Derived categories and Deligne-Lusztig varieties II

TL;DR

This paper advances the understanding of blocks of finite groups of Lie type by establishing Morita equivalences via cohomological invariance of Deligne-Lusztig varieties, leading to new insights into derived categories and representation theory.

Contribution

It extends the Jordan decomposition of blocks, proves invariance of cohomology parts under certain subgroup variations, and shows these equivalences are splendid Rickard equivalences.

Findings

Blocks are Morita equivalent to subgroups associated with isolated elements.

Cohomology parts are invariant under variations of parabolic subgroups.

Deligne-Lusztig induced representations can generate the derived category.

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. 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 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 equivalence preserves defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations.