Some Examples Of Rickard Complexes

After a presentation of Broué’s conjecture for principal blocks with an abelian defect group, we describe a Rickard complex for GL2(q) arising from the `-adic cohomology of a DeligneLusztig 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 ` a prime number. Let P be a Sylow `-subgroup of G and assume P is abelian. Let H = NG(P ). Let O be the ring of integers of a finite unramified extension K of Q`, 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 ∆P = {(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 ∆P -projective permutation modules such that : C ⊗A C ' B in K(B ⊗ B) C ⊗B C ' A in K(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