VALUES OF THE FOURIER TRANSFORMS OF GREEN FUNCTIONS AT REGULAR ELEMENTS

DAVID G. A. JACKSON

1. Introduction

Let G be a connected reductive algebraic group defined over a finite field 𝔽_{q} whose characteristic p is a very good prime for G and let F be the corresponding Frobenius endomorphism of G. Let 𝔤 be the Lie algebra of G.

In 1976, Springer [28] showed that if X ∈ 𝔤 is nilpotent and B_{X}^{G} denotes the variety of Borel subgroups of G whose Lie algebras contain X then one can define an action of the Weyl group W of G on the ℓ-adic cohomology groups H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) which will here be called the Lusztig-Springer action. For w ∈ W and X ∈ 𝔤^{F} nilpotent, Springer gave a definition of the Green functions Q_{w}(X) in terms of certain trigonometric sums and showed that

Q_{w}(X) = ∑_{i⩾0}Tr(F w, H^{2i}(B_{X}, \overline{ℚ}_{ℓ})). (1.1)

It was then shown by Kazhdan [16], Lusztig [22] and Shoji [25] that the Green functions Q_{w} defined by means of the Deligne-Lusztig representation R_{T_{w}}^{G}(1) are also given by equation (1.1) (cf. Shoji [24, §13] and [25, Theorem 5.5]).

In 1981, Lusztig [21] showed how to define an action of W on H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) in the case where X is not necessarily nilpotent. Following Lehrer [19], we may thus define an Ad G-invariant function Q̃_{w} on 𝔤^{F} by

Q̃_{w}(X) = ∑_{i⩾0}Tr(F w, H^{2i}(B_{X}, \overline{ℚ}_{ℓ}))

for any X ∈ 𝔤^{F}.

In 1996, Lehrer [18] undertook a study of Ad G-invariant complex valued functions on 𝔤^{F} (simply called invariant functions in [18]) and, in particular, studied the operation F of Fourier transform on such functions. Lehrer showed that if C(𝔤^{F})_{nil} is the space of nilpotently supported invariant functions then Springer’s definition [28, 5.1–2] expressed the Green functions in terms of the values at strongly regular semisimple elements of the Fourier transforms of functions in C(𝔤^{F})_{nil}. From

FOURIER TRANSFORMS OF GREEN FUNCTIONS 2

the Springer correspondence (see, for example, Shoji [24, §6]) an important subspace of C(𝔤^{F})_{nil} is the space spanned by the Green functions. Lehrer [18, Proposition 6.6] has proved that, for good characteristic, FQ_{w} = ε(w)q^{−r/2}Q̃_{w}, where r is the rank of G and ε is the alternating character of W. The computation of FQ_{w} is thus equivalent to that of Q̃_{w}.

The aim of the current paper is to compute the values of the functions Q̃_{w} at regular elements of 𝔤^{F}. The main result can be stated as follows.

Main Theorem. Let G be a connected reductive group and suppose the characteristic p is very good for G. If X ∈ 𝔤^{F} is regular and J ⊆ Π, x ∈ G and w_{1} ∈ W are such that C_{G}(X_{s})^{∘} = ^{x}L_{J}, x^{−1}F (x) = ẇ_{1} and w_{1}F (J) = J then

Q̃_{w}^{𝔤}(X) = Ind_{W_{J} w_{1}F}^{W F} 1(F w) = |W_{J}|^{−1}|{v ∈ W | ^{v}(F w) ∈ W_{J}w_{1}F }|.

The computation is achieved by generalizing a result of Kato [15] which states (in the case where G is defined over ℂ) that the action of W on H^{i}(B_{X}^{G}, ℂ) can be obtained by inducing the action of W_{H} on H^{i}(B_{X}^{H}, ℂ) where H = C_{G}(X_{s})^{∘} and W_{H} is the Weyl group of H. In section 3, we prove a generalization of this result which relates the actions of W and F on H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) and those of W_{H} and F on H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ}). We take the opportunity here to expand on Kato’s proof and fill in some technical details. One of these details relies upon the result that the double coset L_{J} ẇP_{K} is closed in G (cf. Proposition 3.18).

A preliminary version of this paper is contained in the author’s PhD thesis [14]. This paper was completed under an ARC Research Project in Group representation theory and cohomology of algebraic varieties (Chief Investigator Gus Lehrer). The author would like to thank Bob Howlett and Jean Michel for many helpful suggestions and Matthew Douglass for providing a short proof of Proposition 3.18.

2. Notation and Background

For the general theory of reductive algebraic groups refer to Borel [3], Borel & Tits [4] and Springer [29]. We will often abbreviate the term “reductive algebraic” to merely “reductive”. Lower case gothic characters will be used to denote Lie algebras so, for example, 𝔤 (resp. 𝔥, 𝔟 etc.) denotes the Lie algebra of G (resp. H, B etc.). If H is a reductive group then H^{∘} will denote the connected component containing the identity and for X ∈ 𝔥

\begin{aligned}C_{H}(X) & = \{h ∈ H | Ad h(X) = X\} \\ C_{𝔥}(X) & = \{Y ∈ 𝔥 | [Y, X] = 0\}. \end{aligned}

FOURIER TRANSFORMS OF GREEN FUNCTIONS 3

Let T be a maximal torus of G contained in a Borel subgroup B. Let Φ be the root system of G with respect to T and Π the set of simple roots with respect to the pair (T, B). The coroot system will be denoted by Φ^{∨}, the root lattice by ℤΦ and the weight lattice Hom(ℤΦ^{∨}, ℤ) by Ω. If β = ∑_{α∈Π} n_{α}α is a root then we will define the height ht(β) of β to be the integer ∑_{α∈Π} n_{α}. For r ∈ Φ, let X_{r} be the one-parameter subgroup of G corresponding to root r. If Φ and Π are defined with respect to a rational maximal torus T contained in the rational maximal torus B then the Frobenius endomorphism induces a permutation of Φ, which we shall also denote by F, given by F (X_{r}) = X_{F (r)}. Then F permutes both Φ^{+} and Π. We shall let W be the Weyl group N_{G}(T)/T and denote the set of irreducible characters of W by Ŵ. We shall denote by ẇ a representative in N = N_{G}(T) of the element w of W. The reflection corresponding to the root r will be denoted by s_{r}. For w ∈ W, we will denote by U_{w} the group generated by those X_{r} for which r is a positive root but w(r) is a negative root. Whenever the Bruhat decomposition is invoked we will have in mind the version of Carter [9, Theorem 2.5.14]. If w, w′ ∈ W we will write ^{wF} w′ = wF (w′)w^{−1}. If J ⊆ Π then the root system Φ_{J} and groups W_{J}, P_{J}, L_{J}, U_{J}, B_{J}, N_{J}, etc. are as defined in Carter [9, Chapter 2]. A G-conjugate of a group of the form L_{J} will be called a Levi subgroup of G. We denote by D_{J} = {w ∈ W | w(J) ⊆ Φ^{+}} the set of distinguished (right) coset representatives of W_{J} in W (cf. Carter [8, 2.5.9]).

The group G acts on 𝔤 via the adjoint action. A function f : 𝔤^{F} → ℂ will be called simply an invariant function if it is invariant under Ad G^{F}. The space of invariant functions will be denoted by C(𝔤^{F}). The general properties of invariant functions are given in Lehrer [19].

We shall fix a nontrivial additive character ψ : 𝔽_{q}^{+} → ℂ^{×} of 𝔽_{q}. We may then define the Fourier transform F_{𝔤} = F : C(𝔤^{F}) → C(𝔤^{F}) by

Ff(X) = |𝔤^{F} |^{−1/2} ∑_{Y ∈𝔤^{F}} ψ(⟨X, Y ⟩)f(Y)

where f ∈ C(𝔤^{F}), X ∈ 𝔤^{F} and ⟨, ⟩:𝔤 × 𝔤 → \overline{𝔽}_{q} is an Ad G-invariant non-degenerate bilinear form defined over 𝔽_{q}. The existence of such a form in general imposes conditions on the characteristic (cf. Springer & Steinberg [30, Lemma I.5.3]).

An element g (resp. subgroup H) of G that is fixed by F will be said to be a rational element (resp. subgroup) of G and G^{F} is the group of rational points of G. We will call an Ad G^{F} -orbit in 𝔤^{F} a rational orbit.

