SOME EXAMPLES OF RICKARD COMPLEXES

RAPHAËL ROUQUIER

Talk given at the Constanta Conference on Representation Theory October 6, 1995

Abstract. After a presentation of Broué’s conjecture for principal blocks with an abelian defect group, we describe a Rickard complex for GL_{2}(q) arising from the ℓ-adic cohomology of a Deligne-Lusztig variety, in accordance with the explicit form given by Broué to his conjecture in the case of Chevalley groups in non natural characteristic.

1. Overview of Broué’s conjecture

Let G be a finite group and ℓ a prime number. Let P be a Sylow ℓ-subgroup of G and assume P is abelian. Let H = N_{G}(P). Let O be the ring of integers of a finite unramified extension K of Q_{ℓ}, such that KG and KH are split. Let A and B be the principal blocks of G and H over O. Let us denote by H^{∘} the group opposite to H. Similarly, B^{∘} denotes the algebra opposite to B. We put ΔP = {(x, x^{−1})|x ∈ P } ≤ G × H^{∘}. The sign ⊗ means ⊗_{O}. Finally, if M is an O-module, we put KM = K ⊗ M.

Conjecture 1. The blocks A and B are Rickard equivalent. More precisely, there is a complex C of (left) A ⊗ B^{∘}-modules which are direct summands of relatively ΔP -projective permutation modules such that :

C^{∗} ⊗_{A} C ≃ B in K^{b}(B ⊗ B^{∘})
C ⊗_{B} C^{∗} ≃ A in K^{b}(A ⊗ A^{∘}).

For the sake of simplicity and for the lack of a final form of the conjecture in the general case, we have stated the conjecture for principal blocks only. The original statement of the conjecture [Br1] makes no assumption on C. That C should be of this special type (“ splendid ”) appeared in [Ri4].

The conjecture is known (to the author) to hold in the following cases :

• P cyclic [Ri1, Li, Rou] ;

• G = G(F_{q}) the group of rational points of a connected reductive algebraic group, when ℓ|q − 1 but ℓ does not divide the order of the Weyl group [Pu] ;

• G = A_{5} and ℓ = 2 [Ri4] ;

• G = SL_{2}(8) and ℓ = 2 [Rou] ;

• G is ℓ-solvable.

For Chevalley groups, when ℓ is not the natural characteristic of the group, there is a very precise conjecture of Broué giving a candidate for C, in terms of ℓ-adic cohomology of certain Deligne-Lusztig varieties [Br-Ma]. The aim of the second part is to present the simplest case of this conjecture.

2. A Geometrical construction for GL_{2}(q), ℓ|q + 1

Let q be a prime power. Consider the affine curve X with equation (xy^{q} − x^{q}y)^{q−1} = −1 over an algebraic closure F̄_{q} of F_{q}. The group G = GL_{2}(F_{q})^{1} acts naturally on the affine plane over F̄_{q} and this induces an action of G on X. There is also an action of the group of rational points T ≃ F_{q^{2}}^{∗} of a Coxeter torus of GL_{2}(F̄_{q}) by scalar multiplication (in the isomorphism above, F acts on T as x ↦ x^{q} on F_{q^{2}}^{∗}). Finally, the variety X is defined over F_{q}, with corresponding Frobenius endomorphism F.

This variety is actually the Deligne-Lusztig variety associated to the non trivial element of the Weyl group^{2} [De-Lu, 2.2].

Let ℓ|q + 1 be an odd prime and O be the ring of integers of a finite unramified extension K of Q_{ℓ}, such that KG and KH are split, where H = N_{G}(T) = T ⋊ W and |W | = 2. Let P be the ℓ-Sylow subgroup of T.

Our object of study is the complex RΓ_{c}(X, O) (O is the constant ℓ-adic sheaf) giving rise to the compact support ℓ-adic cohomology : this is an object in the derived category of (OG) ⊗ (OT)^{∘}-modules. Actually, we will consider the finer invariant C = Λ_{c}(X, O) in the homotopy category of (OG)⊗(OT)^{∘}-modules, as defined by J.Rickard [Ri3]. This is a complex of direct summands of permutation O(G × T^{∘})-modules. Note that there is an action of the Frobenius F on C, giving rise to a right action of OT ⋊ F.

1In the talk, the case of SL_{2}(q) had been considered, where the same methods apply.

2This is a Coxeter variety, i.e., the variety associated to a Coxeter element of the Weyl group. These varieties should be studied by the author in a future paper.

Let e be the sum of the ℓ-blocks with positive defect of OG. Define A = OGe and B = OH.

Proposition 1. ^{3} The action of OT ⋊ F on C factors through an action of an algebra isomorphic to B. The action of OG on C factors through an action of A : the complex C is then a complex of direct summands of relatively ΔP -projective permutation modules. We have

C^{∗} ⊗_{A} C ≃ B in K^{b}(B ⊗ B^{∘}) and
C ⊗_{B} C^{∗} ≃ A in K^{b}(A ⊗ A^{∘}).

Proof. Since X is an affine curve, the cohomology groups H_{c}^{i}(X, O) are zero for i = 0 and i > 2. Since X is in addition smooth, the cohomology groups H_{c}^{1}(X, O) and H_{c}^{2}(X, O) are free as O-modules [SGA4\frac{1}{2}, Arcata, III.§3].

Since both G and T act freely on X, the complex C is perfect (i.e. isomorphic to a bounded complex of projective modules) as an object of D^{b}(OG) and as an object of D^{b}(OT) [De-Lu, (proof of) 3.5].

