Derived categories and Deligne-Lusztig varietiesII

Abstract

This paper is a continuation and a completion of [BoRo1]. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group -this is the modular version of a fundamental result of Lusztig, and the best approximation of the character-theoretic Jordan decomposition that can be obtained via Deligne-Lusztig varieties. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups.We also bring in local block theory methods: we show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of [BoRo1], the finer homotopy equivalence was unknown. As a consequence, the equivalences preserve defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations. An additional new feature is an extension of the results to disconnected reductive algebraic groups, which is required to handle local subgroups.

Rationality of blocks of quasi-simple finite groups

“…Section 3 studies the effect of ℓ-adic field automorphisms on characters of finite reductive groups in characteristic different from ℓ. In Section 4 we refine the results of the previous section to the setting of the key reduction theorem of Bonnafé-Dat-Rouquier [5,Section 7]. Theorems 4.4 and 4.5 are then used to prove almost all of the results for blocks of the finite groups of Lie type in non-defining characteristic in Section 5; the remaining cases in type E 8 are dealt with on an ad hoc basis in Section 5.1.…”

On the Bonnafé–Dat–Rouquier Morita equivalence

“…Using an argument from the unpublished note [3] we show that the extended bimodule induces a Morita equivalence and from this we can deduce the validity of [2,Theorem 7.7] in this case.…”

“…In this final section we prove that the extended bimodule induces a Morita equivalence. The following section borrows arguments from [3].…”

“…Note however that [2,Theorem 7.7] was announced without the assumption that this bimodule extends. As the proof of [2,Proposition 7.3] is incomplete, this assumption is necessary at the moment. One case where this assumption is easily seen to be satisfied is when the quotient subgroup N F /L F is cyclic, see Lemma 3 below.…”

On imprimitive representations of finite reductive groups in non-defining characteristic

“…In this section, we shall show that this Morita equivalence is compatible with Harish-Chandra induction which also implies that it preserves and reflects the property of being imprimitive. To do so we shall need a result by Bonnafé, Dat and Rouquier from [1] which gives a sufficient condition for Lusztig induction to depend only on the Levi subgroup (and not on the parabolic subgroup).…”