Deligne-Lusztig varieties and period domains over finite fields

TL;DR

This paper demonstrates that the Drinfeld halfspace uniquely serves as both a Deligne-Lusztig variety and a period domain over finite fields, using cohomology theorems and affineness criteria.

Contribution

It establishes the uniqueness of the Drinfeld halfspace as a Deligne-Lusztig variety that is also a period domain over finite fields, linking cohomology results with geometric properties.

Findings

Drinfeld halfspace is the only DL-variety that is a period domain over finite fields.

A cohomology vanishing theorem for DL-varieties is compared with a non-vanishing theorem for period domains.

An affineness criterion for DL-varieties is discussed.

Abstract

We prove that the Drinfeld halfspace is essentially the only Deligne-Lusztig variety which is at the same time a period domain over a finite field. This is done by comparing a cohomology vanishing theorem for DL-varieties, due to Digne, Michel, and Rouquier, with a non-vanishing theorem for PD, due to the first author. We also discuss an affineness criterion for DL-varieties.