ALVIS-CURTIS DUALITY AS AN EQUIVALENCE OF DERIVED CATEGORIES

MARC CABANES AND JEREMY RICKARD

Abstract. Let G be the group of rational points of a connected reductive group defined over a finite field of characteristic p, and let O be any commutative ring in which p is invertible. We prove that the duality operation of Alvis, Curtis, Kawanaka and Lusztig on the characters of G is induced by a self-equivalence of the derived category of OGmodules, which was conjectured by Broué. We also prove that this equivalence of derived categories is compatible with Harish-Chandra induction and truncation.

1. Introduction

Let G be a finite reductive group of characteristic p.

Alvis-Curtis duality, introduced independently by Alvis, Curtis, Kawanaka, and Lusztig around 1980, is an isometric involution

D_{G} : K_{0}(ℂG) ⟶ K_{0}(ℂG)

of the group of generalized characters of G (see for instance [7], Chapter 8).

In the paper [2] in which he introduced the notion of perfect isometries and his conjectures on derived equivalence of blocks, Broué showed that in all characteristics other than p, D_{G} gives a perfect isometry. This would be a consequence of D_{G} being induced by a self-equivalence of the derived category D^{b}(mod(OG)) for a complete discrete valuation ring O with residue field of characteristic different from p, and Broué conjectured that this was the case.

In this paper, we shall show that this conjecture of Broué is correct, and in fact the equivalence of derived categories is defined over any commutative coefficient ring O in which p is invertible. Our proof uses a complex, or more precisely a coefficient system, of ℤ[1/p]G-bimodules which gives a version over ℤ[1/p] of the complex of ℂG-modules introduced by Curtis in [3]. As in the approach by Deligne and Lusztig in [5], the main step of our proof consists of determining the image of a module which is the Harish-Chandra induction of a cuspidal module (see Theorem 3.1). For that purpose, an essential ingredient is the fact that the Harish-Chandra induction and truncation functors are independent of the choice of a parabolic subgroup ([8], [6]) and the related statements on the bimodule inducing these functors (see Lemma 3.4).

We also prove the structural analogue over ℤ[1/p] of the character-theoretic theorem of Curtis stating that Alvis-Curtis duality commutes with Harish-Chandra induction (see for instance [7], Theorem 8.11).

1991 Mathematics Subject Classification. Primary 20C33; Secondary 20J05, 18E30.

We shall use standard notation involving derived categories and chain homotopy categories for which Weibel’s book [11] is a good reference. For background concerning equivalences of derived categories in representation theory, see the lecture notes of König and Zimmermann [9].

2. Preliminaries

2.1. Notation. Let us start by fixing some notation. Throughout this paper, G will be a connected reductive group defined over a finite field 𝔽_{q}, where q is a power of the prime p, and F will be the Frobenius endomorphism of G corresponding to the given 𝔽_{q}-structure. By G we shall denote the finite group G^{F} of 𝔽_{q}-rational points of G.

We shall fix a quasi-split rational maximal torus T of G and a rational Borel subgroup B containing T. We denote W = W (G, T)^{F} and let S ⊆ W be the set of generators associated with the choice of B. Recall that S is in bijection with the set of orbits of the Frobenius map F on the set of simple reflections of W (G, T), and that the conjugacy classes of rational parabolic subgroups of G are parametrized by subsets of S; given I ⊆ S, there is a unique member P_{I} of the corresponding class of parabolics that contains B. Let U_{I} be the unipotent radical of P_{I} and let L_{I} be the Levi complement that contains T. Thus P_{∅} = B, and P_{S} = G.

To refer to the finite groups of 𝔽_{q}-rational points of these various subgroups of G, we shall use the same notation, but in ordinary rather than in bold type. Thus B = B^{F}, L_{I} = L_{I}^{F}, and so on. We shall be working mainly with the split BN-pair of G defined by B, N = N_{G}(T)^{F}, and S ([4], Section 69).

We shall be considering representations of these finite groups over a commutative coefficient ring which contains a multiplicative inverse to p. Throughout the paper, O will usually denote such a ring. Often it will be sufficient to deal with the coefficient ring ℤ[1/p], which we shall denote by R.

For A any p-subgroup of G, denote by [A] the idempotent |A|^{−1} ∑_{a∈A} a ∈ RG.

In Sections 3 and 4, we shall restrict to the case where O is a field with characteristic different from p, which we shall denote by k.

We shall need to consider both left and right modules for group algebras, but we shall adopt the convention that when we do not specify which side of a module M the action is on, we intend M to be a left module. By Mod(OH) we mean the abelian category of left OH-modules for a finite group H. In Section 7.1 we shall need to consider the abelian category mod(OH) of finitely generated left OH-modules for a commutative Noetherian ring O.

2.2. Harish-Chandra induction functors and associated bimodules. Let P be a rational parabolic subgroup of G with unipotent radical U and rational Levi complement L, and let P, U and L be the corresponding subgroups of the finite group G. One denotes by

R_{L}^{G} : Mod(OL) ⟶ Mod(OG)

the functor R_{L}^{G} = Ind_{P}^{G} Infl_{L}^{P} where Infl_{L}^{P} is the inflation functor making OL-modules into U-trivial OP -modules. This functor is called Harish-Chandra induction. Since p is invertible in O, one may identify R_{L}^{G} with the tensor product functor OG[U]⊗_{OL}? ([8], Section 1), where OG[U] is considered as an OG-OL-bimodule. In the same way, the Harish-Chandra truncation functor

∗R_{L}^{G} : Mod(OL) ⟶ Mod(OG)

is identified with the tensor product functor [U].OG⊗_{OG}?. Recall that, if M is an OGmodule, then ^{∗}R_{L}^{G}M can be seen as the OL-module M^{U} of fixed points of M under the action of U.

The two functors are induced by bimodules that are projective as right (and as left) modules, so they are exact. They are moreover both left and right adjoint to each other.

The notation R_{L}^{G} and ^{∗}R_{L}^{G} originates from the case O = ℂ where the functors do not depend on the choice of P (see for instance [7], Proposition 6.1). In the general case of a ring O where p is invertible, Howlett-Lehrer and Dipper-Du proved independently that the same result holds: different parabolic subgroups with the same Levi complement L give isomorphic functors ([8], Theorem 1.1 or [6], Theorem 5.2). We shall be mostly interested in the case L = L_{I}, in which case we shall, to be definite, use the parabolic subgroup P_{I} to define the functors.

