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

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 ${\mathrm{GL}}_n$, for all elements $w$ in the Weyl group. As a consequence, we obtain the $\mathrm{mod}{p}^m$ and integral $p$-adic étale 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 $\mathrm{mod}{p}^m$ and integral $p$-adic étale 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 ${\mathrm{GL}}_n$. 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.

40 pages

Cited by in corpus (1)

• Geometry and cohomology of compactified Deligne--Lusztig varieties