Homology of the Steinberg Variety

and Weyl Group Coinvariants

J. Matthew Douglass and Gerhard Röhrle ^{1}

Received: September 1, 2008 Revised: July 31, 2009

Communicated by Wolfgang Soergel

Abstract. Let G be a complex, connected, reductive algebraic group with Weyl group W and Steinberg variety Z. We show that the graded Borel-Moore homology of Z is isomorphic to the smash product of the coinvariant algebra of W and the group algebra of W.

2000 Mathematics Subject Classification: Primary 20G05; Secondary 20F55 Keywords and Phrases: Borel-Moore Homology, Steinberg Variety, Coinvariant algebra, Weyl group

1. Introduction

Suppose G is a complex, reductive algebraic group, B is the variety of Borel subgroups of G. Let 𝔤 be the Lie algebra of G and N the cone of nilpotent elements in 𝔤. Let T^{∗}B denote the cotangent bundle of B. Then there is a moment map, μ_{0} : T^{∗}B → N. The Steinberg variety of G is the fibered product T^{∗}B ×_{N} T^{∗}B which we will identify with the closed subvariety

Z = { (x, B′, B′′) ∈ N × B × B | x ∈ Lie(B′) ∩ Lie(B′′) }

of N × B × B. Set n = dim B. Then Z is a 2n-dimensional, complex algebraic variety. If V = ⊕_{i≥0}V_{i} is a graded vector space, we will frequently denote V by V_{•}. Similarly, if X is a topological space, then H_{i}(X) denotes the i^{th} rational Borel-Moore homology of X and H_{•}(X) = ⊕_{i≥0}H_{i}(X) denotes the total Borel-Moore homology of X. Fix a maximal torus, T, of G, with Lie algebra 𝔱, and let W = N_{G}(T)/T be the Weyl group of (G, T). In [6] Kazhdan and Lusztig defined an action of W × W on H_{•}(Z) and they showed that the representation of W × W on the top-dimensional homology of Z, H_{4n}(Z), is equivalent to the two-sided regular representation of W. Tanisaki [11] and, more recently, Chriss and Ginzburg [3] have strengthened the connection between H_{•}(Z) and W by defining a ℚalgebra structure on H_{•}(Z) so that H_{i}(Z) ∗ H_{j}(Z) ⊆ H_{i+j−4n}(Z). Chriss and Ginzburg [3, §3.4] have also given an elementary construction of an isomorphism between H_{4n}(Z) and the group algebra ℚW. Let Z_{1} denote the “diagonal” in Z:

1The authors would like to thank their charming wives for their unwavering support during the preparation of this paper

Z_{1} = { (x, B′, B′) ∈ N × B × B | x ∈ Lie(B′) }.

In this paper we extend the results of Chriss and Ginzburg [3, §3.4] and show in Theorem 2.3 that for any i, the convolution product defines an isomorphism H_{i}(Z_{1}) ⊗ H_{4n}(Z) ⟶^{∼} H_{i}(Z). It then follows easily that with the convolution product, H_{•}(Z) is isomorphic to the smash product of the coinvariant algebra of W and the group algebra of W. Precisely, for 0 ≤ i ≤ n let Coinv_{2i}(W) denote the degree i subspace of the rational coinvariant algebra of W, so Coinv_{2i}(W) may be identified with the space of degree i, W -harmonic polynomials on 𝔱. If j is odd, define Coinv_{j}(W) = 0. Recall that the smash product, Coinv(W)#ℚW, is the ℚalgebra whose underlying vector space is Coinv(W)⊗_{ℚ} ℚW with multiplication satisfying (f_{1}⊗ϕ_{1})·(f_{2}⊗ϕ_{2}) = f_{1}ϕ_{1}(f_{2})⊗ϕ_{1}ϕ_{2} where f_{1} and f_{2} are in Coinv(W), ϕ_{1} and ϕ_{2} are in ℚW, and ℚW acts on Coinv(W) in the usual way. The algebra Coinv(W)#ℚW is graded by (Coinv(W)#ℚW)_{i} = Coinv_{i}(W)#ℚW and we will denote this graded algebra by Coinv_{•}(W)#ℚW. In Theorem 2.5 we construct an explicit isomorphism of graded algebras H_{4n−•}(Z) ≅ Coinv_{•}(W)#ℚW. This paper was motivated by the observation, pointed out to the first author by Catharina Stroppel, that the argument in [3, 8.1.5] can be used to show that H_{•}(Z) is isomorphic to the smash product of ℚW and Coinv_{•}(W). The details of such an argument have been carried out in a recent preprint of Namhee Kwon [8]. This argument relies on some deep and technical results: the localization theorem in K-theory proved by Thomason [12], the bivariant Riemann-Roch Theorem [3, 5.11.11], and the Kazhdan-Lusztig isomorphism between the equivariant K-theory of Z and the extended, affine, Hecke algebra [7]. In contrast, and also in the spirit of Kazhdan and Lusztig’s original analysis of H_{4n}(Z), and the analysis of H_{4n}(Z) in [3, 3.4], our argument uses more elementary notions and is accessible to readers who are not experts in equivariant K-theory and to readers who are not experts in the representation theory of reductive, algebraic groups. Another approach to the Borel-Moore homology of the Steinberg variety uses intersection homology. Let μ: Z → N be projection on the first factor. Then, as in [3, §8.6], H_{•}(Z) ≅ Ext_{D(N)}^{4n−•} (Rμ_{∗}ℚ_{N} , Rμ_{∗}ℚ_{N}). The Decomposition Theorem of Beilinson, Bernstein, and Deligne can be used to decompose Rμ_{∗}ℚ_{N} into a direct sum of simple perverse sheaves Rμ_{∗}ℚ_{N} ≅ ⊕_{x,ϕ} IC_{x,ϕ}^{n_{x,ϕ}} where x runs over a set of orbit representatives in N, for each x, ϕ runs over a set of irreducible representations of the component group of Z_{G}(x), and IC_{x,ϕ} denotes an intersection complex (see [2] or [9, §4,5]). Chriss and Ginzburg have used this construction to describe an isomorphism H_{4n}(Z) ≅ ℚW and to in addition give a description of the projective, indecomposable H_{•}(Z)-modules. It follows from Theorem 2.3 that H_{i}(Z) ≅ Coinv_{4n−i}(W) ⊗ H_{4n}(Z) and so

(1.1)

Coinv_{i}(W) ⊗ H_{4n}(Z) ≅ Ext_{D(N)}^{4n−i} (Rμ_{∗}ℚ_{N} , Rμ_{∗}ℚ_{N})

≅ ⨁_{x,ϕ} ⨁_{y,ψ} Ext_{D(N)}^{4n−i} (IC_{x,ϕ}^{n_{x,ϕ}}, IC_{y,ψ}^{n_{y,ψ}}) .

In the special case when i = 0 we have that

Coinv_{0}(W) ⊗ H_{4n}(Z) ≅ End_{D(N)} (Rμ_{∗}ℚ_{N}) ≅ ⨁_{x,ϕ} End_{D(N)} (IC_{x,ϕ}^{n_{x,ϕ}}) .

The image of the one-dimensional vector space Coinv_{0}(W) in End_{D(N)} (Rμ_{∗}ℚ_{N}) is the line through the identity endomorphism and

ℚW ≅ H_{4n}(Z) ≅ ⊕_{x,ϕ} End_{D(N)} (IC_{x,ϕ}^{n_{x,ϕ}}) is the Wedderburn decomposition of ℚW as a direct sum of minimal two-sided ideals. For i > 0 we have not been able to find a nice description of the image of Coinv_{i}(W) in the right-hand side of (1.1). The rest of this paper is organized as follows: in §2 we set up our notation and state the main results; in §3 we construct an isomorphism of graded vector spaces between Coinv_{•}(W) ⊗ ℚW and H_{4n−•}(Z); and in §4 we complete the proof that this isomorphism is in fact an algebra isomorphism when Coinv_{•}(W) ⊗ ℚW is given the smash product multiplication. Some very general results about graphs and convolution that we need for the proofs of the main theorems are proved in an appendix. In this paper ⊗ = ⊗_{ℚ}, if X is a set, then δ_{X}, or just δ, will denote the diagonal embedding of X in X × X, and for g in G and x in 𝔤, g · x denotes the adjoint action of g on x.

2. Preliminaries and Statement of Results

Fix a Borel subgroup, B, of G with T ⊆ B and define U to be the unipotent radical of B. We will denote the Lie algebras of B and U by 𝔟 and 𝔲 respectively. Our proof that H_{•}(Z) is isomorphic to Coinv_{•}(W)#ℚW makes use of the specialization construction used by Chriss and Ginzburg in [3, §3.4] to establish the isomorphism between H_{4n}(Z) and ℚW. We begin by reviewing their construction. The group G acts diagonally on B × B. Let O_{w} denote the orbit containing (B, wBw^{−1}). Then the rule w ↦ O_{w} defines a bijection between W and the set of G-orbits in B × B. Let π_{Z}: Z → B × B denote the projection on the second and third factors and for w in W define Z_{w} = π_{Z}^{−1}(O_{w}). For w in W we also set 𝔲_{w} = 𝔲 ∩ w · 𝔲. The following facts are well-known (see [10] and [9, §1.1]):

• Z_{w} ≅ G ×^{B∩^{w}B} 𝔲_{w}.

• dim Z_{w} = 2n.
• The set \{ \overline{Z_{w}} | w ∈ W \} is the set of irreducible components of Z.

Define

𝔤̃ = { (x, B′) ∈ 𝔤 × B | x ∈ Lie(B′) },
Ñ = { (x, B′) ∈ N × B | x ∈ Lie(B′) }, and
Ẑ = { (x, B′, B′′) ∈ 𝔤 × B × B | x ∈ Lie(B′) ∩ Lie(B′′) },

and let μ: 𝔤̃ → 𝔤 denote the projection on the first factor. Then Ñ ≅ T^{∗}B, μ(Ñ) = N, Z ≅ Ñ ×_{N} Ñ, and Ẑ ≅ 𝔤̃ ×_{𝔤} 𝔤̃. Let π̂: Ẑ → B × B denote the projection on the second and third factors and for w in W define Ẑ_{w} = π̂^{−1}(O_{w}). Then it is well-known that dim Ẑ_{w} = dim 𝔤 and that the closures of the Ẑ_{w}’s for w in W are the irreducible components of Ẑ (see [9, §1.1]). Next, for (x, gBg^{−1}) in 𝔤̃, define ν(x, gBg^{−1}) to be the projection of g^{−1} ·x in 𝔱. Then μ and ν are two of the maps in Grothendieck’s simultaneous resolution:

\xymatrix{𝔤̃ \ar[r]^{μ} \ar[d]_{ν} & 𝔤 \ar[d] \\ 𝔱 \ar[r] & 𝔱/W}

It is easily seen that if μ̂: Ẑ → 𝔤 is the projection on the first factor, then the square

\xymatrix{Ẑ \ar[r]^{μ̂} \ar[d] & 𝔤 \ar[d]^{δ_{𝔤}} \\ 𝔤̃ × 𝔤̃ \ar[r]_{μ×μ} & 𝔤 × 𝔤}

is cartesian, where the vertical map on the left is given by (x, B′, B′′) ↦ ((x, B′), (x, B′′)). We will frequently identify Ẑ with the subvariety of 𝔤̃ × 𝔤̃ consisting of all pairs ((x, B′), (x, B′′)) with x in Lie(B′) ∩ Lie(B′′). For w in W, let Γ_{w^{−1}} = { (h, w^{−1} · h) | h ∈ 𝔱 } ⊆ 𝔱 × 𝔱 denote the graph of the action of w^{−1} on 𝔱 and define

Λ_{w} = Ẑ ∩ (ν × ν)^{−1} (Γ_{w^{−1}}) = { (x, B′, B′′) ∈ Ẑ | ν(x, B′′) = w^{−1}ν(x, B′) }.

In the special case when w is the identity element in W, we will denote Λ_{w} by Λ_{1}.

The spaces we have defined so far fit into a commutative diagram with cartesian squares:

(2.1)  \xymatrix{Λ_{w} \ar[r] \ar[d] & Ẑ \ar[r]^{μ̂} \ar[d] & 𝔤 \ar[d]^{δ_{𝔤}} \\ (ν × ν)^{−1} (Γ_{w^{−1}}) \ar[r] \ar[d] & 𝔤̃ × 𝔤̃ \ar[r]_{μ×μ} \ar[d]_{ν×ν} & 𝔤 × 𝔤 \\ Γ_{w^{−1}} \ar[r] & 𝔱 × 𝔱}

Let ν_{w}: Λ_{w} → Γ_{w^{−1}} denote the composition of the leftmost vertical maps in (2.1), so ν_{w} is the restriction of ν × ν to Λ_{w}. For the specialization construction, we consider subsets of Ẑ of the form ν_{w}^{−1}(S′) for S′ ⊆ Γ_{w^{−1}}. Thus, for h in 𝔱 we define Λ_{w}^{h} = ν_{w}^{−1}(h, w^{−1}h). Notice in particular that Λ_{w}^{0} = Z. More generally, for a subset S of 𝔱 we define Λ_{w}^{S} = ∐_{h∈S} Λ_{w}^{h}. Then, Λ_{w}^{S} = ν_{w}^{−1}(S′), where S′ is the graph of w^{−1} restricted to S. Let 𝔱_{reg} denote the set of regular elements in 𝔱. Fix a one-dimensional subspace, ℓ, of 𝔱 so that ℓ ∩ 𝔱_{reg} = ℓ \ {0} and set ℓ^{∗} = ℓ \ {0}. Then Λ_{w}^{ℓ} = Λ_{w}^{ℓ^{∗}} ∐ Λ_{w}^{0} = Λ_{w}^{ℓ^{∗}} ∐ Z. We will see in Corollary 3.6 that the restriction of ν_{w} to Λ_{w}^{ℓ^{∗}} is a locally trivial fibration with fibre G/T. Thus, using a construction due to Fulton and MacPherson ([4, §3.4], [3, §2.6.30]), there is a specialization map

lim: H_{•+2}(Λ_{w}^{ℓ^{∗}}) ⟶ H_{•}(Z).

Since Λ_{w}^{ℓ^{∗}} is an irreducible, (2n+1)-dimensional variety, if [Λ_{w}^{ℓ^{∗}}] denotes the fundamental class of Λ_{w}^{ℓ^{∗}}, then H_{4n+2}(Λ_{w}^{ℓ^{∗}}) is one-dimensional with basis {[Λ_{w}^{ℓ^{∗}}]}. Define λ_{w} = lim([Λ_{w}^{ℓ^{∗}}]) in H_{4n}(Z). Chriss and Ginzburg [3, §3.4] have proved the following theorem.

Theorem 2.2. Consider H_{•}(Z) endowed with the convolution product.

(a) For 0 ≤ i, j ≤ 4n, H_{i}(Z) ∗ H_{j}(Z) ⊆ H_{i+j−4n}(Z). In particular, H_{4n}(Z) is a subalgebra of H_{•}(Z).

(b) The element λ_{w} in H_{4n}(Z) does not depend on the choice of ℓ.

(c) The assignment w ↦ λ_{w} extends to an algebra isomorphism

α: ℚW ⟶^{≅} H_{4n}(Z).

Now consider

Z_{1} = { (x, B′, B′) ∈ N × B × B | x ∈ Lie(B′) }.

Then Z_{1} may be identified with the diagonal in Ñ × Ñ. It follows that Z_{1} is closed in Z and isomorphic to Ñ.

Since Ñ ≅ T^{∗}B, it follows from the Thom isomorphism in Borel-Moore homology [3, §2.6] that H_{i+2n}(Z_{1}) ≅ H_{i}(B) for all i. Since B is smooth and compact, H_{i}(B) ≅ H^{2n−i}(B) by Poincaré duality. Therefore, H_{4n−i}(Z_{1}) ≅ H^{i}(B) for all i. The cohomology of B is well-understood: there is an isomorphism of graded algebras, H^{•}(B) ≅ Coinv_{•}(W). It follows that H_{j}(Z_{1}) = 0 if j is odd and H_{4n−2i}(Z_{1}) ≅ Coinv_{2i}(W) for 0 ≤ i ≤ n. In §3 below we will prove the following theorem.

Theorem 2.3. Consider the Borel-Moore homology of the variety Z_{1}.

(a) There is a convolution product on H_{•}(Z_{1}). With this product, H_{•}(Z_{1}) is a commutative ℚ-algebra and there is an isomorphism of graded ℚalgebras

β: Coinv_{•}(W) ⟶^{≅} H_{4n−•}(Z_{1}).

(b) If r: Z_{1} → Z denotes the inclusion, then the direct image map in Borel-Moore homology, r_{∗}: H_{•}(Z_{1}) ⟶ H_{•}(Z), is an injective ring homomorphism.

(c) If we identify H_{•}(Z_{1}) with its image in H_{•}(Z) as in (b), then the linear transformation given by the convolution product

H_{i}(Z_{1}) ⊗ H_{4n}(Z) ⟶^{∗} H_{i}(Z)

is an isomorphism of vector spaces for 0 ≤ i ≤ 4n.

The algebra Coinv_{•}(W) has a natural action of W by algebra automorphisms, and the isomorphism β in Theorem 2.3(a) is in fact an isomorphism of W - algebras. The W -algebra structure on H_{•}(Z_{1}) is described in the next theorem, which will be proved in §4.

Theorem 2.4. If w is in W and H_{•}(Z_{1}) is identified with its image in H_{•}(Z), then

λ_{w} ∗ H_{i}(Z_{1}) ∗ λ_{w^{−1}} = H_{i}(Z_{1}).

Thus, conjugation by λ_{w} defines a W -algebra structure on H_{•}(Z_{1}). With this W -algebra structure, the isomorphism β: Coinv_{•}(W) ⟶^{≅} H_{4n−•}(Z_{1}) in Theorem 2.3(a) is an isomorphism of W -algebras.

Recall that the algebra Coinv(W)#ℚW is graded by (Coinv(W)#ℚW)_{i} = Coinv_{i}(W)⊗ℚW. Then combining Theorem 2.2(c), Theorem 2.3(c), and Theorem 2.4 we get our main result.

Theorem 2.5. The composition

Coinv_{•}(W)#ℚW ⟶^{β⊗α} H_{4n−•}(Z_{1}) ⊗ H_{4n}(Z) ⟶^{∗} H_{4n−•}(Z)

is an isomorphism of graded ℚ-algebras.

3. Factorization of H_{•}(Z)

Proof of Theorem 2.3(a). We need to prove that H_{•}(Z_{1}) is a commutative ℚ-algebra and that Coinv_{•}(W) ≅ H_{4n−•}(Z_{1}). Let π: Ñ → B by π(x, B′) = B′. Then π may be identified with the vector bundle projection T^{∗}B → B and so the induced map in cohomology π^{∗}: H^{i}(B) → H^{i}(Ñ) is an isomorphism. The projection π determines an isomorphism in Borel-Moore homology that we will also denote by π^{∗} (see [3,

§2.6.42]). We have π^{∗}: H_{i}(B) ⟶^{≅} H_{i+2n}(Ñ).

For a smooth m-dimensional variety X, let pd: H^{i}(X) ⟶ H_{2m−i}(X) denote the Poincaré duality isomorphism. Then the composition

H_{2n−i}(B) ⟶^{pd^{−1}} H^{i}(B) ⟶^{π^{∗}} H^{i}(Ñ) ⟶^{pd} H_{4n−i}(Ñ)

is an isomorphism. It follows from the uniqueness construction in [3, §2.6.26] that

pd ∘ π^{∗} ∘ pd^{−1} = π^{∗}: H_{2n−i}(B) ⟶ H_{4n−i}(Ñ)

and so π^{∗} ∘ pd = pd ∘ π^{∗}: H^{i}(B) ⟶ H_{4n−i}(Ñ). Recall that Coinv_{j}(W) = 0 if j is odd and Coinv_{2i}(W) is the degree i subspace of the coinvariant algebra of W. Let bi: Coinv_{•}(W) ⟶ H^{•}(B) be the Borel isomorphism (see [1, §1.5] or [5]). Then with the cup product, H^{•}(B) is a graded algebra and bi is an isomorphism of graded algebras. Define β: Coinv_{i}(W) → H_{4n−i}(Z_{1}) to be the composition

Coinv_{i}(W) ⟶^{bi} H^{i}(B) ⟶^{π^{∗}} H^{i}(Ñ) ⟶^{pd} H_{4n−i}(Ñ) ⟶^{δ_{∗}} H_{4n−i}(Z_{1})

where δ = δÑ. Then β is an isomorphism of graded vector spaces and

β = δ_{∗} ∘ pd ∘ π^{∗} ∘ bi = δ_{∗} ∘ π^{∗} ∘ pd ∘ bi.

The algebra structure of H^{•}(B) and H^{•}(Ñ) is given by the cup product, and π^{∗}: H^{•}(B) → H^{•}(Ñ) is an isomorphism of graded algebras. Since Ñ is smooth, as in [3, §2.6.15], there is an intersection product defined on H_{•}(Ñ) using Poincaré duality and the cup product on H^{•}(Ñ). Thus, pd: H^{•}(Ñ) → H_{4n−•}(Ñ) is an algebra isomorphism. Finally, it is observed in [3, §2.7.10] that δ_{∗}: H_{•}(Ñ) → H_{•}(Z_{1}) is a ring homomorphism and hence an algebra isomorphism. This shows that β is an isomorphism of graded algebras and proves Theorem 2.3(a).

Proof of Theorem 2.3(b). To prove the remaining parts of Theorem 2.3, we need a linear order on W. Suppose |W | = N. Fix a linear order on W that extends the Bruhat order. Say W = {w_{1}, …, w_{N}}, where w_{1} = 1 and w_{N} is the longest element in W. For 1 ≤ j ≤ N, define Z_{j} = ∐_{i=1}^{j} Z_{w_{i}}. Then, for each j, Z_{j} is closed in Z, Z_{w_{j}} is open in Z_{j}, and Z_{j} = Z_{j−1} ∐ Z_{w_{j}}. Notice that Z_{N} = Z and Z_{1} = Z_{w_{1}}. Similarly, define Ẑ_{j} = ∐_{i=1}^{j} Ẑ_{w_{i}}. Then each Ẑ_{j} is closed in Ẑ, Ẑ_{w_{j}} is open in Ẑ_{j}, and Ẑ_{j} = Ẑ_{j−1} ∐ Ẑ_{w_{j}}.

We need to show that r_{∗}: H_{•}(Z_{1}) ⟶ H_{•}(Z) is an injective ring homomorphism. Let res_{j}: H_{i}(Z_{j}) → H_{i}(Z_{w_{j}}) denote the restriction map in Borel-Moore homology induced by the open embedding Z_{w_{j}} ⊆ Z_{j} and let r_{j}: H_{i}(Z_{j−1}) ⟶ H_{i}(Z_{j}) denote the direct image map in Borel-Moore homology induced by the closed embedding Z_{j−1} ⊆ Z_{j}. Then there is a long exact sequence in homology

\xymatrix{⋯ \ar[r] & H_{i}(Z_{j−1}) \ar[r]^{r_{j}} & H_{i}(Z_{j}) \ar[r]^{^{res^{j}}} & H_{i}(Z_{w_{j}}) \ar[r]^{∂} & H_{i−1}(Z_{j−1}) \ar[r] & ⋯}

It is shown in [3, §6.2] that ∂ = 0 and so the sequence

(3.1)  \xymatrix{0 \ar[r] & H_{i}(Z_{j−1}) \ar[r]^{r_{j}} & H_{i}(Z_{j}) \ar[r]^{^{res^{j}}} & H_{i}(Z_{w_{j}}) \ar[r] & 0}

is exact for every i and j. Therefore, if r: Z_{j} → Z denotes the inclusion, then the direct image r_{∗}: H_{i}(Z_{j}) → H_{i}(Z) is an injection for all i. (The fact that r depends on j should not lead to any confusion.) We will frequently identify H_{i}(Z_{j}) with its image in H_{i}(Z) and consider H_{i}(Z_{j}) as a subset of H_{i}(Z). Thus, we have a flag of subspaces 0 ⊆ H_{i}(Z_{1}) ⊆ ⋯ ⊆ H_{i}(Z_{N−1}) ⊆ H_{i}(Z). In particular, r_{∗}: H_{i}(Z_{1}) → H_{i}(Z) is an injection for all i. It follows from [3, Lemma 5.2.23] that r_{∗} is a ring homomorphism. This proves part (b) of Theorem 2.3.

Proof of Theorem 2.3(c). We need to show that the linear transformation given by the convolution product H_{i}(Z_{1})⊗H_{4n}(Z) → H_{i}(Z) is an isomorphism of vector spaces for 0 ≤ i ≤ 4n. The proof is a consequence of the following lemma.

Lemma 3.2. The image of the convolution map ∗ : H_{i}(Z_{1})⊗H_{4n}(Z_{j}) ⟶ H_{i}(Z) is precisely H_{i}(Z_{j}) for 0 ≤ i ≤ 4n and 1 ≤ j ≤ N.

Assuming that the lemma has been proved, taking j = N, we conclude that the convolution product in H_{•}(Z) induces a surjection H_{i}(Z_{1})⊗H_{4n}(Z) ⟶ H_{i}(Z). It is shown in [3, §6.2] that dim H_{•}(Z) = |W |^{2} and so dim H_{•}(Z_{1}) ⊗ H_{4n}(Z) = |W |^{2} = dim H_{•}(Z). Thus, the convolution product induces an isomorphism H_{i}(Z_{1}) ⊗ H_{4n}(Z) ≅ H_{i}(Z). The rest of this section is devoted to the proof of Lemma 3.2. To prove Lemma 3.2 we need to analyze the specialization map, lim: H_{•+2}(Λ_{w}^{ℓ^{∗}}) → H_{•}(Z), beginning with the subvarieties Λ_{w}^{ℓ} and Λ_{w}^{ℓ^{∗}} of Λ_{w}.

Subvarieties of Λ_{w}. Suppose that ℓ is a one-dimensional subspace of 𝔱 with ℓ^{∗} = ℓ \ {0} = ℓ ∩ 𝔱_{reg}. Recall that 𝔲_{w} = 𝔲 ∩ w · 𝔲 for w in W.

