ})) ≃ (V_{ψ}^{∗} ⊗ H^{i}(ℬ) ⊗ V_{χ})^{W} .  □

Corollary 4.7. For χ, ψ ∈ Irr(W), \underline{Hom}^{i}(IC_{χ}, IC_{ψ}) vanishes if i is odd, and is pure of weight i if i is even.

Proof. This follows from the previous theorem and the well-known fact that H^{i}(ℬ) vanishes if i is odd and is pure of weight i if i is even. □

5. Orthogonal Decomposition of D_{m}^{b}(N)

For a G-stable locally closed subvariety Y ⊂ N, and let D_{Spr}(Y) ⊂ D_{m}^{b}(Y) be the full triangulated subcategory generated by objects IC_{χ}|_{Y} with O_{χ} ⊂ Y. On the other hand, let D_{Spr}^{⊥}(Y) ⊂ D_{m}^{b}(Y) be the full triangulated subcategory generated by simple perverse sheaves IC(O, L)|_{Y} with O ⊂ Y but IC(O, L) ∉ Spr.

Theorem 5.1. For any G-stable locally closed subvariety u : Y ↪ N, we have

(5.1)  D_{m}^{b}(Y) ≃ D_{Spr}(Y) ⊕ D_{Spr}^{⊥}(Y).

Moreover, if s : Z ↪ Y is the inclusion of a smaller G-stable locally closed subvariety, the functors s^{∗} and s^{!} respect this decomposition: we have

(5.2)

s^{∗}(D_{Spr}(Y)), s^{!}(D_{Spr}(Y)) ⊂ D_{Spr}(Z),
s^{∗}(D_{Spr}^{⊥}(Y)), s^{!}(D_{Spr}^{⊥}(Y)) ⊂ D_{Spr}^{⊥}(Z).

Proof. If s : Z → Y is an open embedding, then (5.2) is obvious. Since the inclusion of any locally closed subvariety can be factored as a closed embedding followed by an open embedding, we henceforth treat (5.2) only in the case where Z is closed in Y. Let n_{Y} denote the number of nilpotent orbits in \overline{Y} ∖ Y. We will prove (5.1) and (5.2) simultaneously by induction on n_{Y}. Note that (5.1) is equivalent to the assertion that for F ∈ D_{Spr}(Y) and F′ ∈ D_{Spr}^{⊥}(Y), we have \underline{Hom}(F, F′) = \underline{Hom}(F′, F) = 0. We can further reduce to the case where F and F′ are shifts of simple perverse sheaves. That is, (5.1) is equivalent to the statement that if F and F′ are simple perverse sheaves with F ∈ D_{Spr}(Y) and F′ ∈ D_{Spr}^{⊥}(Y), then

(5.3)  \underline{Hom}^{i}(F, F′) = \underline{Hom}^{i}(F′, F) = 0  for all i ≥ 0.

We begin by proving (5.3) in the case where n_{Y} = 0, i.e., when Y is closed in N. In fact, since u_{∗} : D_{m}^{b}(Y) → D_{m}^{b}(N) is faithful for any closed Y ⊂ N, we may reduce to the case where Y = N. Since F ∈ Spr, F is a direct summand of A, and it suffices to show that \underline{Hom}^{i}(A, F′) = \underline{Hom}^{i}(F′, A) = 0. Since A ≃ Ψ(1_{ℬ} ), we have by adjunction that

\underline{Hom}^{i}(A, F′) ≃ \underline{Hom}^{i}(1_{ℬ} , Φ′(F′)) and \underline{Hom}^{i}(F′, A) ≃ \underline{Hom}^{i}(Φ(F), 1_{ℬ} ).

Since Φ(F′) = Φ′(F′) = 0 by Theorem 3.1 and Proposition 4.1, the desired vanishing holds. Suppose now that (5.1) is known to hold whenever n_{Y} ≤ k. Let us prove (5.2). Since Z is assumed to be a closed subvariety of Y, we clearly have n_{Z} ≤ n_{Y}; in particular, we know that D_{m}^{b}(Z) ≃ D_{Spr}(Z) ⊕ D_{Spr}^{⊥}(Z). Therefore, given F ∈ D_{Spr}(Y), we have a canonical decomposition s^{∗}F ≃ (s^{∗}F)_{Spr} ⊕ (s^{∗}F)_{Spr}^{⊥} with (s^{∗}F)_{Spr} ∈ D_{Spr}(Z) and (s^{∗}F)_{Spr}^{⊥} ∈ D_{Spr}^{⊥}(Z). We wish to prove that (s^{∗}F)_{Spr}^{⊥} = 0. If that is not the case, there certainly exists some simple perverse sheaf F′ ∈ D_{Spr}^{⊥}(Z) and some i ∈ ℤ such that \underline{Hom}^{i}((s^{∗}F)_{Spr}^{⊥}, F′) ≠ 0. We also know that \underline{Hom}^{•}((s^{∗}F)_{Spr}, F′) = 0, so it follows that

0 ≠ \underline{Hom}^{i}((s^{∗}F)_{Spr}^{⊥}, F′) ≃ \underline{Hom}^{i}(s^{∗}F, F′) ≃ \underline{Hom}^{i}(F, s_{∗}F′).