If F acts on the group H then a, b ∈ H are said to be F -conjugate if there exists x ∈ H such that b = x^{−1}aF (x). We may form the semidirect product H ⋊ ⟨F ⟩ by

FOURIER TRANSFORMS OF GREEN FUNCTIONS 4

defining F · h = F (h) · F for h ∈ H. Then the elements a, b ∈ H are F -conjugate if and only if the elements aF, bF ∈ H ⋊ ⟨F ⟩ are conjugate by an element of H. We shall denote by H^{1}(F, H) the set of H-classes of elements of HF.

Let V be an algebraic variety defined over 𝔽_{q} and assume that an algebraic group G defined over 𝔽_{q} acts on V by an action defined over 𝔽_{q}. Let O be an F -stable orbit and X, Ad g(X) ∈ O^{F}. If c ∈ Stab_{G}(X)/ Stab_{G}(X)^{∘} is the image of g^{−1}F (g) under the canonical map then we say that Ad g(X) is c-twisted with respect to X. By Digne & Michel [12, 3.21], c is determined up to F -conjugacy.

If X ∈ 𝔤, we will denote the semisimple and nilpotent parts of X by X_{s} and X_{n} respectively.

Let B^{G} be the variety of Borel subgroups of G. For X ∈ 𝔤 let B_{X}^{G} = {B′ ∈ B^{G} | X ∈ Lie B′}. We will simply write B and B_{X} if no confusion arises. Let 𝔤_{rs} be the dense open subset of 𝔤 consisting of all regular semisimple elements of 𝔤. Define

\begin{aligned}𝔤̃ & = \{(X, B′) | X ∈ 𝔤, B′ ∈ B_{X}^{G}\}, \\ 𝔤̃_{rs} & = \{(X, B′) | X ∈ 𝔤_{rs}, B′ ∈ B_{X}^{G}\}. \end{aligned}

Let p_{𝔤} : 𝔤̃ → 𝔤 and p_{𝔤_{rs}} : 𝔤̃_{rs} → 𝔤_{rs} be the projections onto their first entries.

If X is an algebraic variety defined over 𝔽_{q} and ℓ is a prime not equal to p then we can define the ℓ-adic cohomology groups with compact support H_{c}^{i}(X, \overline{ℚ}_{ℓ}). For details of their definition and proofs of the following facts see, for example, Srinivasan [31, Chapter V] and Digne & Michel [12, Chapter 10]. Any finite morphism f : X → X induces a linear endomorphism of H_{c}^{i}(X, \overline{ℚ}_{ℓ}) for any i via the inverse image f^{∗}. In the case of the Frobenius morphism, we will also denote by F the endomorphism of H_{c}^{i}(X, \overline{ℚ}_{ℓ}) given by (F^{∗})^{−1}.

If X is of dimension n then H_{c}^{i}(X, \overline{ℚ}_{ℓ}) = 0 unless 0 ⩽ i ⩽ 2n. If X is an affine space of dimension n, the ℓ-adic cohomology groups H_{c}^{i}(X, \overline{ℚ}_{ℓ}) all vanish except H_{c}^{2n}(X, \overline{ℚ}_{ℓ}), which is one dimensional. The linear endomorphism induced by F on H_{c}^{2n}(X, \overline{ℚ}_{ℓ}) is then simply multiplication by q^{n}. Let f : X → X′ be a continuous map of \overline{𝔽}_{q}-schemes and let S and S′ denote the categories of complexes of \overline{ℚ}_{ℓ}-sheaves on X and X′ respectively. We will consider a \overline{ℚ}_{ℓ}-sheaf as a complex concentrated in degree zero. Denote the direct and inverse image functors by f_{∗} : S → S′ and f^{∗} : S′ → S respectively. We will repeatedly use the fact that the inverse image of a constant sheaf is also a constant sheaf. The right derived functor of f will be denoted by R^{•}f_{∗}. If \overline{ℚ}_{ℓ}_{X} denotes the constant sheaf on X then the fibre of R^{i}f_{∗}(\overline{ℚ}_{ℓ}_{X}) at a point x ∈ X′ is H_{c}^{i}(f^{−1}(x), \overline{ℚ}_{ℓ}). We will frequently omit the subscript X if the context is clear. We will use the term local system synonymously with that of locally constant sheaf. We will need the following version of the Base Change Theorem.

The Proper Base Change Theorem. Let

\begin{matrix}X′ & ⟶^{f′} & Y′ \\ ↓^{g′} & & ↓^{g} \\ X & ⟶^{f} & Y \end{matrix}

be a cartesian diagram of morphisms of schemes such that f : X → Y is a proper morphism. Let S be a torsion \overline{ℚ}_{ℓ}-sheaf on X. Then the base change morphism

g^{∗}(R^{•}f_{∗}(S)) → R^{•}f′_{∗}(g′^{∗}S)

is an isomorphism for all g : Y′ → Y.

We will also need to use the intersection cohomology functor. Good accounts are to be found in Shoji [24] and Goresky & Macpherson [13]. Let D(X) be the derived category of complexes of \overline{ℚ}_{ℓ}-sheaves on X. For a complex K in D(X), H^{i}K will denote the ith cohomology sheaf of K. We denote by D_{c}^{b}(X) the full subcategory of D(X) consisting of bounded complexes K such that H^{i}K are constructible for any i. For K ∈ D_{c}^{b}(X), K[i] means the ith shift so that H^{i}(K[j]) = H^{i+j}K. We regard a constructible sheaf T on X as a complex in D_{c}^{b}(X) concentrated in degree 0. Let D : D_{c}^{b}(X) → D_{c}^{b}(X) be the Verdier dual operator. A complex K ∈ D_{c}^{b}(X) is said to be a perverse sheaf if

(i) dim supp H^{i}K ⩽ −i, (ii) dim supp H^{i}DK ⩽ −i for any i. Let MX denote the full subcategory of D_{c}^{b}(X) consisting of perverse sheaves (for more details on perverse sheaves see Beilinson, Bernstein and Deligne [2]). If Y is a locally closed smooth subvariety of dimension d of X and T is a local system on Y then the intersection cohomology complex IC(\overline{Y} , T ) is an object in D_{c}^{b}(\overline{Y}). The complex K = IC(\overline{Y} , T )[d] is characterized by the following properties.

(i) H^{i}K = 0 if i < −d, (ii) H^{−d}K|Y = T , (iii) dim supp H^{i}K < −i if i > −d, (iv) dim supp H^{i}DK < −i if i > −d. If i : \overline{Y} → X is the inclusion then we may consider an object in M\overline{Y} as lying in MX by applying i_{∗}. We hence get a functor IC(X, ·) which associates to a local system on Y a perverse sheaf on X.

FOURIER TRANSFORMS OF GREEN FUNCTIONS 6

3. Generalization of a result of Kato

In this section G is an connected reductive group, B is any F -stable Borel subgroup and T is any F -stable maximal torus of B.

Lemma 3.1 (cf. Carter [9, Prop. 5.9.2]). Suppose p is good for G. If X is a semisimple element of 𝔤 then C_{G}(X)^{∘} is G-conjugate to the standard Levi subgroup L_{J} for some J ⊆ Π.

Proof. We may assume that X ∈ 𝔱. Since p is good, {α ∈ Φ | α(X) = 0} is a ℚ-closed root subsystem of Φ (cf. Slodowy [26, 3.5,3.15]) and hence, according to Bourbaki [5, VI, §1, 1.8, Proposition 24], is of the form Φ_{J′} for some simple system Π′ of roots and some J′ ⊆ Π′. For some w ∈ W, w(Π′) = Π and J = w(J′). Then ẇC_{G}(X)^{∘} = L_{J}.

Lemma 3.2. Let J ⊆ Π. If w ∈ D_{J}^{−1} with ^{wF} W_{J} = W_{J} then wF (J) = J.

Proof. If w ∈ W is such that ^{wF} W_{J} = W_{J} then, for α ∈ Φ_{J}, the reflection ^{wF} s_{α} = s_{wF (α)} lies in W_{J}. As is the proof of Digne & Michel [12, Proposition 4.2], this implies that wF (α) ∈ Φ_{J}. Hence wF (Φ_{J}) = Φ_{J}. If in addition w ∈ D_{J}^{−1} then w^{−1}(J) ⊆ F (Φ_{J}^{+}). Consideration of heights then shows that w^{−1}(J) = F (J).

Proposition 3.3 (cf. Digne & Michel [12, Prop. 4.3]). The G^{F} -classes of rational Levi subgroups of G are parametrized by W -classes of cosets W_{J}wF where J ⊆ Π and wF are such that ^{wF} W_{J} = W_{J}. If L is a rational Levi subgroup and W_{J}wF is an element of the corresponding W -class then there exists g ∈ G such that L = ^{g}L_{J} and g^{−1}F (g) = ẇ. According to Lemma 3.2, we may assume that w is chosen so that wF (J) = J.