Lemma 3.3. The variety Λ_{w}^{ℓ} ∩ Ẑ_{w} is the G-saturation in Ẑ of the subset

{ (h + n, B, wBw^{−1}) | h ∈ ℓ, n ∈ 𝔲_{w} }.

Proof. By definition,

Λ_{w}^{ℓ} = Λ_{w}^{ℓ^{∗}} ∐ Λ_{w}^{0} = { (x, B′, B′′) ∈ Ẑ | ν(x, B′′) = w^{−1}ν(x, B′) ∈ w^{−1}(ℓ) }.

Suppose that h is in 𝔱_{reg} and (x, g_{1}Bg_{1}^{−1}, g_{2}Bg_{2}^{−1}) is in Λ_{w}^{h}. Then g_{1}^{−1}·x = h+n_{1} and g_{2}^{−1} · x = w^{−1}h + n_{2} for some n_{1} and n_{2} in 𝔲. Since h is regular, there are elements u_{1} and u_{2} in U so that u_{1}^{−1}g_{1}^{−1} · h = h and u_{2}^{−1}g_{2}^{−1} · h = w^{−1}h. Then x = g_{1}u_{1} · h = g_{2}u_{2}w^{−1} · h and so g_{1}u_{1} = g_{2}u_{2}w^{−1}t for some t in T. Therefore, (x, g_{1}Bg_{1}^{−1}, g_{2}Bg_{2}^{−1}) = g_{1}u_{1} ·(h, B, wBw^{−1}). Thus, Λ_{w}^{h} is contained in the G-orbit of (h, B, wBw^{−1}). Since ν is G-equivariant, it follows that Λ_{w}^{h} is G-stable and so Λ_{w}^{h} is the full G-orbit of (h, B, wBw^{−1}). Therefore, Λ_{w}^{ℓ^{∗}} is the G-saturation of { (h + n, B, wBw^{−1} | h ∈ ℓ^{∗}, n ∈ 𝔲_{w} } and Λ_{w}^{h} ⊆ Ẑ_{w} for h in ℓ^{∗}. We have already observed that Λ_{w}^{0} = Z and so

Λ_{w}^{ℓ} ∩ Ẑ_{w} = (Λ_{w}^{ℓ^{∗}} ∩ Ẑ_{w}) ∐ (Λ_{w}^{0} ∩ Ẑ_{w}) = Λ_{w}^{ℓ^{∗}} ∐ Z_{w}.

It is easy to see that Z_{w} is the G-saturation of { (n, B, wBw^{−1}) | n ∈ 𝔲_{w} } in Z. This proves the lemma. □

Corollary 3.4. The variety Λ_{w}^{ℓ} ∩ Ẑ_{w} is a locally trivial, affine space bundle over O_{w} with fibre isomorphic to ℓ + 𝔲_{w}, and hence there is an isomorphism Λ_{w}^{ℓ} ∩ Ẑ_{w} ≅ G ×^{B∩^{w}B} (ℓ + 𝔲_{w}).

Proof. It follows from Lemma 3.3 that the map given by projection on the second and third factors is a G-equivariant morphism from Λ_{w}^{ℓ} onto O_{w} and that the fibre over (B, wBw^{−1}) is { (h + n, B, wBw^{−1}) | h ∈ ℓ, n ∈ 𝔲_{w} }. Therefore, Λ_{w}^{ℓ} ≅ G ×^{B∩^{w}B} (ℓ + 𝔲_{w}). □

Let 𝔤_{rs} denote the set of regular semisimple elements in 𝔤 and define 𝔤̃_{rs} = { (x, B′) ∈ 𝔤̃ | x ∈ 𝔤_{rs} }. For an arbitrary subset S of 𝔱, define

𝔤̃^{S} = ν^{−1}(S) = { (x, B′) ∈ 𝔤̃ | ν(x, B′) ∈ S }.

For w in W, define w̃: G/T × 𝔱_{reg} ⟶ G/T × 𝔱_{reg} by w̃(gT, h) = (gwT, w^{−1}h). The rule (gT, h) ↦ (g · h, gB) defines an isomorphism of varieties

f: G/T × 𝔱_{reg} ⟶^{≅} 𝔤̃_{rs}

and we will denote the automorphism f ∘ w̃ ∘ f^{−1} of 𝔤̃_{rs} also by w̃. Notice that if h is in 𝔱_{reg} and g is in G, then w̃(g · h, gB) = (g · h, gwBw^{−1}g^{−1}).

Lemma 3.5. The variety Λ_{w}^{ℓ^{∗}} is the graph of w̃|𝔤̃^{ℓ^{∗}} : 𝔤̃^{ℓ^{∗}} → 𝔤̃^{w^{−1}(ℓ^{∗})}.

Proof. It follows from Lemma 3.3 that

Λ_{w}^{ℓ^{∗}} = { (g · h, gBg^{−1}, gwBw^{−1}g^{−1}) ∈ 𝔤_{rs} × B × B | h ∈ ℓ^{∗}, g ∈ G }
= { ((g · h, gBg^{−1}), (g · h, gwBw^{−1}g^{−1})) ∈ 𝔤̃ × 𝔤̃ | h ∈ ℓ^{∗}, g ∈ G }.

The argument in the proof of Lemma 3.3 shows that

𝔤̃^{ℓ^{∗}} = { (g · h, gBg^{−1}) | h ∈ ℓ^{∗}, g ∈ G } and by definition w̃(g · h, gB) = (g · h, gwBw^{−1}g^{−1}). Therefore, Λ_{w}^{ℓ^{∗}} is the graph of w̃|𝔤̃^{ℓ^{∗}}. □

Corollary 3.6. The map ν_{w}: Λ_{w}^{ℓ^{∗}} → ℓ^{∗} is a locally trivial fibration with fibre isomorphic to G/T.

Proof. This follows from the lemma and the fact that 𝔤̃^{ℓ^{∗}} ≅ G/T × ℓ^{∗}. □

The specialization map. Suppose that w is in W and that ℓ is a onedimensional subspace of 𝔱 with ℓ^{∗} = ℓ \ {0} = ℓ ∩ 𝔱_{reg}. As in [4] and [3, §2.6.30], lim: H_{i+2}(Λ_{w}^{ℓ^{∗}}) → H_{i}(Z) is the composition of three maps, defined as follows. As a vector space over ℝ, ℓ is two-dimensional. Fix an ℝ-basis of ℓ, say {v_{1}, v_{2}}. Define P to be the open half plane ℝ_{>0}v_{1}⊕ℝv_{2}, define I_{>0} to be the ray ℝ_{>0}v_{1}, and define I to be the closure of I_{>0}, so I = ℝ_{≥0}v_{1}. Since P is an open subset of ℓ^{∗}, Λ_{w}^{P} is an open subset of Λ_{w}^{ℓ^{∗}} and so there is a restriction map in Borel-Moore homology res: H_{i+2}(Λ_{w}^{ℓ^{∗}}) → H_{i+2}(Λ_{w}^{P}). The projection map from P to I_{>0} determines an isomorphism in Borel-Moore homology ψ: H_{i+2}(Λ_{w}^{P}) → H_{i+1}(Λ_{w}^{I_{>0}}). Since I = I_{>0} ∐{0}, we have Λ_{w}^{I} = Λ_{w}^{I_{>0}} ∐ Λ_{w}^{0} = Λ_{w}^{I_{>0}} ∐ Z, where Z is closed in Λ_{w}^{I}. The connecting homomorphism of the long exact sequence in Borel-Moore homology arising from the partition Λ_{w}^{I} = Λ_{w}^{I_{>0}} ∐ Z is a map

∂: H_{i+1}(Λ_{w}^{I_{>0}}) → H_{i}(Z).

By definition, lim = ∂ ∘ ψ ∘ res. Now fix j with 1 ≤ j ≤ N and set w = w_{j}. Consider the intersection Λ_{w}^{I}∩Ẑ_{j} = (Λ_{w}^{I_{>0}}∩Ẑ_{j}) ∐(Z∩Ẑ_{j}). Then Z∩Ẑ_{j} is closed in Λ_{w}^{I} ∩ Ẑ_{j} and by construction, Λ_{w}^{I_{>0}} ⊆ Ẑ_{j} and Z ∩ Ẑ_{j} = Z_{j}. Thus, Λ_{w}^{I} ∩ Ẑ_{j} = Λ_{w}^{I_{>0}} ∐ Z_{j}. Let ∂_{j}: H_{i+1}(Λ_{w}^{I_{>0}}) → H_{i}(Z_{j}) be the connecting homomorphism of the long exact sequence in Borel-Moore homology arising from this partition. Because the long exact sequence in Borel-Moore homology is natural, we have a commutative square:

[figure: H_{i+1}(Λ_{w}^{I_{>0}}); ∂; H_{i}(Z); H_{i+1}(Λ_{w}^{I_{>0}}); ∂_{j}; H_{i}(Z_{j}); r_{∗}]

This proves the following lemma.

