Mod $\ell$ cohomology of some Deligne-Lusztig varieties for $\operatorname{GL}_n(q)$

TL;DR

This paper investigates the mod $\, ext{ell}$ cohomology of specific Deligne-Lusztig varieties for $ ext{GL}_n(q)$, establishing torsion-free conditions and explicit computations relevant to Broué's conjecture.

Contribution

It proves torsion-freeness of cohomology groups under certain conditions and verifies a partial-tilting property crucial for Broué's conjecture in this context.

Findings

Cohomology groups are torsion-free under specified conditions.

Explicit computation of cohomology groups is achieved.

Cohomology complex satisfies partial-tilting condition.

Abstract

In this article, we study the mod $\ell$ cohomology of some Deligne-Lusztig varieties for $\operatorname{GL}_n(q)$. We prove that the cohomology groups of these varieties are torsion-free under some conditions on the characteristic. Under the torsion-free assumption we can compute the cohomology groups explicitly and we prove that the cohomology complex satisfies partial-tilting condition, which is one of the necessary conditions in the geometric version of Brou\'{e}'s abelian defect group conjecture.