Definition 3.4. Suppose p is good for G. If X ∈ 𝔤^{F} then, according to Lemma 3.1, C_{G}(X_{s})^{∘} is a rational Levi subgroup. By Proposition 3.3, there is a corresponding W -class of cosets W_{J}wF such that^{wF} W_{J} = W_{J}. We will call the W -class of W_{J}wF the genus of X. We will frequently abuse notation and say simply that X has genus W_{J}wF.

Notes. There are several points worth making concerning this definition of genus. Carter [7] defines what it means for two rational semisimple elements of the group to be “of the same genus”. We can make a similar definition in the case of rational semisimple elements of the Lie algebra. It then turns out that the use of the term “genus” is consistent with that of Carter in that two rational semisimple elements

FOURIER TRANSFORMS OF GREEN FUNCTIONS 7

are “of the same genus” according to Carter if and only if they have the same genus as defined above.

Deriziotis [10] shows that (in the split case which we will assume for simplicity) the equivalence classes of rational semisimple elements (in G) “of the same genus” can be parametrized by pairs (J, C) (which he calls families) where J is a proper subset of Π ∪ {−δ_{0}}, δ_{0} is the highest root and C is a conjugacy class in N_{W} (W_{J})/W_{J}. Note that there is a difference in the behaviours of centralizers of semisimple elements of G and of 𝔤. The situation is illustrated by the group G = Sp_{4}(q) with p good. Let Π = {a, b} with a the long root so that δ_{0} = a + 2b. The connected centralizer of the semisimple element s = diag(−1, 1, 1, −1) ∈ G is a group of type A_{1} × A_{1} corresponding to J = {a, −a − 2b}. However, there is no semisimple element of 𝔤 with connected centralizer of this type.

Lehrer [20] uses the phrase “regular semisimple element of type w” instead of “regular semisimple element of genus wF ”. Under the assumptions that the centralizers of semisimple elements are connected and G is F -split, Lehrer [17] defines a map ω from the set of F -stable semisimple conjugacy classes of G to the set of conjugacy classes of W. The relationship between this map and the notion of genus is given by Lemma 3.6 which follows.

The following definition of the map ω generalizes the definition of Lehrer [17, §5] to the non-split case.

Definition 3.5 (cf. Lehrer [17, §5]). Let G be a connected reductive group and T a maximally split maximal torus of G. If X ∈ 𝔤^{F}, let T′ be a maximally split maximal torus of C_{G}(X_{s})^{∘} which is w-twisted in G with respect to T. Then wF is determined up to W -conjugacy and we may define a map ω : 𝔤^{F} → H^{1}(W, F) by sending X to the W -class of wF.

Lemma 3.6. Suppose p is good for G. Let G be the set of W -classes of cosets W_{J}wF where J ⊆ Π and ^{wF} W_{J} = W_{J}. Let α : 𝔤^{F} → G be the genus map. If C is a W -class of cosets W_{J}wF as above then we can choose w so that wF (J) = J. If ν : G → H^{1}(W, F) sends C to the W -class containing wF then the following diagram commutes.

\xymatrix{𝔤^{F} \ar[r]^{α} \ar[dr]_{ω} & G \ar[d]^{ν} \\  & H^{1}(W, F)}

FOURIER TRANSFORMS OF GREEN FUNCTIONS 8

Proof. Let X have genus W_{J}wF so that C_{G}(X_{s})^{∘} = ^{g}L_{J} with g^{−1}F (g) = ẇ and wF (J) = J. Then^{g}T is an F -stable maximal torus contained in the F -stable Borel subgroup^{g}B_{J} of^{g}L_{J} and hence^{g}T is a maximally split maximal torus in^{g}L_{J} which is w-twisted in G with respect to T. Hence ω(X) is the W -class containing wF which is just ν(W_{J}wF). The proof also shows that ν is well-defined.

Lemma 3.7 (cf. Carter [9, Prop. 3.5.2]). Let X ∈ 𝔤 be semisimple and T a maximal torus of G. Then X ∈ 𝔱 if and only if T ⊆ C_{G}(X)^{∘}.

Proposition 3.8. Let p be good for G and X be regular in 𝔤. Then there exist g ∈ G and J ⊆ Π such that C_{G}(X_{s})^{∘} = ^{g}L_{J} and X_{n} ∈ Ad(g)(𝔟_{J}) and

B_{X} = {^{gẇ}B | w ∈ D_{J}^{−1}}.

As a consequence |B_{X}| = |W |/|W_{J}|.

Proof. By Lemma 3.1, C_{G}(X_{s})^{∘} = ^{g′}L_{J} for some J ⊆ Π, g′ ∈ G. Now X_{n} ∈ C_{𝔤}(X_{s}) = Lie(C_{G}(X_{s})^{∘}) = Lie(^{g′}L_{J}) by Borel [3, 9.1]. According to Borel [3, Prop. 14.25], X_{n} is contained in a Borel subalgebra of Ad g′(𝔩_{J}). Hence we can find h ∈ L_{J} such that X_{n} ∈ Ad(g′h)(𝔟_{J}). Setting g = g′h completes the first part of the proof.

Since X is regular, X_{n} is regular in C_{𝔤}(X_{s}) = Ad g(𝔩_{J}) (cf. Steinberg [32, Corollary 3.7]) and hence Ad(g^{−1})X_{n} is regular in 𝔩_{J}. By Springer [27, Lemma 5.8], if we write Ad(g^{−1})X_{n} = ∑_{α∈Φ_{J}^{+}} X_{α} with X_{α} ∈ 𝔤_{α} for α ∈ Φ_{J}^{+} then we have X_{α} ≠ 0 for α ∈ J. Now suppose that ^{gx}B ∈ B_{X} for some x ∈ G. Then X ∈ Lie(^{gx}B) and hence X_{n} ∈ Lie(^{gx}B). Write x = uẇb where u ∈ U_{w^{−1}}, w ∈ W and b ∈ B. Then Ad u^{−1}(Ad g^{−1}(X_{n})) ∈ Ad ẇ(𝔟) = ⨁_{α∈Φ^{+}} 𝔤_{w(α)} and, by the Chevalley commutator relations,

Ad u^{−1}(Ad g^{−1}(X_{n})) = ∑_{α∈J} X_{α} + ∑_{ht(α)>1} Y_{α}

for some Y_{α} ∈ 𝔤_{α}. Since X_{α} ≠ 0 for α ∈ J we must then have J ⊆ w(Φ^{+}) and hence w ∈ D_{J}^{−1}. Thus B_{X} ⊆ {^{gẇ}B | w ∈ D_{J}^{−1}}.

Conversely, let w ∈ D_{J}^{−1}. Then ^{ẇ^{−1}}X_{α} = X_{w^{−1}(α)} ⊆ B for all α ∈ Φ_{J}^{+}. Since B_{J} = ⟨T, X_{α} | α ∈ Φ_{J}^{+}⟩ and ^{ẇ^{−1}}T = T we thus have B_{J} ⊆ ẇB. Now ^{g}T ⊆ ^{g}L_{J} = C_{G}(X_{s})^{∘} and hence, by Lemma 3.7, X_{s} ∈ Lie(^{g}T). Thus

X = X_{s} + X_{n} ∈ Lie(^{g}B_{J}) ⊆ Lie(^{gẇ}B)

and ^{gẇ}B ∈ B_{X}. The last statement follows since |D_{J}| = |W |/|W_{J}|.

Lemma 3.9 (cf. Springer & Steinberg [30, III, 1.8(a)]). The variety 𝔤̃ is irreducible and we have dim 𝔤̃ = dim 𝔤.

FOURIER TRANSFORMS OF GREEN FUNCTIONS 9

Proof. Since N_{B}(𝔟) = B, we can write 𝔤̃ ≅ {(X, gB) | X ∈ Ad g(𝔟)}. Its inverse image in 𝔤 × G is {(X, g) | X ∈ Ad g(𝔟)}. This is the image of the injective morphism f : 𝔟 × G → 𝔤 × G given by (X, g) ↦ (Ad g(X), g). The irreducibility of 𝔤̃ then follows from that of 𝔟 × G. We also see that

dim 𝔤̃ = dim im f − dim B = dim(𝔟 × G) − dim B = dim 𝔤.

Definition 3.10. Let H be an F -stable Levi subgroup of G. For the rest of this paper J ⊆ Π, x ∈ G and w_{1} ∈ W will be such that H = ^{x}L_{J}, x^{−1}F (x) = ẇ_{1} and w_{1}F (J) = J. Define

𝔥_{c} = {X ∈ 𝔥 | C_{G}(X_{s})^{0} ⊆ H}.

