On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $G{L}_n$

Abstract

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a $p$-adic version of this theory started to emerge: there are $p$-adic Deligne-Lusztig spaces, whose cohomology encodes representation theoretic information for $p$-adic groups - for instance, it partially realizes local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic $\ell \ne p$ has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain $p$-adic Deligne-Lusztig spaces to Vignéras's modular local Langlands correspondence for ${\mathbf{GL}}_n$.

26 pages, minor changes, final version

References in corpus (1)

• Existence and properties of geometric quotients