Lemma 3.7. Fix j with 1 ≤ j ≤ N and set w = w_{j}. Then the map ∂: H_{i+1}(Λ_{w}^{I_{>0}}) ⟶ H_{i}(Z) factors as r_{∗} ∘ ∂_{j} where ∂_{j}: H_{i+1}(Λ_{w}^{I_{>0}}) ⟶ H_{i}(Z_{j}) is the connecting homomorphism of the long exact sequence arising from the partition Λ_{w}^{I} ∩ Ẑ = Λ_{w}^{I_{>0}} ∐ Z_{j}.

It follows from the lemma that lim: H_{i+2}(Λ_{w}^{ℓ^{∗}}) ⟶ H_{i}(Z) factors as

(3.8) H_{i+2}(Λ_{w}^{ℓ^{∗}}) ⟶^{res} H_{i+2}(Λ_{w}^{P}) ⟶^{ψ} H_{i+1}(Λ_{w}^{I_{>0}}) ⟶^{∂_{j}} H_{i}(Z_{j}) ⟶^{r_{∗}} H_{i}(Z).

Define lim_{j}: H_{i+2}(Λ_{w}^{ℓ^{∗}}) ⟶ H_{i}(Z_{j}) by lim_{j} = ∂_{j} ∘ ψ ∘ res.

Specialization and restriction. As above, fix j with 1 ≤ j ≤ N and a one-dimensional subspace ℓ of 𝔱 with ℓ^{∗} = ℓ \ {0} = ℓ ∩ 𝔱_{reg}. Set w = w_{j}. Recall the restriction map res_{j}: H_{i}(Z_{j}) → H_{i}(Z_{w}) from (3.1).

Lemma 3.9. The composition res_{j} ∘ lim_{j}: H_{i+2}(Λ_{w}^{ℓ^{∗}}) ⟶ H_{i}(Z_{w}) is surjective for 0 ≤ i ≤ 4n.

Proof. Using (3.8), res_{j} ∘ lim_{j} factors as

H_{i+2}(Λ_{w}^{ℓ^{∗}}) ⟶^{res} H_{i+2}(Λ_{w}^{P}) ⟶^{ψ} H_{i+1}(Λ_{w}^{I_{>0}}) ⟶^{∂_{j}} H_{i}(Z_{j}) ⟶^{res_{j}} H_{i}(Z_{w}).

Lemma 3.11 below shows that res is always surjective and the map ψ is an isomorphism, so we need to show that the composition res_{j} ∘ ∂_{j} is surjective. Consider Λ_{w}^{I} ∩ Ẑ_{w} = (Λ_{w}^{I} ∩ Ẑ_{j}) ∩ Ẑ_{w} = Λ_{w}^{I_{>0}} ∐ Z_{w}. Then Λ_{w}^{I_{>0}} is open in Λ_{w}^{I} ∩ Ẑ_{w} and we have a commutative diagram of long exact sequences

[figure: ⋯; H_{i+1}(Λ_{w}^{I} ∩ Ẑ_{w}); H_{i+1}(Λ_{w}^{I_{>0}}); ∂_{w}; H_{i}(Z_{w}); ⋯; ⋯; H_{i+1}(Λ_{w}^{I} ∩ Ẑ_{j}); H_{i+1}(Λ_{w}^{I_{>0}}); ∂_{j}; H_{i}(Z_{j}); _{res_{j}}; ⋯]

where ∂_{w} is the connecting homomorphism of the long exact sequence arising from the partition Λ_{w}^{I} ∩Ẑ_{w} = Λ_{w}^{I_{>0}} ∐ Z_{w}. We have seen at the beginning of this section that res_{j} is surjective and so it is enough to show that ∂_{w} is surjective. Recall that {v_{1}, v_{2}} is an ℝ-basis of ℓ and I = ℝ_{≥0}v_{1}. Define

E_{I} = G ×^{B∩^{w}B} (ℝ_{≥0}v_{1} + 𝔲_{w}) ,
E_{I_{>0}} = G ×^{B∩^{w}B} (ℝ_{>0}v_{1} + 𝔲_{w}) , and
E_{0} = G ×^{B∩^{w}B} 𝔲_{w}.

It follows from Corollary 3.4 that E_{I} ≅ Λ_{w}^{I}, E_{I_{>0}} ≅ Λ_{w}^{I_{>0}}, and E_{0} ≅ Z_{w}, so the long exact sequence arising from the partition Λ_{w}^{I} ∩ Ẑ_{w} = Λ_{w}^{I_{>0}} ∐ Z_{w} may be identified with the long exact sequence arising from the partition E_{I} = E_{I_{>0}} ∐ E_{0}:

\xymatrix{⋯ \ar[r] & H_{i+1}(E_{I}) \ar[r] & H_{i+1}(E_{I_{>0}}) \ar[r]^{∂_{E}} & H_{i}(E_{0}) \ar[r] & ⋯}

Therefore, it is enough to show that ∂_{E} is surjective. In fact, we show that H_{•}(E_{I}) = 0 and so ∂_{E} is an isomorphism. Define E_{ℝ} = G ×^{B∩^{w}B} (ℝv_{1} + 𝔲_{w}). Then E_{ℝ} is a smooth, real vector bundle over G/B∩^{w}B and so E_{ℝ} is a smooth manifold containing E_{I} as a closed subset. We may apply [3, 2.6.1] and conclude that H_{i}(E_{I}) ≅ H^{4n+1−i}(E_{ℝ}, E_{ℝ} \ E_{I}).

Consider the cohomology long exact sequence of the pair (E_{ℝ}, E_{ℝ} \ E_{I}). Since E_{ℝ} is a vector bundle over G/B ∩^{w}B, it is homotopy equivalent to G/B ∩^{w}B. Similarly, E_{ℝ} \E_{I} ≅ G×^{B∩^{w}B} (ℝ_{<0}v_{1} + 𝔲_{w}) and so is also homotopy equivalent to G/B ∩^{w}B. Therefore, H^{i}(E_{ℝ}) ≅ H^{i}(E_{ℝ} \E_{I}) and it follows that the relative cohomology group H^{i}(E_{ℝ}, E_{ℝ}\E_{I}) is trivial for every i. Therefore, H_{•}(E_{I}) = 0, as claimed. This completes the proof of the lemma. □

Corollary 3.10. The specialization map lim_{1} : H_{i+2}(Λ_{1}^{ℓ^{∗}}) ⟶ H_{i}(Z_{1}) is surjective for 0 ≤ i ≤ 4n.

Proof. This follows from Lemma 3.9, because Z_{1} = Z_{w_{1}} and so res_{1} is the identity map. □

The next lemma is true for any specialization map.

Lemma 3.11. The restriction map res: H_{i+2}(Λ_{w}^{ℓ^{∗}}) ⟶ H_{i+2}(Λ_{w}^{P}) is surjective for every w in W and every i ≥ 0.

Proof. There are homeomorphisms Λ_{w}^{ℓ^{∗}} ≅ G/T × ℓ^{∗} and Λ_{w}^{P} ≅ G/T × P. By definition, P is an open subset of ℓ^{∗} and so there is a restriction map res: H_{2}(ℓ^{∗}) → H_{2}(P). This map is a non-zero linear transformation between one-dimensional ℚ-vector spaces so it is an isomorphism. Using the Künneth formula we get a commutative square where the horizontal maps are isomorphisms and the right-hand vertical map is surjective:

\xymatrix{H_{i+2}(Λ_{w}^{ℓ^{∗}}) \ar[r]^{≅} \ar[d]_{res} & H_{i}(G/T) ⊗ H_{2}(ℓ^{∗}) + H_{i+1}(G/T) ⊗ H_{1}(ℓ^{∗}) \ar[d]^{id⊗res +0} \\ H_{i+2}(Λ_{w}^{P}) \ar[r]_{≅} & H_{i}(G/T) ⊗ H_{2}(P)}

It follows that res: H_{i+2}(Λ_{w}^{ℓ^{∗}}) → H_{i+2}(Λ_{w}^{P}) is surjective. □

Proof of Lemma 3.2. Fix i with 0 ≤ i ≤ 4n. We show that the image of the convolution map ∗ : H_{i}(Z_{1}) ⊗ H_{4n}(Z_{j}) ⟶ H_{i}(Z) is precisely H_{i}(Z_{j}) for 1 ≤ j ≤ N using induction on j. For j = 1, H_{4n}(Z_{1}) is one-dimensional with basis {λ_{1}}. It follows from Theorem 2.2(c) that λ_{1} is the identity in H_{•}(Z) and so clearly the image of the convolution map H_{i}(Z_{1}) ⊗ H_{4n}(Z_{1}) ⟶ H_{i}(Z) is precisely H_{i}(Z_{1}). Assume that j > 1 and set w = w_{j}. We will complete the proof using the commutative diagram with exact rows