Lemma 3.11. If p is a very good prime for G then 𝔥_{c} is open dense in 𝔥.

Proof. Let Φ be the root system in G contained in X(𝔱), the character group of 𝔱. Let \overline{𝔽}_{q}[𝔱] be the coordinate ring of 𝔱, identified with the group algebra \overline{𝔽}_{q}[X(𝔱)]. Define an element c_{J} ∈ \overline{𝔽}_{q}[𝔱] by

c_{J} = ∏_{α∈Φ\Φ_{J}} α.

Then c_{J} is W_{J}-invariant and hence c_{J} ∈ \overline{𝔽}_{q}[𝔱]^{W_{J}} ≅ \overline{𝔽}_{q}[𝔱/W_{J}]. The condition on p implies that p does not divide |Ω/ℤΦ| and hence, by Slodowy [26, 3.12], there is a homomorphism

\overline{𝔽}_{q}[𝔱/W_{J}] ≅ \overline{𝔽}_{q}[𝔱]^{W_{J}} ⟶^{∼} \overline{𝔽}_{q}[𝔩_{J}]^{L_{J}} ↪ \overline{𝔽}_{q}[𝔩_{J}].

which induces a morphism χ_{J} : 𝔩_{J} → 𝔱/W_{J} taking an element Y ∈ 𝔩_{J} to 𝔱 ∩ Ad L_{J}(Y_{s}). Then X ∈ 𝔥_{c} if and only if X ∈ 𝔥 and C_{G}(Ad x^{−1}(X_{s}))^{0} ⊆ L_{J}; that is, if and only if c_{J}(χ_{J}(Ad x^{−1}(X))) ≠ 0. Hence

𝔥_{c} = Ad x(χ_{J}^{−1}({X ∈ 𝔱/W_{J} | c_{J}(X) ≠ 0})).

Since p is very good this is a nonempty open subset of 𝔥 (cf. Slodowy [26, 3.14]). Since 𝔥 is an irreducible algebraic variety this implies that 𝔥_{c} is also dense in 𝔥.

Lemma 3.12. We have {B′ ∈ B^{G} | T ⊂ B′} = {ẇB | w ∈ W }.

Proof. Suppose that B′ is a Borel subgroup of G containing T. Then B′ = ^{g}B for some g ∈ G. Since T ⊆ B′, ^{g^{−1}}T is a maximal torus of B, and hence ^{g^{−1}}T = ^{b}T for some b ∈ B. Thus gb ∈ N and gb = ẇt for some w ∈ W, t ∈ T. Then B′ = ^{g}B = ^{gb}B = ^{ẇt}B = ẇB.

Lemma 3.13. We have D_{J}^{−1} = {w ∈ W | ẇB ∩ L_{J} = B_{J}}.

Proof. We have

w ∈ D_{J}^{−1} ⟺ J ⊆ w(Φ^{+}) ⟺ Φ_{J}^{+} ⊆ w(Φ^{+}) ⟺ w(Φ^{+}) ∩ Φ_{J} = Φ_{J}^{+}.

If L_{J} ∩ ẇB = B_{J} then B_{J} ⊆ ẇB implying that Φ_{J}^{+} ⊆ w(Φ^{+}) and hence that w ∈ D_{J}^{−1}.

Conversely, suppose that w ∈ D_{J}^{−1} so that w(Φ^{+}) ∩ Φ_{J} = Φ_{J}^{+}. If α ∈ Φ_{J}^{+} then α ∈ Φ_{J}, α ∈ w(Φ^{+}) and hence X_{α} ∈ L_{J} ∩ ẇB. Since T ⊆ L_{J} ∩ ẇB, we thus have B_{J} ⊆ L_{J} ∩ ẇB. On the other hand, suppose that ẇbẇ^{−1} ∈ L_{J} ∩ ẇB with b ∈ B. By the Bruhat decomposition in L_{J}, write ẇbẇ^{−1} = b_{1}ẇ_{1}b_{2} with b_{1}, b_{2} ∈ B_{J} and w_{1} ∈ W_{J}. Then

(ẇ^{−1}b_{1}ẇ)(ẇ^{−1}ẇ_{1}ẇ)(ẇ^{−1}b_{2}ẇ) = b.

Now both ẇ^{−1}b_{1}ẇ and ẇ^{−1}b_{2}ẇ lie in B since w^{−1}(Φ_{J}^{+}) = Φ^{+} ∩ w^{−1}(Φ_{J}). Thus, by the Bruhat decomposition in G, we must have w^{−1}w_{1}w = 1 and hence w_{1} = 1 implying that ẇbẇ^{−1} ∈ B_{J}. Thus L_{J} ∩ ẇB ⊆ B_{J}.

Lemma 3.14. If (X, B′) ∈ p_{𝔤}^{−1}(𝔥_{c}) then B′ = ^{xmẇ}B for some m ∈ L_{J} and w ∈ D_{J}^{−1}. In this expression, w and the coset mB_{J} are uniquely determined.

Proof. If (X, B′) ∈ p_{𝔤}^{−1}(𝔥_{c}) then X ∈ 𝔥_{c} and X_{s} ∈ 𝔱′ = Lie(T′) for some maximal torus T′ of B′. According to Lemma 3.7, this implies that T′ ⊆ C_{G}(X_{s})^{0} ⊆ H and hence that ^{x^{−1}}T′ ⊆ L_{J}. Thus, for some m ∈ L_{J}, ^{m^{−1}x^{−1}}T′ = T and hence m^{−1}x^{−1}B′ ⊇ T. Lemma 3.12 then implies that ^{m^{−1}x^{−1}}B′ = ẇB for some w ∈ W which by choice of m we may assume to lie in D_{J}^{−1}.

Now suppose that ^{xm_{1}ẇ_{1}}B = ^{xm_{2}ẇ_{2}}B for some m_{1}, m_{2} ∈ L_{J}, w_{1}, w_{2} ∈ D_{J}^{−1}. If we let m = m_{1}^{−1}m_{2} then ẇ_{1}^{−1}mẇ_{2} ∈ B. This implies that ẇ_{1} ∈ L_{J} ẇ_{2}B and hence that Bẇ_{1}B ⊆ BL_{J} ẇ_{2}B. Now L_{J} = B_{J}N_{J}B_{J} ⊆ BN_{J}B. Because of the BN-pair relation

Bẇ_{α}B · BẇB ⊆ BẇB ∪ Bẇ_{α}ẇB

we must have

BL_{J} ẇ_{2}B ⊆ ⋃_{w∈W_{J}} Bẇẇ_{2}B.

By the Bruhat decomposition we then have w_{1}w_{2}^{−1} ∈ W_{J} and hence w_{1} = w_{2}. Thus m = m_{1}^{−1}m_{2} ∈ ẇ_{1}B ∩ L_{J} = B_{J} by Lemma 3.13 and m_{1}B_{J} = m_{2}B_{J}.

FOURIER TRANSFORMS OF GREEN FUNCTIONS 11

Corollary 3.15. If (X, B′) ∈ p_{𝔤}^{−1}(𝔥_{c}) and B′ = ^{xmẇ}B for some m ∈ L_{J} and w ∈ D_{J}^{−1} then B′ ∩ H = ^{xm}B_{J} ∈ B^{H}.

Proof. We have B′ ∩ H = ^{xmẇ}B ∩ H = ^{xm}(ẇB ∩ L_{J}) = ^{xm}B_{J} by Lemma 3.13 and this is a Borel subgroup of H.

Proposition 3.16 (cf. Kato [15, 2.6]). With notation as above, define a map

ϕ : p_{𝔥}^{−1}(𝔥_{c}) × W_{J}\W → p_{𝔤}^{−1}(𝔥_{c})

by

ϕ((X,^{xm}B_{J}), W_{J}w) = (X,^{xmẇ}B)

where m ∈ L_{J} and w ∈ D_{J}^{−1}. Then ϕ is well-defined and is an H-equivariant isomorphism where H acts on p_{𝔥}^{−1}(𝔥_{c}) and p_{𝔤}^{−1}(𝔥_{c}) (resp. W_{J}\W) by conjugation (resp. the trivial action).

Proof. It is routine to show that ϕ is well-defined and H-equivariant. To show that ϕ is an isomorphism we explicitly construct an inverse

ψ : p_{𝔤}^{−1}(𝔥_{c}) → p_{𝔥}^{−1}(𝔥_{c}) × W_{J}\W.