The representation of G×T^{∘} on H_{c}^{2}(X, O) is isomorphic to the permutation representation on the connected components of X. Its character is

∑_{α∈Irr(F_{q}^{∗})} det_{α} ⊗α

where det_{α} is the character α ∘ det of G. The Frobenius morphism F acts with the eigenvalue q on H_{c}^{2}(X, O).

The character of the KG ⊗ (KT)^{∘}-module H_{c}^{1}(X, K) is [Di-Mi2, 15.9] :

∑_{α∈Irr(F_{q}^{∗})} St_{α} ⊗α + ∑_{\begin{matrix}ω∈Irr(F_{q^{2}}^{∗})/W, \\ ω^{q−1}≠1 \end{matrix}}[q − 1]_{ω} ⊗ (ω + ω^{q})

where St_{α} = St · det_{α}, St is the Steinberg character of G and

{[q − 1]_{ω}}_{ω∈Irr(F_{q^{2}}^{∗})/W, ω^{q−1}≠1}

3The proposition actually holds for O replaced by Z_{ℓ}, as suggested by K.W.Roggenkamp.

is the set of irreducible characters of G with degree q − 1. The Frobenius F acts with the eigenvalue 1 on the G-isotypic component with character St_{α} and with eigenvalues \sqrt{−q} and −\sqrt{−q} on the component with character [q − 1]_{ω} (this is a consequence of Lefschetz formula, [Di-Mi1, V.1.3]). From this description of the character of H^{∗}(C), it follows that OG acts on C through A.

Let σ ∈ KT ⋊ F defined by

σ = \frac{i}{q − 1}(2F − 1 − q) + \frac{i − 1}{\sqrt{−q}}F

where i = \frac{1}{q^{2}−1} ∑_{t∈T} t^{q−1} and where \sqrt{−q} ∈ O is chosen such that ℓ|1 − \sqrt{−q}. Note that σ is actually in OT ⋊ F, since

σ = \frac{−1}{(q − 1)^{2}} ∑_{t∈T} t^{q−1} − \frac{1}{\sqrt{−q}} F + \frac{1 − \sqrt{−q}}{\sqrt{−q}(q − 1)^{2}(1 + \sqrt{−q})} ∑_{t∈T} t^{q−1}F.

For t ∈ T, we have σt = t^{q}σ since F t = t^{q}F. Now, we see that σ acts trivially on H_{c}^{2}(X, K), with eigenvalue −1 on the G-isotypic components of H_{c}^{1}(X, K) with character St_{α} and with eigenvalues 1 and −1 on the G-isotypic components with character [q−1]_{ω} ; in particular, σ^{2} acts trivially on H_{c}^{∗}(X, K). Finally, the image in End_{K^{b}(KA)}(KC) of the sub-algebra of KT ⋊ F generated by T and σ is isomorphic to KH. But it is clear that End_{K^{b}(KA)}(KC) is isomorphic to KH : this means that the image in End_{K^{b}(KA)}(KC) of the sub-algebra of KT ⋊F generated by T and σ is actually the image of KT ⋊F. Hence, we have proven the analog of the proposition where scalars are extended to K.

For α ∈ Irr(F_{q}^{∗}), let e_{α} be the sum of the blocks of G containing the characters [q − 1]_{ω} with ω^{q+1} = α and the characters det_{α} and St_{α}. Let C_{α} = e_{α}C. Since H^{i}(C) is free over O and C is perfect in D^{b}(OG), it is isomorphic to a complex of projective modules 0 → C^{1} ⟶^{φ} C^{2} → 0 as an OG-module. Let us choose C^{1} such that Im φ has no projective direct summand. Put C_{α}^{i} = e_{α}C^{i}. Then, C_{α}^{2} is a projective cover of det_{α}, since H^{2}(C_{α}) ≃ det_{α}. Now, C_{α} splits as

0 → C′_{α}^{1} → C_{α}^{2} → 0 ⊕ 0 → C”_{α}^{1} → 0 → 0

where C′_{α}^{1} → C_{α}^{2} is the beginning of a projective resolution of det_{α}. From this description and from the knowledge of the characters of C′_{α}^{1}, C”_{α}^{1} and C_{α}^{2}, it follows that C is a tilting complex for A.

Let B′ be the image of the sub-algebra of OT ⋊F generated by T and σ in End_{K^{b}(A)}(C). The algebra B′ is isomorphic to OH and C is perfect in D^{b}(B′), since it is perfect in D^{b}(OT).

A proof similar to the one above shows that C is a tilting complex for B′. Now, by [Br2, théorème 2.3], this implies that C is a two-sided tilting complex for A ⊗ B^{∘}, i.e., the isomorphisms of the proposition hold in the derived categories and a priori not in the homotopy categories. Note that we have obtained that B′ is the whole of End_{K^{b}(A)}(C), hence B′ is the image of OT ⋊ F in End_{K^{b}(A)}(C).

If S is a non trivial ℓ-subgroup of G×T^{∘} which is not conjugate to ΔP, then S acts freely on X, hence by [Ri3, Corollary 3.3], C is a complex of direct summands of relatively ΔP -projective permutation modules. If S is a non trivial ℓ-subgroup of G × T^{∘}, then the fixed points set X^{S} has dimension zero, hence Λ_{c}(X^{S}, O) is concentrated in degree 0. Hence, by [Ri3, Theorem 4.2], C is homotopic to a bounded complex of modules which are all projective A ⊗ B^{∘}-modules, except C^{0} ; this implies that the isomorphisms of the proposition hold indeed in the homotopy category [Ri2, (proof of) Corollary 5.5].

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