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

Abstract

In this article, we study the mod $\ell$ cohomology of some Deligne-Lusztig varieties for ${\mathrm{GL}}_n(q)$. 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 Broué's abelian defect group conjecture.

Final version

References in corpus (1)

• Translation by the full twist and Deligne-Lusztig varieties