Deligne-Lusztig restriction of a Gelfand-Graev module

TL;DR

This paper investigates how Deligne-Lusztig restriction affects Gelfand-Graev modules in the context of finite groups of Lie type, revealing that the restriction results in a shifted Gelfand-Graev module.

Contribution

It introduces a new understanding of the behavior of Gelfand-Graev modules under Deligne-Lusztig restriction using Deodhar's decomposition.

Findings

Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module.

Utilizes Deodhar's decomposition of double Schubert cells.

Provides insights into the cohomology of Deligne-Lusztig varieties associated to tori.

Abstract

Using Deodhar's decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module.