Let (X, B′) ∈ p_{𝔤}^{−1}(𝔥_{c}). By Lemma 3.14, we have B′ = ^{xmẇ}B for some m ∈ L_{J} and w ∈ D_{J}^{−1} and in this expression w and the coset mB_{J} are uniquely determined. By Corollary 3.15, B′ ∩ H = ^{xm}B_{J} is a Borel subgroup of H. Hence we may define ψ by

ψ(X,^{xmẇ}B) = ((X,^{xm}B_{J}), W_{J}w).

It is clear that ϕ and ψ are mutually inverse.

Corollary 3.17. Let the notation be as above. For w ∈ D_{J}^{−1} let

Y_{w} = {(X,^{xmẇ}B) | X ∈ 𝔥_{c}, m ∈ L_{J}, X ∈ Ad(xmẇ)(𝔟)}.

Then p_{𝔤}^{−1}(𝔥_{c}) = ∐_{w∈D_{J}^{−1}} Y_{w} is a decomposition into irreducible components and for each w ∈ D_{J}^{−1} the variety Y_{w} ≅ p_{𝔥}^{−1}(𝔥_{c}).

Proof. All the statements follow immediately from Proposition 3.16 except that the decomposition is into irreducible components. Since the decomposition is a finite disjoint union, it suffices to show that Y_{w} is closed in p_{𝔤}^{−1}(𝔥_{c}).

Consider the map π : p_{𝔤}^{−1}(𝔥_{c}) → G/B taking (X,^{xmẇ}B) to mẇB where m ∈ L_{J}. Then Y_{w} = π^{−1}(L_{J} ẇB/B). By Proposition 3.18 (the statement of which follows), L_{J} ẇB is closed in G and hence L_{J} ẇB/B is closed in G/B. Thus Y_{w} is closed in p_{𝔤}^{−1}(𝔥_{c}) as required.

Proposition 3.18. If J, K ⊆ Π and w ∈ W then L_{J} ẇP_{K} is closed in G.

Proof. We may assume that w is a distinguished coset representative in W_{J}\W/W_{K}, that is, w ∈ D_{J}^{−1} ∩ D_{K}. If b ∈ B_{J} then ẇ^{−1}bẇ ∈ B ⊆ P_{K} and so lbẇP_{K} = lẇP_{K} for all l ∈ L_{J}, b ∈ B_{J}. Thus the morphism L_{J} → L_{J} ẇP_{K}/P_{K} sending l to lẇP_{K} factors through L_{J} → L_{J}/B_{J}, so lB_{J} ↦ lẇP_{K} is a well-defined morphism of varieties L_{J}/B_{J} → G/P_{K}. Since L_{J}/B_{J} is complete, its image L_{J} ẇP_{K}/P_{K} is closed in G/P_{K} whence the result.

Proposition 3.19. If p is very good for G then the variety p_{𝔤}^{−1}(𝔥_{c}) is smooth and dim p_{𝔤}^{−1}(𝔥_{c}) = dim 𝔥.

Proof. Let Y_{w} be as in the statement of Corollary 3.17. Since 𝔥_{c} is open in 𝔥, p_{𝔥}^{−1}(𝔥_{c}) is open in 𝔥̃. Since 𝔥̃ is smooth this implies that p_{𝔥}^{−1}(𝔥_{c}) and hence Y_{w} is smooth for each w ∈ D_{J}^{−1}. By Corollary 3.17, we thus find that p_{𝔤}^{−1}(𝔥_{c}) is smooth.

Since the Y_{w} are all isomorphic to p_{𝔥}^{−1}(𝔥_{c}) we have dim p_{𝔤}^{−1}(𝔥_{c}) = dim p_{𝔥}^{−1}(𝔥_{c}). Now p_{𝔥}^{−1}(𝔥_{c}) is open in 𝔥̃ and since 𝔥̃ is irreducible this implies that p_{𝔥}^{−1}(𝔥_{c}) is dense in 𝔥̃. Hence dim p_{𝔥}^{−1}(𝔥_{c}) = dim 𝔥̃ = dim 𝔥.

Lemma 3.20 (cf. Kato [15, §2]). Let (X, B′) ∈ 𝔤̃_{rs}. Then there exists a unique element (Y, gT) of 𝔱_{reg}×G/T satisfying (Ad g(Y),^{g}B) = (X, B′) where 𝔱_{reg} = 𝔱∩𝔤_{rs}.

Proof. Choose g′ ∈ G so that ^{g′^{−1}}B′ = B. Then there exists b ∈ B such that Y = Ad(bg′^{−1})(X) ∈ 𝔱. Hence, if g = g′b^{−1}, (Ad g(Y),^{g}B) = (X, B′) and we have thus proved existence. To prove uniqueness suppose that Ad g(Y) = Ad g′(Y′) = X and ^{g}B = ^{g′}B = B′ with Y, Y′ ∈ 𝔱. Then g^{−1}g′ ∈ N_{G}(B) = B and, by Springer-Steinberg [30, II, 3.16], Y′ = Ad ẇ(Y) for some w ∈ W. Hence Ad(g^{−1}g′ẇ)(Y) = Y and g^{−1}g′ẇ ∈ C_{G}(Y). Since X is regular semisimple, C_{G}(Y) ⊆ N and hence g^{−1}g′ ∈ B ∩ N = T. Thus gT = g′T and ẇ ∈ C_{G}(Y) implying that Y′ = Y.

The Weyl group W = N_{G}(T)/T acts on 𝔤̃_{rs} from the right via (Ad g(Y),^{g}B)·w = (Ad g(Y),^{gẇ}B). Define the local system T^{𝔤} = p_{𝔤_{rs}∗}\overline{ℚ}_{ℓ}. The right action of W on 𝔤̃_{rs} induces a left action of W on T^{𝔤}. A result of Lusztig (cf. Shoji [24, Prop. 4.2] and Lusztig [21, §3]) states that IC(𝔤, T^{𝔤}) ≅ p_{𝔤∗}\overline{ℚ}_{ℓ} in M𝔤. Via the intersection cohomology functor, the action of W on T^{𝔤} induces an action of W on p_{𝔤∗}\overline{ℚ}_{ℓ} which in turn induces an action of W on the fibre of p_{𝔤∗}\overline{ℚ}_{ℓ} above X ∈ 𝔤_{rs}. But this fibre is isomorphic to H^{∗}(p_{𝔤}^{−1}(X), \overline{ℚ}_{ℓ}) = H^{∗}(B_{X}, \overline{ℚ}_{ℓ}). The action of W on the ℓ-adic cohomology groups H^{∗}(B_{X}, \overline{ℚ}_{ℓ}) defined in this way will be called the Lusztig-Springer action of W.

FOURIER TRANSFORMS OF GREEN FUNCTIONS 13

The analogous result to Lemma 3.20 for 𝔥̃ states that if (X, B′) ∈ 𝔥̃ then there exists m ∈ L_{J} such that B′ = ^{xm}B_{J} and X ∈ Ad(xm)(𝔱). Thus W_{J} acts on p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) from the right via

(X,^{xm}B_{J}) · w = (X,^{xmẇ}B_{J}).

This induces a left action of W_{J} on T^{𝔥}. For reasons that will be apparent later on we will henceforth denote the Frobenius morphisms on 𝔥 and p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) and the induced action on T^{𝔥} by F′.

The proofs of the following two lemmas follow by some simple calculations.

Lemma 3.21. We have the following commutative diagram

p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) × W_{J}\W ⟶^{ϕ} p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥)
↓^{F′×w_{1}F}  ↓^{F}
p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) × W_{J}\W ⟶^{ϕ} p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥)

Lemma 3.22. Let w ∈ W, w′ ∈ D_{J}^{−1} and suppose that w′w = w_{2}w_{3} with w_{2} ∈ W_{J} and w_{3} ∈ D_{J}^{−1}. Then the following diagram commutes

p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) × {W_{J}w′} ⟶^{ϕ} p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥)
↓^{w_{2}×w}  ↓^{w}
p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) × {W_{J}w_{3}} ⟶^{ϕ} p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥)

Let W̃ = W ⋊ ⟨F ⟩ and W̃_{J} be the semidirect product W_{J} ⋊ ⟨F′⟩ with F′ · w = w_{1}F w·F′ for w ∈ W_{J}. We can define an injective group homomorphism ι : W̃_{J} ↪ W̃ by ι(w) = w for w ∈ W_{J} and ι(F′) = w_{1}F. We may thus regard \overline{ℚ}_{ℓ}[W̃] as a right \overline{ℚ}_{ℓ}[W̃_{J}]-module via ι. It is easily seen that for the right action of W on p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) we have the equation w · F = F · F (w). Since the induced action of F on T^{𝔤}|_{𝔤_{rs}∩𝔥} is given by (F^{∗})^{−1} = (F^{−1})^{∗}, the left action of W̃ on T^{𝔤}|_{𝔤_{rs}∩𝔥} satisfies the equation F · w = F (w) · F. Similarly the left action of W̃_{J} on T^{𝔥}|_{𝔤_{rs}∩𝔥} satisfies the equation F′ · w = ^{w_{1}F} w · F′. Hence T^{𝔤}|_{𝔤_{rs}∩𝔥} has the structure of a left \overline{ℚ}_{ℓ}[W̃]-module and T^{𝔥}|_{𝔤_{rs}∩𝔥} has the structure of a left \overline{ℚ}_{ℓ}[W̃_{J}]-module. The relationship between these two modules is given as follows.