2.3. Alvis-Curtis duality and the associated coefficient system. We define our functor X_{O}⊗_{OG}? in terms of homology of local coefficient systems. For the basic definitions and terminology, see, for example, Benson’s book [1], Chapter 7.

Let Δ/G be the set of parabolic subgroups of G containing B, partially ordered by the opposite relation to inclusion of subgroups (this coincides with the quotient of the building by the conjugation action of G, but this interpretation is not essential for our purpose). Since the elements of Δ/G are in bijection with subsets of S by I ↦ P_{I}, it can be considered as a simplicial complex where the i-dimensional simplices are the parabolics P_{I} such that |S \ I| = i + 1. We shall regard G itself as a (−1)-dimensional simplex.

Recall that R = ℤ[1/p].

Let D be the coefficient system on Δ/G which associates with P_{I} the RG-RG-bimodule D_{I} = RG[U_{I}] ⊗_{RP_{I}} [U_{I}].RG. If one has an inclusion in Δ/G (i.e., a reverse inclusion of parabolic subgroups containing B), then one defines a restriction map as follows. For I ⊆ J ⊆ S, define

ϕ_{IJ} : D_{I} → D_{J}

by

x ⊗_{RP_{I}} y ↦ x ⊗_{RP_{J}} y

for x ∈ RG[U_{I}], y ∈ [U_{I}].RG.

It is easily checked that this is a map of RG-RG-bimodules and that ϕ_{JK}ϕ_{IJ} = ϕ_{IK} whenever I ⊆ J ⊆ K ⊆ S.

We now define a complex X_{R} as C̃_{∗}(Δ/G, D), the reduced chain complex with coefficients in D, with degrees renumbered so as to produce a cochain complex with degrees varying from 0 to |S|. If O is any ring where p is invertible, we set X_{O} = X_{R} ⊗_{R} O. Then X_{O} is a cochain complex of OG-OG-bimodules

⋯ → 0 → X_{O}^{0} → X_{O}^{1} → ⋯ → X_{O}^{|S|} → 0 → ⋯

with X_{O}^{i} = ⨁_{I} OG[U_{I}] ⊗_{OP_{I}} [U_{I}].OG where I ranges over the subsets of S of cardinality i. In particular, X_{O}^{|S|} = OG.

Note first that the complex above is defined in an intrinsic way from the subgroup B, since the P_{I}’s are just the subgroups of G containing B, |I| is determined by the height of P_{I} in this poset, and U_{I} is just the maximal normal p-subgroup of P_{I}. Since the outcome would be the same with a G-conjugate of B, and since the conjugates of B may be characterized as the normalizers of Sylow p-subgroups of G, we see that X_{R} is defined in an intrinsic way from the abstract structure of G.

The functor

X_{O}⊗_{OG}? : Mod(OG) ⟶ C^{b}(Mod(OG))

extends to an endofunctor of C^{b}(Mod(OG)), also given by taking the tensor product with X_{O}. This functor preserves contractibility of complexes, so it induces an endofunctor of the chain homotopy category K^{b}(Mod(OG)). Finally, since the terms of X_{O} are projective as right OG-modules, the functor X_{O}⊗_{OG}? is exact, and so it induces an endofunctor of the derived category D^{b}(Mod(OG)). We shall prove in Section 5 that this functor

X_{O}⊗_{OG}? : D^{b}(Mod(OG)) ⟶ D^{b}(Mod(OG))

is a self-equivalence.

The following property is clear from the definition of D.

Lemma 2.1. If we consider D as a coefficient system of O[G × G]-modules in the usual way, via (g, g′)x = gxg′^{−1}, it is stable under the automorphism of G × G that interchanges the two factors. Hence X_{O} is also stable under this automorphism.

As an easy consequence of this, we get the following result, which will mean that we can understand the functor given by tensoring with the dual of X_{O} (i.e., the adjoint functor to the one described above) in terms of the original functor. For clarity, let us make a distinction between two kinds of dual: if M is a left O-free OG-module, M^{∗} will be the O-linear dual of M, regarded as a left OG-module via the Hopf algebra structure of OG; on the other hand, if A and B are O-algebras and N is an O-free A-B-bimodule, then N^{∨} will be the O-linear dual of N, regarded as a B-A-bimodule.

Corollary 2.2. If M is any OG-module, then there is an isomorphism

(X_{O}^{∨} ⊗_{OG} M)^{∗} ≅ X_{O} ⊗_{OG} M^{∗}

of complexes of OG-modules. If M is finitely generated and free as a O-module, then

X_{O}^{∨} ⊗_{OG} M ≅ (X_{O} ⊗_{OG} M^{∗})^{∗}.

Let us now relate our tensor product functor to the usual duality operation D_{G}. If M in an OG-module, the complex X_{O} ⊗_{OG} M is a cochain complex concentrated in degrees 0 to |S|, with (X_{O} ⊗_{OG}M)^{i} = ⨁_{I} OG[U_{I}]⊗_{OP_{I}} [U_{I}].OG⊗_{OG}M where the sum is over I ⊆ S of cardinality i. We clearly have OG[U_{I}]⊗_{OP_{I}} [U_{I}].OG ≅ OG[U_{I}]⊗_{OL_{I}} [U_{I}].OG as bimodules, so OG[U_{I}] ⊗_{OP_{I}} [U_{I}].OG ⊗_{OG} M ≅ R_{L_{I}}^{G}^{∗}R_{L_{I}}^{G} M. If O is a domain of characteristic zero and M is O-free and finitely generated, then the trace character of X_{O} ⊗_{OG} M equals ∑_{I⊆S}(−1)^{|I|}R_{L_{I}}^{G} ∘ ^{∗}R_{L_{I}}^{G} (χ_{M}) where χ_{M} is the trace character of M. This is the formula defining the Alvis-Curtis duality operation on characters ([7], Definition 8.8).

