On the Cohomology of Deligne-Lusztig Varieties

TL;DR

This paper proposes a conjecture on the degrees of unipotent characters in the cohomology of Deligne-Lusztig varieties, supporting Broué's conjecture by establishing integer and parity properties, and verifying perverse equivalences in specific cases.

Contribution

It introduces a conjecture on unipotent character degrees in Deligne-Lusztig varieties and demonstrates its implications for Broué's conjecture, including integer and parity properties and verification of perverse equivalences.

Findings

Conjecture on unipotent character degrees is consistent with known cases.

Proved the degree is an integer and has correct parity.

Verified perverse equivalences for certain unipotent blocks.

Abstract

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 necessary piece of data in the geometric version of Brou\'e's abelian defect group conjecture, and can be used to verify this geometric conjecture in new cases. The geometric version of Brou\'e'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, and verify that it induces a perverse equivalence for all unipotent blocks of groups of Lie type with cyclic defect groups, whenever the shape of the Brauer tree is known (i.e., not E7 and E8). It has also been used to find perverse equivalences for some non-cyclic cases. This paper is a contribution to the conjectural description of the exact form of a derived equivalence proving Brou\'e's conjecture for groups of Lie type.