Proposition 3.23. Let the notation be as above. Then

T^{𝔤}|_{𝔤_{rs}∩𝔥} ≅ \overline{ℚ}_{ℓ}[W̃] ⊗_{\overline{ℚ}_{ℓ}[W̃_{J}]} T^{𝔥}|_{𝔤_{rs}∩𝔥}

as left \overline{ℚ}_{ℓ}[W̃]-modules.

Proof. We have the following commutative diagram in which π_{1} is projection onto the first entry and each square is Cartesian.

p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) × W_{J}\W ⟶^{ϕ} p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) ⟶^{i_{1}} 𝔤̃_{rs}
↓^{p_{𝔥rs}∘π_{1}}  ↓^{p_{𝔤rs}}  ↓^{p_{𝔤rs}}
𝔤_{rs} ∩ 𝔥  ⟶^{id} 𝔤_{rs} ∩ 𝔥 ⟶^{i_{2}} 𝔤_{rs}

Then disregarding the actions of W and Frobenius, we have

T^{𝔤}|_{𝔤rs∩𝔥} = i_{2}^{∗}p_{𝔤_{rs}∗}\overline{ℚ}_{ℓ}

≅ (id ∘i_{2})^{∗}p_{𝔤_{rs}∗}\overline{ℚ}_{ℓ}

≅ (p_{𝔥_{rs}} ∘ π_{1})_{∗}(i_{1} ∘ ϕ)^{∗}\overline{ℚ}_{ℓ} by change of base

≅ p_{𝔥_{rs}∗}π_{1∗}\overline{ℚ}_{ℓ}

≅ p_{𝔥_{rs}∗}( ⨁_{wW_{J} ∈W/W_{J}}[\overline{ℚ}_{ℓ}_{p{−1 \atop 𝔥rs} (𝔤rs∩𝔥)}]_{wW_{J}}) by definition of π_{1}_{∗}

≅ ⨁_{wW_{J} ∈W/W_{J}}[p_{𝔥_{rs}∗}\overline{ℚ}_{ℓ}]_{wW_{J}} .

We have used W/W_{J} in the summation since the right action of W on 𝔤̃_{rs} induces a left action on the corresponding local system. By applying base change to

p_{𝔥_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) ⟶ 𝔥̃_{rs}
↓^{p_{𝔥rs}} ↓^{p_{𝔥rs}}
𝔤_{rs} ∩ 𝔥 ⟶ 𝔥_{rs}

we deduce that

T^{𝔤}|_{𝔤_{rs}∩𝔥} ≅ ⨁_{wW_{J} ∈W/W_{J}}[T^{𝔥}|_{𝔤_{rs}∩𝔥}]_{wW_{J}} . (3.1)

Let w′ ∈ D_{J} and w ∈ W. Write ww_{3} = w′w_{2} with w_{2} ∈ W_{J} and w_{3} ∈ D_{J}. Under the bijection W_{J}\W ≅ W/W_{J} which takes W_{J}w to w^{−1}W_{J}, the map W_{J}w′^{−1} ↦ W_{J}w_{3}^{−1} given by right multiplication by w corresponds to the map w′W_{J} ←w_{3}W_{J} given by left multiplication by w. Lemma 3.22 thus implies that the action of w on the summands in Equation (3.1) is given by

[T^{𝔥}|_{𝔤_{rs}∩𝔥}]_{w_{3}W_{J}} → [T^{𝔥}|_{𝔤_{rs}∩𝔥}]_{w_{3}W_{J}} → [T^{𝔥}|_{𝔤_{rs}∩𝔥}]_{w′W_{J}}

where the first map is that induced by the action of w_{2} and the second map is the canonical isomorphism.

FOURIER TRANSFORMS OF GREEN FUNCTIONS 15

Similarly Lemma 3.21 implies that the action of F on T^{𝔤}|_{𝔤_{rs}∩𝔥} is determined by its action on the summands,

[T^{𝔥}|_{𝔤_{rs}∩𝔥}]_{w′W_{J}} → [T^{𝔥}|_{𝔤_{rs}∩𝔥}]_{w′W_{J}} → [T^{𝔥}|_{𝔤_{rs}∩𝔥}]_{F (w′)w{−1 \atop 1}W_{J}}

where the first map is that induced by F′ and the second map is the canonical isomorphism. We are then done by virtue of the relation F w′ = F (w′)w_{1}^{−1}ι(F′) in W̃.

Proposition 3.24. Let the notation be as above. If p is very good for G then IC(𝔤, T^{𝔤})|_{𝔥_{c}} ≅ IC(𝔥_{c}, T^{𝔤}|_{𝔤_{rs}∩𝔥}) and IC(𝔥, T^{𝔥})|_{𝔥_{c}} ≅ IC(𝔥_{c}, T^{𝔥}|_{𝔤_{rs}∩𝔥}).

Proof. We will prove the existence of the first isomorphism. The proof of the existence of the second is entirely analogous. We will follow the proof of Shoji [24, Proposition 4.2]. Let d = dim 𝔥_{c} and K = IC(𝔤, T^{𝔤})|_{𝔥_{c}}[d]. We have to show that this complex satisfies the following four conditions (cf. Shoji [24, 3.3.1]):

(i) H^{i}K = 0 if i < −d, (ii) H^{−d}K|Y = T , (iii) dim supp H^{i}K < −i if i > −d, (iv) dim supp H^{i}DK < −i if i > −d. From the isomorphism IC(𝔤, T^{𝔤}) ≅ p_{𝔤∗}\overline{ℚ}_{ℓ} in M𝔤 and using the proper base change theorem for

p_{𝔤}^{−1}(𝔥_{c}) ⟶ 𝔤̃
↓^{p_{𝔤}} ↓^{p_{𝔤}}
𝔥_{c} ⟶ 𝔤

we have K ≅ p_{𝔤∗}\overline{ℚ}_{ℓ}[d]. Hence H^{i}K ≅ R^{i}p_{𝔤∗}\overline{ℚ}_{ℓ}[d]. Thus (i) follows from the fact that the stalk (H^{i}K)_{X} at X ∈ 𝔥_{c} is isomorphic to H^{i+d}(p_{𝔤}^{−1}(X), \overline{ℚ}_{ℓ}) and is hence zero for i < −d. By using the proper base change theorem for

p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) ⟶ p_{𝔤}^{−1}(𝔥_{c})
↓^{p_{𝔤rs}} ↓^{p_{𝔤}}
𝔤_{rs} ∩ 𝔥 ⟶ 𝔥_{c}

and

p_{𝔤_{rs}}^{−1}(𝔤_{rs} ∩ 𝔥) ⟶ 𝔤̃_{rs}
↓^{p_{𝔤rs}} ↓^{p_{𝔤rs}}
𝔤_{rs} ∩ 𝔥 ⟶ 𝔤_{rs}

FOURIER TRANSFORMS OF GREEN FUNCTIONS 16

we see that H^{−d}K|_{𝔤_{rs}∩𝔥} = p_{𝔤_{rs}∗}\overline{ℚ}_{ℓ} = T^{𝔤}|_{𝔤_{rs}∩𝔥} verifying (ii). From Proposition 3.19, p_{𝔤}^{−1}(𝔥_{c}) is smooth and dim p_{𝔤}^{−1}(𝔥_{c}) = d. Hence D(\overline{ℚ}_{ℓ}_{p{−1 \atop 𝔤}(𝔥c)}[d]) ≅ \overline{ℚ}_{ℓ}_{p{−1 \atop 𝔤}(𝔥c)}[d]. Since 𝔤̃ is closed in 𝔤 × B^{G}, p_{𝔤} : 𝔤̃ → 𝔤 is proper and the diagram

p_{𝔤}^{−1}(𝔥_{c}) ⟶ 𝔤̃
↓^{p_{𝔤}} ↓^{p_{𝔤}}
𝔥_{c} ⟶ 𝔤

shows that p_{𝔤} : p_{𝔤}^{−1}(𝔥_{c}) → 𝔥_{c} is also proper (see Mumford [23, II. 7(iii)]). This implies that DK ≅ K since the Verdier dual operator commutes with the direct image for a proper map. It is hence enough to show (iii). According to Proposition 3.16, p_{𝔤}^{−1}(X) ≅ B_{X}^{𝔥} × W_{J}\W for X ∈ 𝔥_{c} and hence the condition R^{i}p_{𝔤∗}(\overline{ℚ}_{ℓ})_{X} ≠ 0 implies that H^{i}(B_{X}^{𝔥}, \overline{ℚ}_{ℓ}) ≠ 0 and thus i ⩽ 2 dim B_{X}^{𝔥}. Hence the support of H^{i}K is contained in the set {X ∈ 𝔥 | i + d ⩽ 2 dim B_{X}^{𝔥}} which has dimension less than −i for i > −d by Shoji [24, Corollary 1.7(i)], thus proving (iii).