Finally, let us relate the functor X_{O}⊗_{OG}? to Curtis’ construction [3]. The building Δ of G is a simplicial complex whose poset of simplices is opposite to the poset of proper parabolic subgroups of G, and we shall regard G itself as corresponding to a (-1)-dimensional simplex. Given an OG-module M, a G-equivariant coefficient system F_{M} of O-modules can be constructed on the building by attaching the R_{u}(P)-fixed points M^{R_{u}(P)} to the simplex corresponding to a parabolic subgroup P ≤ G with unipotent radical R_{u}(P), with the natural inclusion M^{R_{u}(P)} ⟶ M^{R_{u}(P′)} as the restriction map when P ≤ P′. We also attach M itself to the (-1)-dimensional simplex corresponding to G. Since one has N_{G}(P) = N_{G}(R_{u}(P)) = P for any parabolic subgroup, it is clear that our X_{O} ⊗_{OG} M is just the reduced chain complex C̃_{∗}(Δ, F_{M}) of the building with coefficients in F_{M}, with the same re-indexing as before to get a cochain complex with degrees increasing from 0 to |S|, and viewed as a complex of O-modules with a natural G-action induced by the conjugation action of G on the set of parabolic subgroups ([1], Definition 7.3.1).

In particular, if M = O is the trivial module, then X_{O} ⊗_{OG} M is related to the reduced simplicial chain complex of Δ with constant coefficients O, and therefore has a single non-zero cohomology group, since Δ has the homotopy type of a bouquet of spheres of the same dimension, by the Solomon-Tits theorem ([4], Theorem 66.33).

3. Harish-Chandra inductions of cuspidal modules

We keep the notation of the previous section, but we will now work with a coefficient field k of characteristic different from p.

For any connected reductive group H defined over 𝔽_{q} (so H might be one of the Levi subgroups L_{I} of G), we say that a kH-module V is cuspidal if the Harish-Chandra truncation ^{∗}R_{L}^{H}(V) is zero for the rational Levi complements L of all proper rational parabolic subgroups P < H. In other words, this says that V^{U} = 0 for the unipotent radical U of every proper rational parabolic subgroup P of H. This section is devoted to the proof of the following theorem.

Theorem 3.1. Let k be a field of characteristic different from p, let I_{0} be a subset of S, let V be a cuspidal kL_{I_{0}}-module, and let E = R_{L_{I_{0}}}^{G} (V). Then

X_{k} ⊗_{kG} E ≅ E[−|I_{0}|]

and

X_{k}^{∨} ⊗_{kG} E ≅ E[|I_{0}|]

in the chain homotopy category K^{b}(Mod(kG)).

Let us state what we shall prove in terms of coefficient systems.

Let Y be the coefficient system on Δ/G associating with P_{I} the set

Y_{I} = {P_{I}gP_{I_{0}} : (P_{I}^{g} ∩ P_{I_{0}})U_{I_{0}} = P_{I_{0}}}.

We define ψ_{IJ}(P_{I}gP_{I_{0}}) = P_{J}gP_{I_{0}} when I ⊆ J ⊆ S. This is a coefficient system taking values in the category of finite sets. We make it into a coefficient system ℤY with values in the category of abelian groups in the standard way, replacing each finite set Y_{I} with the free abelian group ℤY_{I}. Setting D_{k} = D ⊗_{R} k, we shall prove

Lemma 3.2. D_{k}⊗_{kG}E ≅ E⊗_{ℤ}ℤY as coefficient systems on Δ/G with values in Mod(kG).

Let us first say how this implies Theorem 3.1.

Using the reflection representation space M of W, the fundamental chamber C ⊆ M and its facets C_{I} (I ⊆ S) of dimension |S| − |I| ([4], Section 66), it is straightforward to check that

w^{−1}C_{I} ↦ P_{I}wP_{I_{0}} (w ∈ W, I ⊆ S)

induces a bijection between Y and the w^{−1}C_{I}’s that are included in the vector space M_{0} spanned by C_{I_{0}}. The restriction map ψ_{IJ} becomes the map sending w^{−1}C_{I} to w^{−1}C_{J}, so those w^{−1}C_{I} are clearly just the simplices in a triangulation of the unit sphere of M_{0}.

The (reduced) chain complex associated to the coefficient system ℤY on Δ/G is thus isomorphic to the (reduced) chain complex, with constant coefficients ℤ, of the sphere, and so its homology is concentrated in degree |S| − |I_{0}| − 1 by classical results on the (reduced) homology of spheres. Since the chain complex associated with ℤY is a complex of free abelian groups, it is actually homotopy equivalent to ℤ[−(|S| − |I_{0}| − 1)]. The first isomorphism of Theorem 3.1 follows once we re-index as mentioned in Section 2.3 to construct X_{k} from D as a cochain complex. The second isomorphism then follows using Corollary 2.2.

The proof of Lemma 3.2 uses two further lemmas.

Let us fix I ⊆ S and g ∈ G. Consider the cuspidal kL_{I_{0}}-module V as a kP_{I_{0}}-module with U_{I_{0}} acting trivially.

Lemma 3.3. If P_{I}gP_{I_{0}} ∉ Y_{I}, then [U_{I}]g.kP_{I_{0}} ⊗_{P_{I_{0}}} V = 0.

Lemma 3.4. If P_{I}gP_{I_{0}} ∈ Y_{I}, then the map

kG ⊗_{P_{I}} kP_{I}[U_{I}]g.kP_{I_{0}}[U_{I_{0}}] → kG[U_{I_{0}}]

such that

x ⊗ y ↦ xy

is an isomorphism.

Proof of Lemmas 3.3 and 3.4. We may assume g ∈ N and that its class modulo T is in the set D_{I,I_{0}} of I-reduced-I_{0} elements of W ([4], Definition 64.39 and Theorem 65.21). Let I′ = I^{g} ∩ I_{0}. Then (P_{I}^{g} ∩ P_{I_{0}})U_{I_{0}} = P_{I′} ([4], Theorem 69.16.(i)). So the condition P_{I}gP_{I_{0}} ∈ Y_{I} is equivalent to I_{0} ⊆ I^{g} (or I′ = I_{0}).

Assume that P_{I}gP_{I_{0}} ∉ Y_{I} and [U_{I}]g.kP_{I_{0}} ⊗_{P_{I_{0}}} V ≠ 0. Since

[U_{I}^{g}].kP_{I_{0}} ⊗_{P_{I_{0}}} V = [U_{I}^{g}][U_{I}^{g} ∩ P_{I_{0}}].kP_{I_{0}} ⊗_{P_{I_{0}}} V,

[U_{I}^{g}].kP_{I_{0}} ⊗_{P_{I_{0}}} V ≠ 0 implies that [U_{I}^{g} ∩ P_{I_{0}}].kP_{I_{0}} ⊗_{P_{I_{0}}} V ≠ 0, or in other words that [U_{I}^{g} ∩ P_{I_{0}}]V ≠ 0. Since U_{I_{0}} acts trivially on V, one has [(U_{I}^{g} ∩ P_{I_{0}})U_{I_{0}}]V ≠ 0. We have (U_{I}^{g} ∩ P_{I_{0}})U_{I_{0}} = U_{I′} ([4], Theorem 69.16.(ii)). Since we are assuming that P_{I}gP_{I_{0}} ∉ Y_{I}, we know that I′ is properly contained in I_{0} and therefore [U_{I′}]V = 0 by the hypothesis of Theorem 3.1. This gives Lemma 3.3.

