The Drinfeld stratification for ${\rm GL}_n$

TL;DR

This paper introduces the Drinfeld stratification for Deligne--Lusztig varieties related to GLn, analyzing their cohomology and conjecturing their role in p-adic group representations.

Contribution

It defines a new stratification called the Drinfeld stratification and studies its cohomology, providing a complete description of the closed stratum and conjectures on its representation-theoretic significance.

Findings

Complete description of the unique closed stratum.

Cohomology analysis of the strata.

Conjectures on the stratification's role in p-adic representations.

Abstract

We define a stratification of Deligne--Lusztig varieties and their parahoric analogues which we call the Drinfeld stratification. In the setting of inner forms of GLn, we study the cohomology of these strata and give a complete description of the unique closed stratum. We state precise conjectures on the representation-theoretic behavior of the stratification. We expect this stratification to play a central role in the investigation of geometric constructions of representations of $p$-adic groups.