Sur la cohomologie des compactifications de vari\'et\'es de Deligne-Lusztig

TL;DR

This paper investigates the étale cohomology of compactified Deligne-Lusztig varieties, providing new results for the general linear group and proposing conjectures for broader classes of reductive groups.

Contribution

It establishes a cohomological result for $GL_d$ compactifications and conjectures its extension to all reductive groups, advancing understanding of these varieties.

Findings

Proved a cohomology result for $GL_d$ compactifications.

Conjectured similar results for all reductive groups.

Enhanced understanding of the étale cohomology of Deligne-Lusztig compactifications.

Abstract

In this article, we study the \'etale cohomology of the compactification of Deligne-Lusztig varieties associated to a Coxeter element. We prove a result for the integral coefficients in the case of general linear group $GL_d$, and we conjecture that the similar result holds for general reductive groups.