Assume P_{I}gP_{I_{0}} ∈ Y_{I}. The map of Lemma 3.4 is clearly well-defined (and is a morphism of kG-kP_{I_{0}}-bimodules). Set P = (P_{I})^{g}.

Let us check first that the dimension of kG ⊗_{kP_{I}} kP_{I}[U_{I}]g.kP_{I_{0}}[U_{I_{0}}], or equivalently kG ⊗_{kP} kP [U_{I}]^{g}.kP_{I_{0}}[U_{I_{0}}], is less than or equal to that of kG[U_{I_{0}}], which is |G : U_{I_{0}}|. Since P_{I_{0}} = (P ∩ P_{I_{0}})U_{I_{0}}, we have

kG ⊗_{kP} kP [U_{I}]^{g}.kP_{I_{0}}[U_{I_{0}}] = kG ⊗_{kP} kP [U_{I}]^{g}[U_{I_{0}}]
= kG ⊗_{kP} kP [U_{I}]^{g}[U_{I_{0}} ∩ P][U_{I_{0}}],

and this is clearly equal to the subspace {x ⊗_{kP} [U_{I_{0}}] : x ∈ kG[U_{I}^{g}(U_{I_{0}} ∩ P)]}, so its dimension is less than or equal to the dimension of kG[U_{I}^{g}(U_{I_{0}} ∩ P_{I}^{g})], which is just |G : U_{I}^{g}(U_{I_{0}} ∩ P_{I}^{g})|. By [4], Theorem 69.16.(ii), U_{I}(^{g}U_{I_{0}} ∩ P_{I}) = U_{^{g}I′} which in turn equals U_{^{g}I_{0}} since P_{I}gP_{I_{0}} ∈ Y_{I}. So, to check our claim it suffices to check that |U_{I_{0}}| = |U_{^{g}I_{0}}|. But this is clear since L_{^{g}I_{0}} = ^{g}L_{I_{0}} and, for any J ⊆ S, U/U_{J} is a Sylow p-subgroup of P_{J}/U_{J} ≅ L_{J} ([4], Proposition 69.9 and Theorem 69.10).

It remains to check the surjectivity of the map defined in Lemma 3.4. Its image is kG[U_{I}]g.kP_{I_{0}}[U_{I_{0}}] = kG[U_{I}]g[U_{I_{0}}].kP_{I_{0}}. So it suffices to check that

(i)  kG[U_{I}]g[U_{I_{0}}] ⊇ kG[U_{I_{0}}].

We now refer to [8], Theorem 2.4. This tells us that

(ii)  kG[U_{I_{0}}] = kG[U_{^{g}I_{0}}]g[U_{I_{0}}].

On the other hand, our hypothesis implies I ⊇ ^{g}I′ = ^{g}I_{0}, so kG[U_{I}] ⊇ kG[U_{^{g}I_{0}}]. Then (ii) above implies (i).

Proof of Lemma 3.2. Let I ⊆ J ⊆ S. As a kP_{I}-kP_{I_{0}}-bimodule, kG = ⨁_{P_{I}gP_{I_{0}}} kP_{I}g.kP_{I_{0}}, so

[U_{I}]E = [U_{I}].kG ⊗_{kP_{I_{0}}} V = ⨁_{P_{I}gP_{I_{0}}} kP_{I}[U_{I}]g.kP_{I_{0}} ⊗_{kP_{I_{0}}} V.

By Lemma 3.3, this is the same as

⨁_{P_{I}gP_{I_{0}}∈Y_{I}} kP_{I}[U_{I}]g.kP_{I_{0}} ⊗_{kP_{I_{0}}} V.

Then

D_{I} ⊗_{kG} E = ⨁_{P_{I}gP_{I_{0}}∈Y_{I}} kG ⊗_{kP_{I}} [U_{I}].kP_{I}g.kP_{I_{0}} ⊗_{kP_{I_{0}}} V.

By Lemma 3.4, for every g the summand kG ⊗_{kP_{I}} [U_{I}].kP_{I}g.kP_{I_{0}} ⊗_{kP_{I_{0}}} V is isomorphic to E = kG[U_{I_{0}}] ⊗_{kP_{I_{0}}} V by the map

x ⊗_{kP_{I}} [U_{I}]a ⊗_{kP_{I_{0}}} v ↦ x[U_{I}]a ⊗_{kP_{I_{0}}} v

for x ∈ kG, a ∈ P_{I}gP_{I_{0}}, v ∈ V. Then D_{I} ⊗_{kG} E ≅ E ⊗_{ℤ} ℤY_{I} by the map

i: x ⊗_{kP_{I}} [U_{I}]a ⊗_{kP_{I_{0}}} v ↦ (x[U_{I}]a ⊗_{kP_{I_{0}}} v) ⊗_{ℤ} (P_{I}aP_{I_{0}})

for x ∈ kG, a ∈ ⋃_{P_{I}gP_{I_{0}}∈Y_{I}} P_{I}gP_{I_{0}}, v ∈ V.

In the same way, we get an isomorphism D_{J} ⊗_{kG} E ≅ E ⊗_{ℤ} ℤY_{J} given by the map

j: y ⊗_{kP_{J}} [U_{J}]b ⊗_{kP_{I_{0}}} v ↦ (y[U_{J}]b ⊗_{kP_{I_{0}}} v) ⊗_{ℤ} (P_{J}bP_{I_{0}})

for y ∈ kG, b ∈ ⋃_{P_{J} gP_{I_{0}}∈Y_{J}} P_{J}gP_{I_{0}}, v ∈ V.