The W (resp. W_{J}) and F actions on T^{𝔤} (resp. T^{𝔥}) induce corresponding actions on IC(𝔤, T^{𝔤})|_{𝔥_{c}} (resp. IC(𝔥, T^{𝔥})|_{𝔥_{c}}) by means of the intersection cohomology functor.

Corollary 3.25. Let the notation be as above and suppose p is very good for G. Then

IC(𝔤, T^{𝔤})|_{𝔥_{c}} ≅ \overline{ℚ}_{ℓ}[W̃] ⊗_{\overline{ℚ}_{ℓ}[W̃_{J}]} IC(𝔥, T^{𝔥})|_{𝔥_{c}}

as left \overline{ℚ}_{ℓ}[W̃]-modules.

Proof. By Proposition 3.23, T^{𝔤}|_{𝔤_{rs}∩𝔥} and \overline{ℚ}_{ℓ}[W̃] ⊗_{\overline{ℚ}_{ℓ}[W̃_{J}]} T^{𝔥}|_{𝔤_{rs}∩𝔥} are isomorphic left \overline{ℚ}_{ℓ}[W̃]-modules. The result then follows from Proposition 3.24 and the uniqueness and functoriality of the intersection cohomology functor.

For the following theorem we recall that G is an arbitrary connected reductive algebraic group over the field 𝔽_{q} of very good characteristic p. Let H = ^{x}L_{J} where J ⊆ Π, x^{−1}F (x) = ẇ_{1} with w_{1}F (J) = J. Let X ∈ 𝔥^{F} with C_{G}(X_{s})^{∘} ⊆ H. Let W (resp. W_{J}) act on H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) (resp. H^{i}(B_{Ad x^{−1}(X)}^{L_{J}}, \overline{ℚ}_{ℓ})) via the Lusztig-Springer action and let W_{J} act on H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ}) by transport of structure via the isomorphism

H^{i}(B_{Ad x^{−1}(X)}^{L_{J}}, \overline{ℚ}_{ℓ}) ≅ H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ}).

Let the F and F′-actions on H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) and H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ}), respectively, be those induced by F : B_{X}^{G} → B_{X}^{G} and F′ : B_{X}^{H} → B_{X}^{H}, respectively. Let W̃ = W ⋊ ⟨F ⟩ with F · w = F (w) · F and W̃_{J} = W_{J} ⋊ ⟨F′⟩ with F′ · w = ^{w_{1}F} w · F′ for w ∈ W_{J}.

Define ι : W̃_{J} ↪ W̃ by ι(w) = w for w ∈ W_{J} and ι(F′) = w_{1} · F. Regard \overline{ℚ}_{ℓ}[W̃] as a right \overline{ℚ}_{ℓ}[W̃_{J}]-module via ι.

Theorem 3.26. With notation as above

H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) ≅ \overline{ℚ}_{ℓ}[W̃] ⊗_{\overline{ℚ}_{ℓ}[W̃_{J}]} H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})

as left \overline{ℚ}_{ℓ}[W̃]-modules.

Proof. The fibres of R^{i}p_{𝔤∗}\overline{ℚ}_{ℓ} and R^{i}p_{𝔥∗}\overline{ℚ}_{ℓ} at X ∈ 𝔥_{c}^{F} are

H^{i}(p_{𝔤}^{−1}(X), \overline{ℚ}_{ℓ}) = H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ})

and

H^{i}(p_{𝔥}^{−1}(X), \overline{ℚ}_{ℓ}) = H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ}).

The result follows by taking fibres at X of the isomorphism of Corollary 3.25.

This result may easily be expressed in terms of “coset induction” (cf. Broué, Malle & Michel [6, §1A] and Aubert [1, Rappels sur les fonctions de F -classes, pp. 35–36]). Let C(W/F) denote the space of complex valued functions on W F which are restrictions of class functions on W̃. We may similarly define C(W_{J}/F′) = C(W_{J}/w_{1}F) upon identifying F′ with its image ι(F′) in W̃. If α ∈ C(W_{J}/w_{1}F) then we define Ind_{W_{J} w_{1}F}^{W F} α ∈ C(W/F) by

Ind_{W_{J} w_{1}F}^{W F} α(wF) = \frac{1}{|W_{J}|} ∑_{\begin{matrix}v∈W \\ ^{v}(wF)∈W_{J} w_{1}F \end{matrix}} α(^{v}(wF)).

Corollary 3.27. With notation as above, if X ∈ 𝔥_{c}^{F} and p is very good then

H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) = Ind_{W_{J} w_{1}F}^{W F} H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})

and

Tr(F w, H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ})) = \frac{1}{|W_{J}|} ∑_{\begin{matrix}v∈W \\ ^{v}(F w)∈W_{J} w_{1}F \end{matrix}} Tr(vF (wv^{−1})w_{1}^{−1} · F′, H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})).

Proof. According to Theorem 3.26, if w_{3} ∈ D_{J} and ww_{3} = w′w_{2} with w′ ∈ D_{J}, w_{2} ∈ W_{J} then the action of F w on H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ}) is determined by the map

[H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})]_{w_{3}W_{J}} → [H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})]_{w_{3}W_{J}} → [H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})]_{F (w′)w_{1}^{−1}W_{J}}

where the first map is that induced by F′w_{2} and the second map is the canonical map. We thus get a contribution to the trace of F w from its action on [H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})]_{w_{3}W_{J}} only if w_{3} = F (w′)w_{1}^{−1}, that is, only if w_{2} = F^{−1}(w_{1}^{−1}w_{3}^{−1})ww_{3} ∈ W_{J}. Hence

\begin{aligned}Tr(F w, H^{i}(B_{X}^{G}, \overline{ℚ}_{ℓ})) & = ∑_{\begin{matrix}w_{3}∈D_{J} \\ F^{−1}(w_{1}^{−1}w_{3}^{−1})ww_{3}∈W_{J} \end{matrix}} Tr(F′ · F^{−1}(w_{1}^{−1}w_{3}^{−1})ww_{3}, H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})) \\ & = ∑_{\begin{matrix}w_{3}∈D_{J} \\ w_{3}^{−1}F (ww_{3})w_{1}^{−1}∈W_{J} \end{matrix}} Tr(w_{3}^{−1}F (ww_{3})w_{1}^{−1} · F′, H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ})). \end{aligned}

The result follows once we realize that, for any u ∈ W_{J}, w_{3}^{−1}F (ww_{3})w_{1}^{−1} ∈ W_{J} if and only if w_{3}^{−1}(F w)w_{3} ∈ W_{J}w_{1}F if and only if (w_{3}u)^{−1}(F w)(w_{3}u) ∈ W_{J}w_{1}F.

Corollary 3.28. Let G be a connected reductive group and suppose the characteristic p is very good for G. If X is regular of genus W_{J}w_{1}F in 𝔤^{F} then

Q̃_{w}^{𝔤}(X) = Ind_{W_{J} w_{1}F}^{W F} 1(F w) = |W_{J}|^{−1}|{v ∈ W | ^{v}(F w) ∈ W_{J}w_{1}F }|.

Proof. Let H = C_{G}(X_{s})^{∘}. Then X ∈ 𝔥_{c}^{F} and according to Proposition 3.8, B_{X}^{H} is a single point. Hence, by Digne and Michel [12, Proposition 10.11], H^{i}(B_{X}^{H}, \overline{ℚ}_{ℓ}) is zero dimensional for i ≠ 0 and 1-dimensional when i = 0. Moreover, F′ acts trivially on H^{0}(B_{X}^{H}, \overline{ℚ}_{ℓ}). We also have B_{X}^{H} ≅ B_{X_{n}}^{H} since X_{s} is central in 𝔥. Thus, by Springer [28, Lemma 7.4] and the fact that the definition of the Lusztig-Springer action differs from Springer’s own by the sign representation of W, W_{J} also acts trivially on H^{0}(B_{X}^{H}, \overline{ℚ}_{ℓ}). The result then follows from Corollary 3.27.

