Homology Class of a Deligne-Lusztig variety and its analogues

TL;DR

This paper studies the homology classes of Deligne-Lusztig varieties and their analogues under conjugation, providing explicit calculations and criteria for class equivalence in the Chow group.

Contribution

It introduces a method to compute classes of these varieties in the Chow group and establishes criteria for when different Weyl group elements yield the same class.

Findings

Calculated classes in the Chow group in terms of Schubert classes

Provided criteria for class equivalence under different Weyl group elements

Extended understanding of Deligne-Lusztig varieties under conjugation actions

Abstract

In this paper we consider Deligne-Lusztig varieties and their analogues when the Frobenius endomorphism is replaced with conjugation by an element in a group, especially a regular semisimple or regular unipotent one. We calculate their classes in the Chow group of the flag variety in terms of Schubert classes. Also we give some sufficient criteria when different elements in the Weyl group result in the same class.