But s_{∗}F′ is clearly a simple perverse sheaf in D_{Spr}^{⊥}(Y), and since F ∈ D_{Spr}(Y), we have a contradiction. Thus, s^{∗}F ∈ D_{Spr}(Z). The proofs of the other assertions in (5.2) are parallel. Now, suppose that (5.1) and (5.2) are both known for n_{Y} ≤ k, and let us prove (5.3) when n_{Y} = k + 1. Let F and F′ be simple perverse sheaves on Y with F ∈ D_{Spr}(Y) and F′ ∈ D_{Spr}^{⊥}(Y). Choose an orbit O that is open in \overline{Y} ∖ Y, and let Ỹ = Y ∪ O. Let s : O → Ỹ and j : Y → Ỹ be the inclusion maps, and consider the objects j_{!∗}F, j_{!∗}F′ ∈ D_{m}^{b}(Ỹ). These are simple perverse sheaves on Ỹ; moreover, we clearly have j_{!∗}F ∈ D_{Spr}(Ỹ) and j_{!∗}F′ ∈ D_{Spr}^{⊥}(Ỹ). Form the distinguished triangle

s_{∗}s^{!}j_{!∗}F′ → j_{!∗}F′ → j_{∗}j^{∗}j_{!∗}F′ → .

Note that j^{∗}j_{!∗}F′ ≃ F′. Next, form the long exact sequence

⋯ →\underline{Hom}^{i}(j_{!∗}F, j_{!∗}F′) →\underline{Hom}^{i}(j_{!∗}F, j_{∗}F′) →\underline{Hom}^{i+1}(j_{!∗}F′, s_{∗}s^{!}j_{!∗}F′) →· · · .

Note that nỸ = n_{Y} − 1, so (5.3) holds on Ỹ by assumption. In particular, we have \underline{Hom}^{•}(j_{!∗}F, j_{!∗}F′) = 0. Since (5.2) also holds by assumption, we have s^{∗}j_{!∗}F ∈ D_{Spr}(O) and s^{!}j_{!∗}F′ ∈ D_{Spr}^{⊥}(O), so

\underline{Hom}^{•}(j_{!∗}F′, s_{∗}s^{!}j_{!∗}F′) = \underline{Hom}^{•}(s^{∗}j_{!∗}F′, s^{!}j_{!∗}F′) = 0.

We conclude that \underline{Hom}^{i}(j_{!∗}F, j_{∗}F′) = 0. But that means \underline{Hom}^{i}(j^{∗}j_{!∗}F, F′) ≃ \underline{Hom}^{i}(F, F′) = 0 as well, as desired. □

One concrete consequence of this theorem is the following.

Corollary 5.2. If L is an irreducible local system on an orbit O ⊂ N that occurs as a composition factor of some cohomology sheaf H^{i}(IC_{χ}|_{O}) with χ ∈ Irr(W), then L ≃ L_{ψ} for some ψ ∈ Irr(W). □

6. Green Functions

The aim of this section is to study the restrictions IC_{χ}|_{O} as O varies over the G-orbits in N. Specifically, we encode information about these restrictions in a family of polynomials p_{χ,ψ}(t), sometimes called Green functions. The main result, Theorem 6.1, gives a way to compute these polynomials, by relating them to the known groups Hom^{i}(IC_{χ}, IC_{ψ}). We begin with some notation. For a variety X, let K(X) denote the quotient of the Grothendieck group of D_{m}^{b}(X) obtained by identifying the classes of simple perverse sheaves of the same weight that become isomorphic when the Weil structure is forgotten. (Thus, K(X) does not detect twists in the Weil structure by a root of unity.) For an object F ∈ D_{m}^{b}(X), let [F] denote its class in K(N). This Grothendieck group is naturally a module over the Laurent polynomial ring ℤ[t, t^{−1}], where multiplication by t corresponds to Tate twist: [F (1)] = t^{−1}[F]. By choosing a square root of the Tate sheaf, we can regard K(X) as a module over ℤ[t^{1/2}, t^{−1/2}], with [F (\frac{1}{2})] = t^{−1/2}[F]. For instance, the group K(pt) is a free ℤ[t^{1/2}, t^{−1/2}]-module of rank 1, generated by the class [1_{pt}] of a 1-dimensional vector space of weight 0. For an orbit O ⊂ N, the group K(O) is a free ℤ[t^{1/2}, t^{−1/2}]-module generated by the classes of irreducible local systems on O. In view of Corollary 5.2, we may write