Definition 3.29. If χ ∈ Ŵ is F -stable then denote by χ̃ an arbitrary (but fixed) extension of χ to W̃. Let C(W/F)_{irr} be the set of such χ̃. For χ̃ ∈ C(W/F)_{irr} define

T (χ̃; W_{J}w_{1}F) = |W_{J}|^{−1} ∑_{w∈W_{J}} χ̃(ww_{1}F).

and

Q̃_{χ} = |W |^{−1} ∑_{w∈W} χ̃(w)Q̃_{w}.

Theorem 3.30. Let G be a connected reductive group and suppose the characteristic p is very good for G. If X ∈ 𝔤^{F} is regular of genus W_{J}w_{1}F and χ ∈ Ŵ is F -stable then

Q̃_{χ}^{𝔤}(X) = T (χ̃; W_{J}w_{1}F).

Proof. From Corollary 3.28,

Q̃_{χ}^{𝔤}(X) = |W |^{−1} ∑_{w∈W} χ̃(F w)Q̃_{w}^{𝔤}(X)

= |W |^{−1}|W_{J}|^{−1} ∑_{\begin{matrix}w,v∈W \\ ^{v}(F w)∈W_{J} w_{1}F \end{matrix}} χ̃(F w)

= |W |^{−1}|W_{J}|^{−1} ∑_{v∈W} ∑_{w_{2}∈W_{J}} χ̃(v^{−1}(w_{2}w_{1}F)v)

= |W |^{−1}|W_{J}|^{−1} ∑_{v∈W} ∑_{w_{2}∈W_{J}} χ̃(w_{2}w_{1}F)

= T (χ̃; W_{J}w_{1}F).

Theorem 3.31. Let G be a connected reductive group and suppose the characteristic p is very good for G. If X ∈ 𝔤^{F} is regular of genus W_{J}w_{1}F then

Q̃_{w}^{𝔤}(X) = ∑_{χ̃∈C(W/F)_{irr}} \overline{T (χ̃; W_{J}w_{1}F)}χ̃(F w).

Proof. From Theorem 3.30, we find that

∑_{χ̃∈C(W/F)_{irr}} \overline{T (χ̃; W_{J}w_{1}F)}χ̃(F w) = ∑_{χ̃∈C(W/F)_{irr}} χ̃(F w)\overline{Q̃_{χ}^{𝔤}(X)}

= ∑_{χ̃∈C(W/F)_{irr}} χ̃(F w)|W |^{−1} ∑_{v∈W} \overline{χ̃(F v)}Q̃_{v}^{𝔤}(X)

= ∑_{v∈W} Q̃_{v}^{𝔤}(X)|W |^{−1} ∑_{χ̃∈C(W/F)_{irr}} χ̃(F w)\overline{χ̃(F v)}

= Q̃_{w}^{𝔤}(X).

In the second line we have used the fact that Q̃_{w}^{𝔤}(X) is a rational number (cf. Lehrer [18, Corollary 6.9]). The fourth line follows from the relations

∑_{χ̃∈C(W/F)_{irr}} χ̃(F w)\overline{χ̃(F v)} = \begin{cases}|W |^{−1}|CW̃(F w)| & if F w ∼W̃ F v, \\ 0 & otherwise, \end{cases}

which, since the number of F -conjugacy classes of W equals the number of F - stable conjugacy classes, in turn follow from the following relations (cf. Digne & Michel [11, §0.4])

|W |^{−1} ∑_{w∈W} χ̃(F w)\overline{ϕ̃(F w)} = \begin{cases}1 & if χ = ϕ, \\ 0 & otherwise, \end{cases}

where χ̃, ϕ̃ ∈ C(W/F)_{irr}.

References

[1] Anne-Marie Aubert. Formule des traces sur les corps finis. In Finite Reductive Groups: Related Structures and Representations, volume 141 of Progress in Mathematics, pages 15– 49. Birkhäuser, Boston, 1997.

[2] A. A. Beilinson, J. Bernstein, and P. Deligne. Faisceaux pervers. Astérisque, 100, 1982.

[3] Armand Borel. Linear Algebraic Groups. Number 126 in Graduate Texts in Mathematics. Springer-Verlag, New-York, second enlarged edition, 1991.

[4] Armand Borel and Jacques Tits. Groupes réductifs. Inst. Hautes Études Sci. Publ. Math., 27:55–152, 1965.

[5] N. Bourbaki. Groupes et Algèbres de Lie. Éléments de Mathématique. Masson, Paris, 1981.

[6] Michel Broué, Gunter Malle, and Jean Michel. Generic blocks of finite reductive groups. Astérisque, 212:7–92, 1993.

[7] R. W. Carter. Centralizers of semisimple elements in the finite classical groups. Proc. London Math. Soc., 42:1–41, 1981.

[8] Roger W. Carter. Simple Groups of Lie Type. John Wiley and Sons, New York, second edition, 1989.

[9] Roger W. Carter. Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. John Wiley and Sons, Chichester, second edition, 1993.

[10] D. I. Deriziotis. Centralizers of semisimple elements in a Chevalley group. Comm. Algebra, 9(19):1997–2014, 1981.

[11] François Digne and Jean Michel. On Lusztig’s parametrization of characters of finite groups of Lie type. Astérisque, 181–182:113–156, 1990.

[12] François Digne and Jean Michel. Representations of Finite Groups of Lie Type. Number 21 in London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991.

[13] M. Goresky and R. MacPherson. Intersection homology II. Invent. Math., 72:77–129, 1983.

[14] D. G. A. Jackson. On nilpotent orbits of type G_{2}. PhD Thesis, Sydney University, 1997.

[15] Shin-Ichi Kato. A realization of irreducible representations of affine Weyl groups. Nederl. Akad. Wetensch. Indag. Math., 86(2):193–201, 1983.

[16] D. Kazhdan. Proof of Springer’s hypothesis. Israel J. Math., 28:272–286, 1977.

[17] G. I. Lehrer. Rational tori, semisimple orbits and the topology of hyperplane complements. Commentarii Math. Helv., 67:226–251, 1992.

[18] G. I. Lehrer. The cohomology of the regular semisimple variety. Sydney University preprint, [19] G. I. Lehrer. The space of invariant functions on a finite Lie algebra. Trans. Amer. Math.

Soc., 348(1):31–50, 1996.

[20] G. I. Lehrer. Fourier transforms, nilpotent orbits, Hall polynomials and Green functions. In Finite Reductive Groups: Related Structures and Representations, volume 141 of Progress in Mathematics, pages 291–309. Birkhäuser, Boston, 1997.

[21] G. Lusztig. Green polynomials and singularities of unipotent classes. Adv. in Math., 42:169– 178, 1981.

[22] G. Lusztig. On the character values of finite Chevalley groups at unipotent elements. J. Algebra, 104:146–194, 1986.

[23] David Mumford. The Red Book of Varieties and Schemes. Number 1358 in Lecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg, 1988.

[24] Toshiaki Shoji. Geometry of orbits and Springer correspondence. Astérisque, 168:61–140, 1988.

[25] Toshiaki Shoji. Character sheaves and almost characters of reductive groups, II. Adv. in Math., 111:314–354, 1995.

[26] Peter Slodowy. Simple Singularities and Simple Algebraic Groups. Number 815 in Lecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg, 1980.

[27] T. A. Springer. Some arithmetical results of semi-simple Lie algebras. Inst. Hautes Études Sci. Publ. Math., 30:115–141, 1966.

[28] T. A. Springer. Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math., 36:173–207, 1976.

[29] T. A. Springer. Linear Algebraic Groups. Birkhäuser, Boston, 1981.

[30] T. A. Springer and R. Steinberg. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups, number 131 in Lecture Notes in Mathematics, part E, pages 167–266. Springer-Verlag, Berlin-Heidelberg, 1986. Second Corrected Printing.

[31] Bhama Srinivasan. Representations of Finite Chevalley Groups. Number 764 in Lecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg, 1979.

[32] Robert Steinberg. Regular elements of semisimple algebraic groups. Inst. Hautes Études Sci. Publ. Math., 25:49–80, 1965.

School of Mathematics and Statistics

University of Sydney Australia 2006