(3.12) \xymatrix{H_{i} ⊗ H_{4n}(Z_{j−1}) \ar@{^{(}->}[r]^{id⊗(r_{j})_{∗}} \ar[d]_{∗} & H_{i} ⊗ H_{4n}(Z_{j}) \ar[d]_{∗} \ar@{->>}[r]^{id⊗res_{j}} & H_{i} ⊗ H_{4n}(Z_{w}) \ar[d]_{∗} \\ H_{i}(Z_{j−1}) \ar@{^{(}->}[r]^{(r_{j})_{∗}} & H_{i}(Z_{j}) \ar@{->>}[r]^{_{res_{j}}} & H_{i}(Z_{w})}

Documenta Mathematica 14 (2009) 339–357

and the Five Lemma, where in the first line H_{i} means H_{i}(Z_{1}). We saw in (3.1) that the bottom row is exact and it follows that the top row is also exact. By induction, the convolution product in H_{•}(Z) determines a surjective map ∗ : H_{i}(Z_{1}) ⊗ H_{4n}(Z_{j−1}) ⟶ H_{i}(Z_{j−1}). To conclude from the Five Lemma that the middle vertical map is a surjection, it remains to define the other vertical maps so that the diagram commutes and to show that the right-hand vertical map is a surjection. First we show that the image of the map H_{i}(Z_{1}) ⊗ H_{4n}(Z_{j}) ⟶ H_{i}(Z_{j}) determined by the convolution product in H_{•}(Z) is contained in H_{i}(Z_{j}). It then follows that the middle vertical map in (3.12) is defined and so by exactness there is an induced map from H_{i}(Z_{1}) ⊗ H_{4n}(Z_{w}) to H_{i}(Z_{w}) so that the diagram (3.12) commutes. Second we show that the right-hand vertical map is a surjection. By Lemma 3.5, Λ_{1}^{ℓ^{∗}} is the graph of the identity map of 𝔤̃^{ℓ^{∗}}, and Λ_{w}^{ℓ^{∗}} is the graph of w̃|𝔤̃^{ℓ^{∗}}. Therefore, Λ_{1}^{ℓ^{∗}} ∘ Λ_{w}^{ℓ^{∗}} = Λ_{w}^{ℓ^{∗}} and there is a convolution product

H_{i+2}(Λ_{1}^{ℓ^{∗}}) ⊗ H_{4n+2}(Λ_{w}^{ℓ^{∗}}) ⟶^{∗} H_{i+2}(Λ_{w}^{ℓ^{∗}}).

Suppose a is in H_{i}(Z_{1}). Then by Corollary 3.10, a = lim_{1}(a_{1}) for some a_{1} in H_{i+2}(Λ_{1}^{ℓ^{∗}}). It is shown in [3, Proposition 2.7.23] that specialization commutes with convolution, so lim(a_{1} ∗ [Λ_{w}^{ℓ^{∗}}]) = lim(a_{1}) ∗ lim([Λ_{w}^{ℓ^{∗}}]) = a ∗ λ_{w}. Also, a_{1} ∗[Λ_{w}^{ℓ^{∗}}] is in H_{i+2}(Λ_{w}^{ℓ^{∗}}) and lim = r_{∗} ∘lim_{j} and so a∗λ_{w} = r_{∗} ∘lim_{j}(a_{1} ∗[Λ_{w}^{ℓ^{∗}}]) is in H_{i}(Z_{j}). By induction, if k < j, then a ∗ λ_{w_{k}} is in H_{i}(Z_{k}) and so a ∗ λ_{w_{k}} is in H_{i}(Z_{k}). Since the set { λ_{w_{k}} | 1 ≤ k ≤ j } is a basis of H_{4n}(Z_{j}), it follows that a ∗ H_{4n}(Z_{j}) ⊆ H_{i}(Z_{j}). Therefore, the image of the convolution map H_{i}(Z_{1}) ⊗ H_{4n}(Z_{j}) ⟶ H_{i}(Z) is contained in H_{i}(Z_{j}). To complete the proof of Lemma 3.2, we need to show that the induced map from H_{i}(Z_{1}) ⊗ H_{4n}(Z_{w}) to H_{i}(Z_{w}) is surjective. Consider the following diagram:

\xymatrix{H_{i+2}(Λ_{1}^{ℓ^{∗}}) ⊗ H_{4n+2}(Λ_{w}^{ℓ^{∗}}) \ar[r]^{∗} \ar[d]_{lim_{1} ⊗ lim_{j}} & H_{i+2}(Λ_{w}^{ℓ^{∗}}) \ar[d]^{_{lim_{j}}} \\ H_{i}(Z_{1}) ⊗ H_{4n}(Z_{j}) \ar[r]^{∗} \ar[d]_{id⊗res_{j}} & H_{i}(Z_{j}) \ar[d]^{_{res_{j}}} \\ H_{i}(Z_{1}) ⊗ H_{4n}(Z_{w}) \ar[r]^{∗} & H_{i}(Z_{w})}

We have seen that the bottom square is commutative. It follows from the fact that specialization commutes with convolution that the top square is also commutative. It is shown in Proposition A.2 that the convolution product H_{i+2}(Λ_{1}^{ℓ^{∗}})⊗H_{4n+2}(Λ_{w}^{ℓ^{∗}}) → H_{i+2}(Λ_{w}^{ℓ^{∗}}) is an injection. Since H_{i+2}(Λ_{1}^{ℓ^{∗}}) is finitedimensional and H_{4n+2}(Λ_{w}^{ℓ^{∗}}) is one-dimensional, it follows that this convolution mapping is an isomorphism. Also, we saw in Lemma 3.9 that res_{j} ∘ lim_{j} is surjective. Therefore, the composition res_{j} ∘ lim_{j} ∘ ∗ is surjective and it follows that the bottom convolution map H_{i}(Z_{1}) ⊗ H_{4n}(Z_{w}) → H_{i}(Z_{w}) is also surjective. This completes the proof of Lemma 3.2.

4. Smash Product Structure

In this section we prove Theorem 2.4. We need to show that λ_{w}∗H_{i}(Z_{1})∗λ_{w^{−1}} = H_{i}(Z_{1}) and that β: Coinv_{•}(W) ⟶^{≅} H_{4n−•}(Z_{1}) is an isomorphism of W - algebras. Suppose that ℓ is a one-dimensional subspace of 𝔱 so that ℓ^{∗} = ℓ\{0} = ℓ∩𝔱_{reg}. Recall that for S ⊆ 𝔱, 𝔤̃^{S} = ν^{−1}(S). By Lemma 3.5, if w is in W, then Λ_{w}^{ℓ^{∗}} is the graph of the restriction of w̃ to 𝔤̃^{ℓ^{∗}}. It follows that there is a convolution product

H_{4n+2}(Λ_{w}^{ℓ^{∗}}) ⊗ H_{i+2}(Λ_{1}^{w^{−1}(ℓ^{∗})}) ⊗ H_{4n+2}(Λ_{w}^{w^{−1}(ℓ^{∗})}) ⟶^{∗} H_{i+2}(Λ_{1}^{ℓ^{∗}}).

Because specialization commutes with convolution, the diagram

\xymatrix{H_{4n+2}(Λ_{w}^{ℓ^{∗}}) ⊗ H_{i+2}(Λ_{1}^{w^{−1}(ℓ^{∗})}) ⊗ H_{4n+2}(Λ_{w}^{w^{−1}(ℓ^{∗})}) \ar[r]^{∗} \ar[d]_{lim ⊗ lim ⊗ lim} & H_{i+2}(Λ_{1}^{ℓ^{∗}}) \ar[d]^{lim} \\ H_{4n}(Z) ⊗ H_{i}(Z_{1}) ⊗ H_{4n}(Z) \ar[r]_{∗} & H_{i}(Z)}

commutes. We saw in Corollary 3.10 that lim_{1} : H_{i+2}(Λ_{1}^{ℓ^{∗}}) → H_{i}(Z_{1}) is surjective. Thus, if c is in H_{i}(Z_{1}), then c = lim(c_{1}) for some c_{1} in H_{i+2}(Λ_{w_{1}}^{w^{−1}(ℓ^{∗})}). Therefore,

λ_{w}∗c∗λ_{w^{−1}} =lim([Λ_{w}^{ℓ^{∗}}])∗lim(c_{1})∗lim([Λ_{w^{−1}}^{w^{−1}(ℓ^{∗})}]) = lim ([Λ_{w}^{ℓ^{∗}}] ∗ c_{1} ∗ [Λ_{w^{−1}}^{w^{−1}(ℓ^{∗})}]) .

