Cohomology and geometry of Deligne--Lusztig varieties for ${\rm GL}_n$

TL;DR

This paper describes the cohomology of Deligne--Lusztig varieties for ${ m GL}_n$, providing new insights into their structure and cohomological properties over various fields, including mod p and p-adic contexts.

Contribution

It offers a comprehensive description of the cohomology groups of smooth compactifications of Deligne--Lusztig varieties for all Weyl group elements in ${ m GL}_n$, including new compactification constructions.

Findings

Computed cohomology groups of compactified Deligne--Lusztig varieties.

Derived mod p^m and p-adic étale cohomology results.

Established pseudo-rational singularities of Zariski closures.

Abstract

We give a description of the cohomology groups of the structure sheaf on smooth compactifications $\overline{X}(w)$ of Deligne--Lusztig varieties $X(w)$ for ${\rm GL}_n$, for all elements $w$ in the Weyl group. As a consequence, we obtain the ${\rm mod}\ p^m$ and integral $p$-adic \'{e}tale cohomology of $\overline{X}(w)$. Moreover, using our result for $\overline{X}(w)$ and a spectral sequence associated to a stratification of $\overline{X}(w)$, we deduce the ${\rm mod}\ p^m$ and integral $p$-adic \'{e}tale cohomology with compact support of $X(w)$. In our proof of the main theorem, in addition to considering the Demazure--Hansen smooth compactifications of $X(w)$, we show that a similar class of constructions provide smooth compactifications of $X(w)$ in the case of ${\rm GL}_n$. Furthermore, we show in the appendix that the Zariski closure of $X(w)$, for any connected reductive group $G$ and any $w$, has pseudo-rational singularities.