It remains to check that j ∘ ϕ_{IJ} = ψ_{IJ} ∘ i. Taking x ∈ kG, a ∈ P_{I}gP_{I_{0}} ∈ Y_{I}, v ∈ V, we have

(ψ_{IJ} ∘ i)(x ⊗_{kP_{I}} [U_{I}]a ⊗_{kP_{I_{0}}} v) = ψ_{IJ}((x[U_{I}]a ⊗_{kP_{I_{0}}} v) ⊗_{ℤ} (P_{I}aP_{I_{0}}))
= (x[U_{I}]a ⊗_{kP_{I_{0}}} v) ⊗_{ℤ} (P_{J}aP_{I_{0}}).

On the other hand,

ϕ_{IJ}(x ⊗_{kP_{I}} [U_{I}]a ⊗_{kP_{I_{0}}} v) = x ⊗_{kP_{J}} [U_{I}]a ⊗_{kP_{I_{0}}} v.

The latter is also equal to

x ⊗_{kP_{J}} [U_{J}][U_{I}]a ⊗_{kP_{I_{0}}} v = |U_{I}|^{−1} ∑_{u∈U_{I}} x ⊗_{kP_{J}} [U_{J}]ua ⊗_{kP_{I_{0}}} v

with P_{J}uaP_{I_{0}} = P_{J}aP_{I_{0}} ∈ Y_{J} for each u. So the image by j is

(|U_{I}|^{−1} ∑_{u∈U_{I}} x[U_{J}]ua ⊗_{kP_{I_{0}}} v) ⊗_{ℤ} (P_{J}aP_{I_{0}}) = (x[U_{I}]a ⊗_{kP_{I_{0}}} v) ⊗_{ℤ} (P_{J}aP_{I_{0}}).

This finishes the proof.

Defining D for P_{I_{0}} instead of G gives a coefficient system of RP_{I_{0}}-RP_{I_{0}}-bimodules on the subsets of I_{0}. We can extend it trivially to a coefficient system on the whole of Δ/G which we denote by D(P_{I_{0}}). We denote the restriction maps of D(P_{0}) by φ_{IJ} for I ⊆ J ⊆ S.

Proposition 3.5. There is a surjective map

π : D ⊗_{RG} RG[U_{I_{0}}] → RG[U_{I_{0}}] ⊗_{RP_{I_{0}}} D(P_{I_{0}})

of coefficient systems of RG-RP_{I_{0}}-bimodules on Δ/G.

Proof. We shall sometimes abbreviate P_{I_{0}} = P_{0}.

Let us define the map π as follows. Let I ⊆ S, and define e = [U_{I}]. Let e_{0} = [U_{I_{0}}]. If I ⊈ I_{0}, let π_{I} = 0. Assume I ⊆ I_{0}. Then

RGe_{0} ⊗_{RP_{0}} D(P_{0}) = RGe_{0} ⊗_{RP_{0}} RP_{0}e ⊗_{RP_{I}} e.RP_{0}.

In

D_{I} ⊗_{RG} RGe_{0} = RGe ⊗_{RP_{I}} e.RG ⊗_{RG} RGe_{0},

let us consider the last tensor product. By the obvious maps of RP_{I}-RP_{0}-bimodules, we have

e.RG ⊗_{RG} RGe_{0} ≅ e.RGe_{0} = ⨁_{P_{I}gP_{0}⊆G} e.RP_{I}g.RP_{0}e_{0},

where the summand for g = 1 gives e.RP_{0}e_{0} = ee_{0}.RP_{0} = e.RP_{0}. So D_{I} ⊗_{RG} RGe_{0} has a summand isomorphic to

RGe ⊗_{RP_{I}} e.RP_{0} ≅ RGe_{0} ⊗_{RP_{0}} RP_{0}e ⊗_{RP_{I}} e.RP_{0}.

This provides us with a surjective map π_{I} defined as follows for x ∈ RG, y, z ∈ G.

π_{I}(xe ⊗_{RP_{I}} ey ⊗_{RG} ze_{0}) = \{ \begin{matrix}xe ⊗_{RP_{0}} e ⊗_{RP_{I}} eyze_{0} & if yz ∈ P_{0} \\ 0 & if yz ∉ P_{0}. \end{matrix}

Let J ⊇ I. We must verify the equality of compositions of maps

(∗)  (RGe_{0} ⊗_{RP_{0}} φ_{IJ}) ∘ π_{I} = π_{J} ∘ (ϕ_{IJ} ⊗_{RG} RGe_{0}).

We may assume that I ⊆ J ⊆ I_{0}, since otherwise D(P_{0})_{J} = 0. Set f = [U_{J}]. Then e = ef = fe and f = fe_{0} = e_{0}f. Take x ∈ RG, y, z ∈ G and consider the element

xe ⊗_{RP_{I}} ey ⊗_{RG} ze_{0} ∈ D_{I} ⊗_{RG} RGe_{0} = RGe ⊗_{RP_{I}} e.RG ⊗_{RG} RGe_{0}.

If yz ∈ P_{0}, then the effects of the two compositions of maps (∗) on this element are respectively

xe ⊗_{RP_{I}} ey ⊗_{RG} ze_{0} ↦ xe ⊗_{RP_{0}} e ⊗_{RP_{I}} eyze_{0} ↦ xe ⊗_{RP_{0}} e ⊗_{RP_{J}} eyze_{0}

and

xe ⊗_{RP_{I}} ey ⊗_{RG} ze_{0} ↦ xe ⊗_{RP_{J}} ey ⊗_{RG} ze_{0} = xef ⊗_{RP_{J}} fey ⊗_{RG} ze_{0}
↦ xef ⊗_{RP_{0}} f ⊗_{RP_{J}} feyze_{0} = xe ⊗_{RP_{0}} f ⊗_{RP_{J}} eyze_{0},

since eyz ∈ RP_{0}. But e ∈ RP_{J}, so xe ⊗_{RP_{0}} f ⊗_{RP_{J}} eyze_{0} = xe ⊗_{RP_{0}} e ⊗_{RP_{J}} eyze_{0}. If yz ∉ P_{0}, then we get

xe ⊗_{RP_{I}} ey ⊗_{RG} ze_{0} ↦ 0 ↦ 0

and

xe ⊗_{RP_{I}} ey ⊗_{RG} ze_{0} ↦ xe ⊗_{RP_{J}} ey ⊗_{RG} ze_{0} = xef ⊗_{RP_{J}} fy ⊗_{RG} ze_{0} ↦ 0.

Thus the two compositions are indeed equal.

4. The main theorem over a field of coefficients

As in the previous section, we shall take O to be a field k of characteristic different from p.

