Cohomology of Deligne–Lusztig Varieties, Broué’s Conjecture, and Brauer Trees

In this paper, we present a conjecture on the degree of unipotent characters in the cohomology of particular Deligne–Lusztig varieties for groups of Lie type, and derive consequences of it. These degrees are a crucial piece of data in the geometric version of Broué’s abelian defect group conjecture, and can be used to verify this geometric conjecture in new cases. The geometric version of Broué’s conjecture should produce a more combinatorially defined derived equivalence, called a perverse equivalence. We prove that our conjectural degree is an integer (which is not obvious) and has the correct parity for a perfect isometry. After giving some evidence for its truth, we then give an application of this conjecture, using the framework of Broué’s abelian defect group conjecture, to produce conjectural Brauer trees for the unipotent blocks of E7 and E8, the last finite groups of Lie type for which Brauer trees are not known. This paper is a contribution both to the understanding of Deligne–Lusztig varieties and to the conjectural description of the exact form of a derived equivalence proving Broué’s conjecture for groups of Lie type.