(6.1) [IC_{χ}|_{O}] = ∑_{{ψ|O_{ψ}=O}} p_{χ,ψ}(t)[L_{ψ}]  for some p_{χ,ψ}(t) ∈ ℤ[t^{1/2}, t^{−1/2}].

(That is, only local systems belonging to D_{Spr}(O) may appear on the righthand side.) We clearly have IC_{χ}|_{O_{χ}} ≃ L_{χ}[dim O_{χ}](\frac{1}{2} dim O_{χ}) and IC_{χ}|_{O} = 0 if O ⊄ \overline{O_{χ}}. In other words:

(6.2) p_{χ,ψ}(t) = \begin{cases}t^{−(dim O_{χ})/2} & if ψ = χ, \\ 0 & if O_{ψ} ⊄ \overline{O_{χ}}, or if O_{ψ} = O_{χ} and ψ ≠ χ. \end{cases}

Our goal is to determine the polynomials p_{χ,ψ}(t). Another description of these polynomials is as follows. Each cohomology sheaf H^{i}(IC_{χ}|_{O}) is, of course, a finite-length object in the category of local systems on O. Let (H^{i}(IC_{χ}|_{O}) : L_{ψ}(j)) denote the multiplicity of L_{ψ}(j) in any composition series. We then have

(6.3)  p_{χ,ψ}(t) = ∑_{i∈ℤ, j∈ \frac{1}{2} ℤ}(−1)^{i}(H^{i}(IC_{χ}|_{O}) : L_{ψ}(j))t^{−j}.

A result of Springer leads to a tremendous simplification of this formula; see Remark 6.2. Actually, p_{χ,ψ}(t) lies in ℤ[t^{−1}] and has nonnegative coefficients (see (6.7)), but we will not require these facts. To state the main result, we require two additional families of polynomials. First, define λ_{χ,ψ}(t) ∈ ℤ[t^{1/2}, t^{−1/2}] as follows:

[RΓ_{c}(O_{χ}, L_{χ} ⊗ L_{ψ})] = λ_{χ,ψ}(t)[1_{pt}]  if O_{χ} = O_{ψ},
(6.4)  λ_{χ,ψ}(t) = 0  if O_{χ} ≠ O_{ψ}.

Second, define polynomials ω_{χ,ψ}(t) by

[𝔻 RHom(IC_{χ}, IC_{ψ})] = ω_{χ,ψ}(t)[1_{pt}].

Recall from Corollary 4.7 that RHom(IC_{χ}, IC_{ψ}) is pure of weight 0 and has vanishing cohomology in odd degrees. The same statements then hold for its dual as well. As with p_{χ,ψ}(t) in (6.7), it follows that

ω_{χ,ψ}(t) = ∑_{i∈ℤ} dim H^{2i}(𝔻 RHom(IC_{χ}, IC_{ψ}))t^{i} = ∑_{i∈ℤ} dim \underline{Hom}^{−2i}(IC_{χ}, IC_{ψ})t^{i}.

The coinvariant algebra of W has the property that the W -action in complementary degrees is related by tensoring with the sign character ε: that is, H^{i}(ℬ) ≃ H^{2d−i}(ℬ) ⊗ ε as W -representations. Using the noncanonical isomorphism (2.1) together with Theorem 4.6, we can rewrite the above formula as

(6.5)  ω_{χ,ψ}(t) = t^{−2d} ∑_{i∈ℤ} dim Hom_{W} (V_{χ} ⊗ V_{ψ} ⊗ ε, H^{2i}(ℬ))t^{i}.

The main result of this section is the following.

Theorem 6.1. The matrices P = (p_{χ,ψ}), Λ = (λ_{χ,ψ}), and Ω = (ω_{χ,ψ}) satisfy

(6.6)  P ΛP^{t} = Ω,

where P^{t} is the transpose of P. In other words, given χ, ψ ∈ Irr(W), we have

∑_{ϕ,ϕ′∈Irr(W)} p_{χ,ϕ}(t)λ_{ϕ,ϕ′}(t)p_{ψ,ϕ′}(t) = ω_{χ,ψ}(t).

Moreover, P and Λ are the unique matrices with entries in ℚ(t^{1/2}) satisfying (6.6), (6.2), and (6.4).

Remark 6.2. This theorem is essentially equivalent to the part of [L, Theorem 24.8] relevant to Spr. The most substantial difference is that in loc. cit., the polynomials p_{χ,ψ}(t) are defined in a slightly different way. Correcting for different normalization conventions (see Remark 6.3), the definition in [L] is

(6.7)  p_{χ,ψ}(t) = ∑_{i∈ℤ}(H^{2i}(IC_{χ}|_{O}) : L_{ψ}(−i))t^{i}.

The equivalence of this formula with (6.3) is implied by an important result of Springer [Sp], which states that for any χ ∈ Irr(W) and any orbit O ⊂ N, the object IC_{χ}|_{O} ∈ D_{m}^{b}(O) is pure of weight 0, and that H^{i}(IC_{χ}|_{O}) = 0 if i is odd. The proof of (6.6) in [L] also relies on Springer’s purity theorem.

Remark 6.3 (Lusztig–Shoji algorithm). The uniqueness asserted in Theorem 6.1 is proved by Lusztig [L] in a very explicit constructive way. This proof, which will not be repeated here, consists primarily of a description of an algorithm for finding P and Λ from knowledge of Ω. Since Ω can be described as in (6.5) using only the representation theory of W, this algorithm can be effectively used to compute the p_{χ,ψ}(t). Generalizations of this algorithm, sometimes called the Lusztig–Shoji algorithm, have been studied in [AA, AH, GM, S3, S4, S5], and a computer implementation is available at [A]. The reader should be aware that Lusztig originally used polynomials p′_{χ,ψ}(t) and ω′_{χ,ψ}(t) following different normalization conventions, while the recent works mentioned above involve polynomials p′′_{χ,ψ}(t) and ω′′_{χ,ψ}(t) following a third convention. The relationship among these is as follows:

p′_{χ,ψ}(t) = t^{\frac{1}{2} dim O_{χ}}p_{χ,ψ}(t)  p′′_{χ,ψ}(t) = t^{d}p_{χ,ψ}(t)
ω′_{χ,ψ}(t) = t^{\frac{1}{2} (dim O_{χ}+dim O_{ψ})}ω_{χ,ψ}(t)  ω′′_{χ,ψ}(t) = t^{2d}ω_{χ,ψ}(t).

For the next three lemmas, let j_{O} : O → N denote the inclusion of an orbit.

Lemma 6.4. For any F, F′ ∈ D_{m}^{b}(N), we have

[𝔻 RHom(F, F′)] = ∑_{O⊂N}[𝔻 RHom(j_{O}^{∗}F, j_{O}^{!}F′)].

Proof. Let X ⊂ N denote the closure of the support of F′, and let O_{0} ⊂ X be an orbit that is open in X. Let h : Y → N be the inclusion of the closed subset Y = X ∖ O_{0}, and let s : (N ∖ Y) → N be the inclusion of its open complement. Consider the distinguished triangle h_{∗}h^{!}F′ → F′ → s_{∗}s^{∗}F′ →. Since s^{∗}F′ ≃ s^{!}F′ is supported on O, we see that s_{∗}s^{∗}F′ is naturally isomorphic to j_{O_{0}∗}j_{O_{0}}^{!}F′. Applying 𝔻 RHom(F, ·) and the usual adjunction properties, we obtain a distinguished triangle

𝔻 RHom(j_{O_{0}}^{∗}F, j_{O_{0}}^{!}F′) → 𝔻 RHom(F, F′) → 𝔻 RHom(h^{∗}F, h^{!}F′) →,

so [𝔻 RHom(F, F′)] = [𝔻 RHom(h^{∗}F, h^{!}F′)] + [𝔻 RHom(j_{O_{0}}^{∗}F, j_{O_{0}}^{!}F′)]. The result then follows by induction on the number of orbits in the support of F′. □

Lemma 6.5. For any orbit O ⊂ N, we have

[j_{O}^{!}IC_{χ}] = t^{− dim O} ∑_{{ψ|O_{ψ}=O}} p_{χ,ψ}(t^{−1})[L_{ψ}].