Since X_{k} is a complex of finitely generated kG-kG-bimodules that are projective as left and as right modules, and since kG is a symmetric algebra, it follows (see, for example, [9], Corollary 9.2.5) that the k-linear dual X_{k}^{∨} of X_{k} induces a functor X_{k}^{∨}⊗_{kG}? that is right adjoint (and also left adjoint) to X_{k}⊗_{kG}?. This is true whether we regard these as endofunctors of the category C^{b}(Mod(kG)) of chain complexes, the chain homotopy category K^{b}(Mod(kG)), or the derived category D^{b}(Mod(kG)). Hence we get a natural map of complexes of kG-kG-bimodules

ϵ : X_{k} ⊗_{kG} X_{k}^{∨} ⟶ kG

that induces the counit of this adjunction.

We shall show that ϵ is an isomorphism in the derived category of kG-kG-bimodules.

The degree |S| term of X_{k} is kG, and so the degree zero term of X_{k} ⊗_{kG} X_{k}^{∨} has a summand isomorphic to kG that is mapped isomorphically by ϵ to kG. Thus we have a short exact sequence

0 ⟶ K_{k} ⟶ X_{k} ⊗_{kG} X_{k}^{∨} ⟶^{ϵ} kG ⟶ 0

of complexes that is split in each degree, where K_{k} is just the kernel of ϵ.

Lemma 4.1. If E is a kG-module Harish-Chandra induced from a finite-dimensional cuspidal module for some Levi subgroup L_{I}, then K_{k} ⊗_{kG} E is chain contractible.

Proof. By standard properties of adjoint functors, the map

ϵ ⊗_{kG} X_{k} : X_{k} ⊗_{kG} X_{k}^{∨} ⊗_{kG} X_{k} ⟶ X_{k}

is split surjective, a splitting map being given by X_{k} ⊗_{kG} η, where

η : kG ⟶ X_{k}^{∨} ⊗_{kG} X_{k}

is the natural map inducing the unit of the adjunction. Thus

ϵ ⊗_{kG} X_{k} ⊗_{kG} E[|I|] : X_{k} ⊗_{kG} X_{k}^{∨} ⊗_{kG} X_{k} ⊗ E[|I|] ⟶ X_{k} ⊗_{kG} E[|I|]

is a split surjection between objects of K^{b}(Mod(kG)) that, by Theorem 3.1, are both isomorphic to E. By the Krull-Schmidt theorem, the map must be an isomorphism. Since X_{k} ⊗_{kG} E[|I|] ≅ E, the map ϵ ⊗_{kG} E is also an isomorphism in K^{b}(Mod(kG)), and hence K_{k} ⊗_{kG} E is chain contractible.

We shall also need the following lemma.

Lemma 4.2. Let K be a bounded complex of projective right A-modules, where A is a finite dimensional algebra over a field. Let E be a set of (left) A-modules such that for every irreducible A-module N there is some E ∈ E with Hom(E, N) ≠ 0. Then K is acyclic if and only if K ⊗_{A} E is acyclic for every E ∈ E.

Proof. Suppose K ≅ K ⊗_{A} A is not acyclic. Let i_{0} be the minimal i such that, for all finitely generated left A-modules E, H^{i+1}(K ⊗_{A} E) = 0.

By the projectivity of K, any short exact sequence 0 → E_{3} → E_{2} → E_{1} → 0 of modules gives rise to an exact sequence of complexes 0 → K ⊗_{A} E_{3} → K ⊗_{A} E_{2} → K ⊗_{A} E_{1} → 0 and then, by the long exact sequence of cohomology, to an exact sequence

H^{i_{0}}(K ⊗_{A} E_{3}) → H^{i_{0}}(K ⊗_{A} E_{2}) → H^{i_{0}}(K ⊗_{A} E_{1}) → H^{i_{0}+1}(K ⊗_{A} E_{3}) = 0.

If H^{i_{0}}(K ⊗_{A} E_{2}) ≠ 0, then either H^{i_{0}}(K ⊗_{A} E_{1}) ≠ 0 or H^{i_{0}}(K ⊗_{A} E_{3}) ≠ 0. There must therefore be an irreducible A-module N such that H^{i_{0}}(K ⊗_{A} N) ≠ 0. By assumption, we can choose the short exact sequence with E_{1} = N and E_{2} ∈ E. Then H^{i_{0}}(K ⊗_{A} E_{2}) = 0 by the hypothesis, and the above exact sequence gives H^{i_{0}}(K ⊗_{A} N) = 0, a contradiction.

Theorem 4.3. With the notation above, ϵ and η are isomorphisms in the derived category of kG-kG-bimodules, and hence

X_{k}⊗_{kG}? : D^{b}(Mod(kG)) ⟶ D^{b}(Mod(kG))

is an equivalence of categories.

Proof. We shall prove that ϵ is an isomorphism by showing that K_{k} is acyclic; the proof for η is similar, with the cokernel of η playing the role of K_{k}.

Let N be an irreducible kG-module, and choose I ⊆ S minimal subject to ^{∗}R_{L_{I}}^{G} (N) ≅ 0̸ (i.e., N^{U_{I}} ≠ 0), so that ^{∗}R_{L_{I}}^{G} (N) is a cuspidal kL_{I}-module. By adjointness, the (nonzero) identity map of ^{∗}R_{L_{I}}^{G} (N) corresponds to a non-zero map R_{L_{I}}^{G}^{∗}R_{L_{I}}^{G} (N) → N. So by Lemma 4.1, the conditions of Lemma 4.2 are satisfied for E the set of kG-modules that are Harish-Chandra induced from finite-dimensional cuspidal modules. Hence the lemma implies that K_{k} is acyclic.

Now X_{k}⊗_{kG}? is an equivalence of categories because we have shown that X_{k} is a ‘twosided tilting complex’ ([9], Definition 6.2.1). To be explicit, a quasi-inverse is given by X_{k}^{∨}⊗_{kG}?, since ϵ and η induce isomorphisms of functors

X_{k} ⊗_{kG} X_{k}^{∨}⊗_{kG}? ≅ kG⊗_{kG}? ≅ id

and

id ≅ kG⊗_{kG}? ≅ X_{k}^{∨} ⊗_{kG} X_{k}⊗_{kG}?.