Since Λ_{w}^{ℓ^{∗}} and Λ_{w^{−1}}^{w^{−1}(ℓ^{∗})} are the graphs of w̃ and w^{−1} respectively, and Λ_{1}^{w^{−1}(ℓ^{∗})}

is the graph of the identity function, it follows that [Λ_{w}^{ℓ^{∗}}] ∗ c_{1} ∗ [Λ_{w^{−1}}^{w^{−1}(ℓ^{∗})}] is in

H_{i+2}(Λ_{1}^{w^{−1}(ℓ^{∗})}) and so by (3.8), λ_{w} ∗ c ∗ λ_{w^{−1}} is in H_{i}(Z_{1}). This shows that λ_{w} ∗ H_{i}(Z_{1}) ∗ λ_{w^{−1}} = H_{i}(Z_{1}) for all i. To complete the proof of Theorem 2.4 we need to show that if w is in W and f is in Coinv_{i}(W), then β(w · f) = λ_{w} ∗ β(f) ∗ λ_{w^{−1}} where w · f denotes the natural action of w on f. To do this, we need some preliminary results. First, since Λ_{1}^{ℓ^{∗}} is the diagonal in 𝔤̃^{ℓ^{∗}} × 𝔤̃^{ℓ^{∗}}, it is obvious that

δ ∘ w^{−1} = (w^{−1} × w^{−1}) ∘ δ: 𝔤̃^{w^{−1}(ℓ^{∗})} ⟶ Λ_{1}^{ℓ^{∗}}.

Therefore,

(4.1)  δ_{∗} ∘ w_{∗}^{−1} = (w^{−1} × w^{−1})_{∗} ∘ δ_{∗}: H_{i}(𝔤̃^{w^{−1}(ℓ^{∗})}) ⟶ H_{i}(Λ_{1}^{ℓ^{∗}})

for all i. (The first δ in (4.1) is the diagonal embedding 𝔤̃^{ℓ^{∗}} ≅ Λ_{1}^{ℓ^{∗}} and the second δ is the diagonal embedding 𝔤̃^{w^{−1}(ℓ^{∗})} ≅ Λ_{1}^{w^{−1}(ℓ^{∗})}.) Next, with ℓ ⊆ 𝔱 as above, 𝔤̃^{ℓ} = 𝔤̃^{ℓ^{∗}} ∐ ν^{−1}(0) = 𝔤̃^{ℓ^{∗}} ∐ Ñ and the restriction of ν: 𝔤̃^{ℓ} → ℓ to 𝔤̃^{ℓ^{∗}} is a locally trivial fibration. Therefore, there is a specialization map lim_{0} : H_{i+2}(𝔤̃^{ℓ^{∗}}) → H_{i}(Ñ). Since δ_{∗}: H_{i+2}(𝔤̃^{ℓ^{∗}}) → H_{i+2}(Λ_{1}^{ℓ^{∗}}) and δ_{∗}: H_{i}(Z) → H_{i}(Z_{1}) are isomorphisms, the next lemma is obvious.

Lemma 4.2. Suppose that ℓ is a one-dimensional subspace of 𝔱 so that ℓ^{∗} = ℓ \ {0} ⊆ 𝔱_{reg}. Then the diagram

\xymatrix{H_{i+2}(𝔤̃^{ℓ^{∗}}) \ar[r]^{δ_{∗}} \ar[d]_{_{lim_{0}}} & H_{i+2}(Λ_{1}^{ℓ^{∗}}) \ar[d]^{_{lim_{1}}} \\ H_{i}(Z) \ar[r]_{δ_{∗}} & H_{i}(Z_{1})}

commutes.

Finally, Ñ ×_{N} N = Ñ and so Z ∘ Ñ = (Ñ ×_{N} Ñ) ∘ (Ñ ×_{N} N) = Ñ ×_{N} N. Thus, there is a convolution action, H_{4n}(Z) ⊗ H_{i}(Ñ) ⟶ H_{i}(Ñ), of H_{4n}(Z) on H_{i}(Ñ). Suppose that w is in W and z is in H^{i}(B). Then π^{∗} ∘pd(z) is in H_{4n−i}(Ñ) and so λ_{w} ∗ (π^{∗} ∘ pd(z)) is in H_{4n−i}(Ñ). It is shown in [3, Proposition 7.3.31] that for y in H_{•}(B), λ_{w} ∗ π^{∗}(y) = ϵ_{w}π^{∗}(w · y) where ϵ_{w} is the sign of w and w · y denotes the action of W on H_{•}(B) coming from the action of W on G/T and the homotopy equivalence G/T ≃ B. It is also shown in [3, Proposition 7.3.31] that pd(w · z) = ϵ_{w}w · pd(z). Therefore,

λ_{w} ∗ (π^{∗} ∘ pd(z)) = ϵ_{w}π^{∗}(w · pd(z)) = ϵ_{w}ϵ_{w}π^{∗} ∘ pd(w · z) = π^{∗} ∘ pd(w · z).

This proves the next lemma.

Lemma 4.3. If w is in W and z is in H_{i}(B), then

λ_{w} ∗ (π^{∗} ∘ pd(z)) = π^{∗} ∘ pd(w · z).

Proof of Theorem 2.4. Fix w in W and f in Coinv_{i}(W). We need to show that λ_{w} ∗ β(f) ∗ λ_{w^{−1}} = β(w · f). Set C = λ_{w} ∗ β(f) ∗ λ_{w^{−1}}. Using the fact that β = δ_{∗} ∘ π^{∗} ∘ pd ∘ bi we compute

C = lim_{1} ([Λ_{w}^{ℓ^{∗}}] ∗ lim_{1}^{−1}(β(f)) ∗ [Λ_{w^{−1}}^{w^{−1}(ℓ^{∗})}])  [3, 2.7.23]
= lim_{1} ∘(w^{−1} × w^{−1})_{∗} ∘ lim_{1}^{−1} ∘β(f)  Proposition A.3
= lim_{1} ∘δ_{∗} ∘ w_{∗}^{−1} ∘ δ_{∗}^{−1} ∘ lim_{1}^{−1} ∘β(f)  (4.1)
= δ_{∗} ∘ lim_{0} ∘w_{∗}^{−1} ∘ δ_{∗}^{−1} ∘ lim_{1}^{−1} ∘δ_{∗} ∘ δ_{∗}^{−1} ∘ β(f)  Lemma 4.2
= δ_{∗} ∘ lim_{0} ∘w_{∗}^{−1} ∘ lim_{0}^{−1} ∘δ_{∗}^{−1} ∘ β(f)  Lemma 4.2

= δ_{∗} ∘ lim_{0} ∘w_{∗}^{−1} ∘ lim_{0}^{−1} ∘π^{∗} ∘ pd ∘ bi(f)

= δ_{∗} ∘ lim_{0} ((lim_{0}^{−1} ∘π^{∗} ∘ pd ∘ bi(f)) ∗ [Λ_{w^{−1}}^{w^{−1}(ℓ^{∗})}])  [3, 2.7.11]
= δ_{∗} ((π^{∗} ∘ pd ∘ bi(f)) ∗ λ_{w^{−1}})  [3, 2.7.23]
= δ_{∗} (λ_{w} ∗ (π^{∗} ∘ pd ∘ bi(f)))  Lemma A.1 and [3, 3.6.11]
= δ_{∗} ∘ π^{∗} ∘ pd(w · bi(f))  Lemma 4.3
= δ_{∗} ∘ π^{∗} ∘ pd ∘ bi(w · f)  bi is W -equivariant

= β(w · f).

This completes the proof of Theorem 2.4.

Appendix A. Convolution and Graphs

In this appendix we prove some general properties of convolution and graphs. Suppose M_{1}, M_{2}, and M_{3} are smooth varieties, dim M_{2} = d, and that Z_{1,2} ⊆ M_{1}×M_{2} and Z_{2,3} ⊆ M_{2}×M_{3} are two closed subvarieties so that the convolution product,

H_{i}(Z_{1,2}) ⊗ H_{j}(Z_{2,3}) ⟶^{∗} H_{i+j−2d}(Z_{1,2} ∘ Z_{2,3}),

in [3, §2.7.5] is defined. For 1 ≤ i, j ≤ 3, let τ_{i,j}: M_{i} × M_{j} → M_{j} × M_{i} be the map that switches the factors. Define Z_{2,1} = τ_{1,2}(Z_{1,2}) ⊆ M_{2} × M_{1} and Z_{3,2} = τ_{2,3}(Z_{2,3}) ⊆ M_{3} × M_{2}. Then the convolution product

H_{j}(Z_{3,2}) ⊗ H_{i}(Z_{2,1}) ⟶^{∗′} H_{i+j−2d}(Z_{3,2} ∘ Z_{2,1})

is defined. We omit the easy proof of the following lemma.

Lemma A.1. If c is in H_{i}(Z_{1,2}) and d is in H_{j}(Z_{2,3}), then (τ_{1,3})_{∗}(c ∗ d) = (τ_{2,3})_{∗}(d) ∗′ (τ_{1,2})_{∗}(c).