Proof. Using Corollary 4.3, we have j_{O}^{!}IC_{χ} ≃ 𝔻j_{O}^{∗}(𝔻IC_{χ}) ≃ 𝔻(IC_{χ}|_{O}). We can therefore obtain a formula for for [j_{O}^{!}IC_{χ}] by applying 𝔻 to the righthand side of (6.1). For any local system L_{ψ} on O, we have 𝔻(L_{ψ}(−i)) ≃ L_{ψ}[2 dim O](dim O + i), so the map [𝔻(·)] : K(O) → K(O) sends t^{i}[L_{ψ}] ↦ t^{− dim O−i}[L_{ψ}]. The result follows. □

Lemma 6.6. We have

[𝔻 RHom(j_{O}^{∗}IC_{χ}, j_{O}^{!}IC_{ψ})] = ∑_{{ϕ,ϕ′|O_{ϕ}=O_{ϕ′} =O}} p_{χ,ϕ}(t)λ_{ϕ,ϕ′}(t)p_{ψ,ϕ′}(t)[1_{pt}].

Proof. Observe that 𝔻 RHom transforms Tate twists according to the formula 𝔻 RHom(F (n), F′(m)) ≃ 𝔻 RHom(F, F′)(n − m). That means that the homomorphism [𝔻 RHom(·, ·)] : K(O)×K(O) → K(pt) is ℤ[t^{1/2}, t^{−1/2}]-linear in the first variable, but antilinear with respect to the involution t^{1/2} ↦ t^{−1/2} in the second variable. Using (6.1) and Lemma 6.5, we find that

[𝔻 RHom(j_{O}^{∗}IC_{χ}, j_{O}^{!}IC_{ψ})] = t^{dim O} ∑_{{ϕ,ϕ′|O_{ϕ}=O_{ϕ′} =O}} p_{χ,ϕ}(t)p_{ψ,ϕ′}(t)[𝔻 RHom(L_{ϕ}, L_{ϕ′})].

It suffices to show that [𝔻 RHom(L_{ϕ}, L_{ϕ′})] = t^{−2 dim O}λ_{ϕ,ϕ′}(t)[1_{pt}]. Using Corollary 4.3, we have that RHom(L_{ϕ}, L_{ϕ′}) ≃ RHom(1_{O}, L_{ϕ}^{∗} ⊗ L_{ϕ′}) ≃ RΓ(O, L_{ϕ} ⊗ L_{ϕ′}), where L_{ϕ}^{∗} denotes the dual local system: L_{ϕ}^{∗} = (𝔻L_{ϕ})[−2 dim O](− dim O). Therefore,

𝔻 RHom(L_{ϕ}, L_{ϕ′}) ≃ RΓ_{c}(O, 𝔻(L_{ϕ} ⊗ L_{ϕ′}) ≃ RΓ(O, L_{ϕ} ⊗ L_{ϕ′})[2 dim O](dim O),

so [𝔻 RHom(L_{ϕ}, L_{ϕ′})] = t^{− dim O}[RΓ_{c}(O, L_{ϕ} ⊗ L_{ϕ′})], as desired. □

Proof of Theorem 6.1. The equation (6.6) follows from Lemmas 6.4 and 6.6, and the uniqueness assertion has been addressed in Remark 6.3. □

References

[A] P. Achar, An implementation of the generalized Lusztig– Shoji algorithm, software available for download from

     http://www.math.lsu.edu/~pramod/, 2008.

[AA] P. Achar and A.-M. Aubert, Springer correspondences for dihedral groups, Transform. Groups 13 (2008), 1–24.

[AH] P. Achar and A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (2008), 27–62.

[BBD] A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171.

[BL] J. Bernstein and V. Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994.

[B] T. Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), 209–216.

[CG] N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkhäuser Boston Inc., Boston, MA, 1997.

[DR] J. M. Douglass and G. Röhrle, Homology of the Steinberg variety and Weyl group coinvariants, Documenta Math. 14 (2009), 339–357.

[GM] M. Geck and G. Malle, On special pieces in the unipotent variety, Experiment. Math. 8 (1999), 281–290.

[L] G. Lusztig, Character sheaves. V, Adv. in Math. 61 (1986), 103–155.

[S1] T. Shoji, On the Green polynomials of a Chevalley group of type F_{4}, Comm. Algebra 10 (1982), 505–543.

[S2] T. Shoji, On the Green polynomials of classical groups, Invent. Math. 74 (1983), 239–267.

[S3] T. Shoji, Green functions associated to complex reflection groups, J. Algebra 245 (2001), 650–694.

[S4] T. Shoji, Green functions associated to complex reflection groups. II, J. Algebra 258 (2002), 563–598.

[S5] T. Shoji, Green functions attached to limit symbols, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 443–467.

[Sp] T. A. Springer, A purity result for fixed point varieties in flag manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), 271–282.

Pramod N. Achar

Department of Mathematics Louisiana State University Baton Rouge LA 70803 USA pramod@math.lsu.edu