In fact, more should be true. In [10] the notion of a ‘splendid’ equivalence is introduced, being an equivalence of derived categories of group algebras or blocks that has various especially good properties. The only aspect of ‘splendour’ that is not immediate for the equivalence that we have just described is that the chain maps ϵ and η that we have shown to be isomorphisms in the derived category are actually chain homotopy equivalences.

Conjecture 4.4. With the notation above, ϵ and η are isomorphisms in the chain homotopy category of complexes of kG-kG-bimodules, and hence

X_{k}⊗_{kG}? : K^{b}(Mod(kG)) ⟶ K^{b}(Mod(kG))

is an equivalence of categories.

To prove this conjecture, one must show that K_{k} is contractible, and not merely acyclic.

5. General coefficient rings

We shall now generalize the results of the previous section to a general commutative coefficient ring O in which p is invertible. Exactly as when we worked over a field k, there is a short exact sequence

0 ⟶ K_{O} ⟶ X_{O} ⊗_{OG} X_{O}^{∨} ⟶^{ϵ} OG ⟶ 0

of complexes of OG-OG-bimodules, split in each degree.

Theorem 5.1. The functor

X_{O}⊗_{OG}? : D^{b}(Mod(OG)) ⟶ D^{b}(Mod(OG))

is an equivalence of categories.

Proof. As in the proof of Theorem 4.3, we need to show that K_{O} is acyclic. It is sufficient to do this for the case O = R = ℤ[1/p], since K_{R} is a bounded complex of free R-modules, and so if it is acyclic then it is contractible as a complex of R-modules, and so, for a general O,

K_{O} ≅ K_{R} ⊗_{R} O

is contractible as a complex of O-modules, and therefore acyclic.

Since K_{R} is a complex of finitely generated free R-modules, its homology groups are finitely generated abelian groups with no p-torsion, and since by Theorem 4.3

K_{ℤ/lℤ} ≅ K_{R} ⊗_{ℤ} ℤ/lℤ

is acyclic for every prime l ≠ p, it follows from the Universal Coefficient Theorem that K_{R} is acyclic.

As in Conjecture 4.4, it should be possible to refine this result.

Conjecture 5.2. The functor

X_{O}⊗_{OG}? : K^{b}(Mod(OG)) ⟶ K^{b}(Mod(OG))

is an equivalence of categories.

In fact, this can be reduced to the previous conjecture.

Proposition 5.3. If Conjecture 4.4 is true, then so is Conjecture 5.2.

Proof. Suppose that K_{k} is contractible as a complex of kG-kG-bimodules for every field k with characteristic different from p. We wish to show that K_{O} is contractible for any coefficient ring O in which p is invertible. Since K_{O} is a bounded complex of permutation O[G×G]-modules, it suffices to prove that the fixed point complex K_{O}^{H} is acyclic for every subgroup H ≤ G × G ([10], Lemma 1.1).

Since change of scalars commutes with fixed point functors for permutation modules, it suffices, as in the proof of Theorem 5.1, to do this for O = R = ℤ[1/p], where the same Universal Coefficient Theorem argument works, since

K_{R}^{H} ⊗_{ℤ} ℤ/lℤ ≅ K_{ℤ/lℤ}^{H}

is acyclic by assumption.

6. A theorem of Curtis type

In this section we check that the equivalence defined above commutes with Harish-Chandra induction.

Let us first recall that, by the remarks following the definition of X_{O} in Section 2.3, the complex X_{O} is defined intrinsically in terms of the abstract group G.

When L is a Levi subgroup of G, let us denote by X_{O}^{(L)} the corresponding complex for L.

The character-theoretic analogue of the following theorem is well-known (see [7],Theorem 8.11, for example).

Theorem 6.1. Let P be a parabolic subgroup of G, with Levi decomposition P = LU_{P}. Let O be any commutative ring in which p is invertible. Let R_{L}^{G} denote the OG-OL-bimodule OG[U_{P}] inducing Harish-Chandra induction of OL-modules. Then

X_{O}^{(G)} ⊗_{OG} R_{L}^{G} ≅ R_{L}^{G} ⊗_{OL} X_{O}^{(L)}

in D^{b}(Mod(OG ⊗ OL^{op})).

Proof. We may choose P containing B, that is P = P_{I_{0}} for some I_{0} ⊆ S. Let e = [U_{P}]. It is sufficient to take O = R = ℤ[1/p], so we need to show that

X_{R}^{(G)} ⊗_{RG} RGe ≅ RGe ⊗_{RL} X_{R}^{(L)}

in the derived category D^{b}(Mod(RG ⊗ RL^{op})).

The map of Proposition 3.5 gives a surjective chain map

X_{R}^{(G)} ⊗_{RG} RGe → RGe ⊗_{RL} X_{R}^{(L)} → 0.

(We may replace P = P_{I_{0}} by L in the tensor products of Proposition 3.5 since U_{P} always acts trivially there.) Let K′_{R} be the kernel of this map. Considered as a map of complexes of RL^{op}-modules, this surjection is split in each degree, since the modules are all projective right RL-modules, so as well as an exact sequence

0 → K′_{R} → X_{R}^{(G)} ⊗_{RG} RGe → RGe ⊗_{RL} X_{R}^{(L)} → 0

in C^{b}(Mod(RL^{op})), we have exact sequences

0 → K′_{k} ⊗_{kL} E → X_{k}^{(G)} ⊗_{kG} kGe ⊗_{kL} E → kGe ⊗_{kL} X_{k}^{(L)} ⊗_{kL} E → 0

in C^{b}(Mod(kG)) for any kL-module E, where k is a field of characteristic different from p and K′_{k} = K′_{R} ⊗_{R} k.

By Lemma 4.2, to show that K′_{k} is acyclic it is sufficient to check that K′_{k} ⊗ E is acyclic for any kL-module of the form E = Ind_{P_{I}}^{P} (V) for I ⊆ I_{0} and V a U_{I}-trivial kP_{I}-module such that V^{U_{J}} = 0 for every J properly contained in I. Set Ẽ := kGe⊗_{kL}E = kG⊗_{kP} E = Ind_{P_{I}}^{G} (V). Now Theorem 3.1 implies that

X_{k}^{(G)} ⊗_{kG} kGe ⊗_{kL} E ≅ X_{k}^{(G)} ⊗_{kG} Ẽ ≅ Ẽ[−|I|]

in D^{b}(Mod(kG)), and X_{k}^{(L)} ⊗_{kL} E ≅ E[−|I|] in D^{b}(Mod(kL)). Since kGe is projective, and hence flat, as a right module, this implies that

kGe ⊗_{kL} X_{k}^{(L)} ⊗_{kL} E ≅ kGe ⊗_{kL} E[−|I|] = Ẽ[−|I|]

in D^{b}(Mod(kG)). Just as in the proof of Theorem 4.3, this implies that K′_{k} ⊗ E is acyclic.