Now suppose X is an irreducible, smooth, m-dimensional variety, Y is a smooth variety, and f: X → Y is a morphism. Then if Γ_{X} and Γ_{f} denote the graphs of id_{X} and f respectively, using the notation in [3, §2.7], we have Γ_{X} ∘ Γ_{f} = Γ_{f} and there is a convolution product ∗: H_{i}(Γ_{X}) ⊗ H_{2m}(Γ_{f}) ⟶ H_{i}(Γ_{f}).

Proposition A.2. The convolution product ∗: H_{i}(Γ_{X})⊗H_{2m}(Γ_{f}) ⟶ H_{i}(Γ_{f}) is an injection.

Proof. For i, j = 1, 2, 3, let p_{i,j} denote the projection of X × X × Y on the i^{th} and j^{th} factors. Then the restriction of p_{1,3} to (Γ_{X} × Y) ∩ (X × Γ_{f}) is the map that sends (x, x, f(x)) to (x, f(x)). Thus, the restriction of p_{1,3} to (Γ_{X} ×Y)∩(X ×Γ_{f}) is an isomorphism onto Γ_{f} and hence is proper. Therefore, the convolution product in homology is defined. Since X is irreducible, so is Γ_{f} and so H_{2m}(Γ_{f}) is one-dimensional with basis [Γ_{f}]. Suppose that c is in H_{i}(Γ_{X}). We need to show that if c ∗ [Γ_{f}] = 0, then c = 0. Fix c in H_{i}(Γ_{X}). Notice that the restriction of p_{1,3} to (Γ_{X} × Y) ∩ (X × Γ_{f}) is the same as the restriction of p_{2,3} to (Γ_{X} × Y) ∩ (X × Γ_{f}). Thus, using the projection formula, we have

c ∗ [Γ_{f}] = (p_{1,3})_{∗} (p_{1,2}^{∗}c ∩ p_{2,3}^{∗}[Γ_{f}])
= (p_{2,3})_{∗} (p_{1,2}^{∗}c ∩ p_{2,3}^{∗}[Γ_{f}])
= ((p_{2,3})_{∗}p_{1,2}^{∗}c) ∩ [Γ_{f}],

where the intersection product in the last line is from the cartesian square:

\xymatrix{Γ_{f} \ar[d] \ar[r]^{=} & Γ_{f} \ar[d] \\ X × Y \ar[r]_{=} & X × Y}

Let p: X × Y → X and q: Γ_{X} → X be the first and second projections, respectively. Then the square

\xymatrix{Γ_{X} × Y \ar[d]_{p_{1,2}} \ar[r]^{p_{2,3}} & X × Y \ar[d]^{p} \\ Γ_{X} \ar[r]_{q} & X}

is cartesian. Thus,

p_{∗} (c ∗ [Γ_{f}]) = p_{∗} (((p_{2,3})_{∗}p_{1,2}^{∗}c) ∩ [Γ_{f}])
= p_{∗} ((p^{∗}q_{∗}c) ∩ [Γ_{f}])
= q_{∗}c ∩ (p|_{Γ_{f}})_{∗}[Γ_{f}]
= q_{∗}c ∩ [X]
= q_{∗}c,

where we have used the projection formula and the fact that (p|_{Γ_{f}})_{∗}[Γ_{f}] = [X]. Now if c∗[Γ_{f}] = 0, then q_{∗}c = 0 and so c = 0, because q is an isomorphism. □

Let Γ_{Y} denote the graph of the identity functions id_{Y}. Then the following compositions and convolution products in Borel-Moore homology are defined:

• Γ_{f} ∘ Γ_{X} = Γ_{f} and so there is a convolution product

H_{i}(Γ_{f}) ⊗ H_{j}(Γ_{X}) ⟶ H_{i+j−m}(Γ_{f}).

• Γ_{Y} ∘ Γ_{f^{−1}} = Γ_{f^{−1}} and so there is a convolution product

H_{i}(Γ_{X}) ⊗ H_{j}(Γ_{f^{−1}}) ⟶ H_{i+j−m}(Γ_{f^{−1}}).

• Γ_{f} ∘ Γ_{f^{−1}} = Γ_{X} and so there is a convolution product

H_{i}(Γ_{f}) ⊗ H_{j}(Γ_{f^{−1}}) ⟶ H_{i+j−m}(Γ_{X}).

Thus, if c is in H_{i}(Γ_{Y}), then [Γ_{f}] ∗ c ∗ [Γ_{f^{−1}}] is in H_{i}(Γ_{X}). Note that the map f^{−1} × f^{−1} : Γ_{Y} → Γ_{X} is an isomorphism, so in particular it is proper.

Proposition A.3. If c is in H_{i}(Γ_{Y}), then [Γ_{f}] ∗ c ∗ [Γ_{f^{−1}}] = (f^{−1} × f^{−1})_{∗}(c).

Proof. We compute ([Γ_{f}] ∗ c) ∗ [Γ_{f^{−1}}], starting with [Γ_{f}] ∗ c. For 1 ≤ i, j ≤ 3 let q_{i,j} be the projection of the subset

Γ_{f} × Y ∩ X × Γ_{Y} = { (x, f(x), f(x)) | x ∈ X }

of X × Y × Y onto the i, j-factors. Then q_{1,3} = q_{1,2}. Therefore, using the projection formula, we see that

[Γ_{f}] ∗ c = (q_{1,3})_{∗} (q_{1,2}^{∗}[Γ_{f}] ∩ q_{2,3}^{∗}c)
= (q_{1,2})_{∗} (q_{1,2}^{∗}[Γ_{f}] ∩ q_{2,3}^{∗}c)
= [Γ_{f}] ∩ (q_{1,2})_{∗}q_{2,3}^{∗}c
= (q_{1,2})_{∗}q_{2,3}^{∗}c.

Next, for 1 ≤ i, j ≤ 3 let p_{i,j} be the projection of the subset

Γ_{f} × X ∩ X × Γ_{f^{−1}} = { (x, f(x), x) | x ∈ X }

of X × Y × X onto the i, j-factors. Then p_{1,3} = (f^{−1} × id) ∘ p_{2,3}. Therefore, using the fact that [Γ_{f}] ∗ c = (q_{1,2})_{∗}q_{2,3}^{∗}c and the projection formula, we have

([Γ_{f}] ∗ c) ∗ [Γ_{f^{−1}}] = (p_{1,3})_{∗} (p_{1,2}^{∗}((q_{1,2})_{∗}q_{2,3}^{∗}c) ∩ p_{2,3}^{∗}[Γ_{f^{−1}}])
= (f^{−1} × id)_{∗}(p_{2,3})_{∗} (p_{1,2}^{∗}((q_{1,2})_{∗}q_{2,3}^{∗}c) ∩ p_{2,3}^{∗}[Γ_{f^{−1}}])
= (f^{−1} × id)_{∗} ((p_{2,3})_{∗}p_{1,2}^{∗}(q_{1,2})_{∗}q_{2,3}^{∗}c ∩ [Γ_{f^{−1}}])
= (f^{−1} × id)_{∗}(p_{2,3})_{∗}p_{1,2}^{∗}(q_{1,2})_{∗}q_{2,3}^{∗}c.

The commutative square

\xymatrix{Γ_{f} × X ∩ X × Γ_{f^{−1}} \ar[r]^{id×id×f} \ar[d]_{id} & Γ_{f} × Y ∩ X × Γ_{Y} \ar[d]^{q_{1,2}} \\ Γ_{f} × X ∩ X × Γ_{f^{−1}} \ar[r]_{p_{1,2}} & Γ_{f}}

is cartesian, so p_{1,2}^{∗}(q_{1,2})_{∗} = (id × id × f)^{∗}. Also, the commutative square

\xymatrix{Γ_{f} × X ∩ X × Γ_{f^{−1}} \ar[r]^{q_{2,3}∘(id×id×f)} \ar[d]_{(f^{−1}×id)∘p_{2,3}} & Γ_{Y} \ar[d]^{f^{−1}×f^{−1}} \\ Γ_{X} \ar[r]_{id} & Γ_{X}}

is cartesian, so (f^{−1} × id)_{∗}(p_{2,3})_{∗}(id × id × f)^{∗}q_{2,3}^{∗} = (f^{−1} × f^{−1})_{∗}. Therefore,

([Γ_{f}] ∗ c) ∗ [Γ_{f^{−1}}] = (f^{−1} × id)_{∗}(p_{2,3})_{∗}p_{1,2}^{∗}(q_{1,2})_{∗}q_{2,3}^{∗}c
= (f^{−1} × id)_{∗}(p_{2,3})_{∗}(id × id × f)^{∗}q_{2,3}^{∗}c
= (f^{−1} × f^{−1})_{∗}c.

This completes the proof of the proposition. □

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J. Matthew Douglass

Department of Mathematics University of North Texas Denton TX USA 76203 douglass@unt.edu Gerhard Röhrle Fakultät für Mathematik Ruhr-Universität Bochum D-44780 Bochum Germany gerhard.roehrle@rub.de 358

Documenta Mathematica 14 (2009)