This shows that K′_{k} is acyclic for any field k of characteristic different from p. Using the Universal Coefficient Theorem as in the proof of Theorem 5.1, we deduce that K′_{R} is acyclic. Thus our claim.

By Lemma 2.1, the corresponding result for Harish-Chandra truncation is also true. As in Conjectures 4.4 and 5.2, it should be possible to refine Theorem 6.1 to the homo-

topy category.

Conjecture 6.2. There is an isomorphism

X_{O}^{(G)} ⊗_{OG} R_{L}^{G} ≅ R_{L}^{G} ⊗_{OL} X_{O}^{(L)}

in K^{b}(Mod(OG ⊗ OL^{op})).

7. Concluding remarks

7.1. Contravariant duality. The equivalences of Theorems 4.3 and 5.1 are covariant functors, and so are probably not worthy of the name ‘duality’, even though they induce an involution on generalized characters. However, it follows from what we have proved and Section 4 of Yekutieli’s paper [12] on dualizing complexes that, if the coefficient ring O is noetherian, the functors

RHom_{OG}(?, X_{O}) : D^{b}(mod(OG))^{op} ⟶ D^{b}(mod(OG^{op}))

and

RHom_{OG}(?, X_{O}) : D^{b}(mod(OG^{op}))^{op} ⟶ D^{b}(mod(OG))

are mutually inverse contravariant equivalences. [Actually, Yekutieli works with a coefficient field k, but his proofs of the results that are relevant here generalize trivially to a general commutative noetherian coefficient ring.]

Identifying OG-modules and OG^{op}-modules in the usual way, the two functors above may be regarded as contravariant functors from D^{b}(mod(OG)) to itself. Lemma 2.1 implies that these two functors are isomorphic, and we shall use the notation

D_{G} : D^{b}(mod(OG))^{op} ⟶ D^{b}(mod(OG))

for either of them.

By the remarks above, we have the following.

Theorem 7.1. Let O be a commutative noetherian ring in which p is invertible. The functor D_{G} is an equivalence of triangulated categories with D_{G} ∘ D_{G} isomorphic to the identity functor.

There is also a version of Theorem 6.1 for this contravariant duality. We shall use D_{L} to denote the contravariant duality functor for D^{b}(mod(OL)).

Theorem 7.2. Let P be a parabolic subgroup of G, with Levi decomposition P = LU_{P}. Let O be a commutative noetherian ring in which p is invertible. Then

D_{G} ∘ R_{L}^{G} ≅ R_{L}^{G} ∘ D_{L}

as contravariant functors from D^{b}(mod(OL)) to D^{b}(mod(OG)).

Proof. To calculate the effect of these functors on an object of D^{b}(mod(OL)), we apply the functors Hom_{OG}(? ⊗_{OL} [U_{P}].OG, X_{O}^{(G)}) and OG[U_{P}] ⊗_{OL} Hom_{OL}(?, X_{O}^{(L)}) to a projective resolution P_{∗} of the object, regarded as a right OL-module. Since O is noetherian, P_{∗} is a complex of finitely generated projective right OL-modules, and so

(i)  Hom_{OG}(P_{∗} ⊗_{OL} [U_{P}].OG, X_{O}^{(G)}) ≅ X_{O}^{(G)} ⊗_{OG} OG[U_{P}] ⊗_{OL} P_{∗}^{∨},

and

(ii)  OG[U_{P}] ⊗_{OL} Hom_{OL}(P_{∗}, X_{O}^{(L)}) ≅ OG[U_{P}] ⊗_{OL} X_{O}^{(L)} ⊗_{OL} P_{∗}^{∨},

where P_{∗}^{∨} is the O-dual of P_{∗}, regarded as a complex of projective left OL-modules. As shown in the proof of Theorem 6.1, there is a natural chain map

X_{O}^{(G)} ⊗_{OG} OG[U_{P}] → OG[U_{P}] ⊗_{OL} X_{O}^{(L)}

that is a split epimorphism in each degree and induces an isomorphism in homology. Taking the tensor product with P_{∗}^{∨}, we get a quasi-isomorphism (i.e., an isomorphism in the derived category) between (i) and (ii).

Hence D_{G} ∘ R_{L}^{G} ≅ R_{L}^{G} ∘ D_{L} as functors taking values in D^{b}(mod(OG)).

There is of course a corresponding statement about Harish-Chandra truncation ^{∗}R_{L}^{G}.

7.2. The Steinberg module. For a field k, usually of characteristic p or zero, the Steinberg module St_{k} for kG is defined as the reduced simplicial homology, with coefficients in k, of the building of G ([4], Definition 66.34). Since, by the Solomon-Tits theorem, the building of G has the homotopy type of a bouquet of (n − 1)-spheres (where n = |S|), this reduced homology is concentrated in the single degree n − 1 ([4], Theorem 66.33). It is consistent to define the Steinberg module St_{O} for OG similarly for a general coefficient ring O.

Applying our functor X_{O}⊗_{OG}? to the trivial module O amounts, up to a suitable reindexing, to calculating the reduced homology of the building of G with coefficients in O, so X_{O} ⊗_{OG} O is isomorphic in D^{b}(Mod(OG)) to St_{O}, the complex consisting of the Steinberg module concentrated in degree zero. Since, by Theorem 5.1, X_{O}⊗_{OG}? is an equivalence of derived categories, the following theorem follows immediately.

Theorem 7.3. There is an isomorphism

Ext_{OG}^{∗}(St_{O}, St_{O}) ≅ H^{∗}(G, O)

of graded rings for any commutative ring O in which p is invertible.

Of course, this is far from true if O is a field of characteristic p, since then St_{O} is a projective OG-module, and so Ext_{OG}^{∗}(St_{O}, St_{O}) is zero in non-zero degrees, which is certainly not the case for H^{∗}(G, O).

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UFR de Mathématiques, Université Paris 7, 2, Place Jussieu, 75005 Paris, FRANCE

   E-mail address: cabanes@math.jussieu.fr

University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENG-LAND

   E-mail address: j.rickard@bristol.ac.uk