Lusztig varieties for regular elements

TL;DR

This paper proves the irreducibility of Lusztig varieties associated with certain regular elements in a reductive group, extending previous results and providing tools for future research in affine Lusztig varieties.

Contribution

It establishes the irreducibility of Lusztig varieties for elements not contained in any standard parabolic subgroup, generalizing prior work on regular semisimple and unipotent elements.

Findings

Irreducibility of intersections with regular conjugacy classes

Irreducibility of Lusztig varieties for specific elements

Extension of previous irreducibility results

Abstract

Let $G$ be a connected reductive group over an algebraically closed field. Let $B$ be a Borel subgroup of $G$ and $W$ be the associated Weyl group. We show that for any $w \in W$ that is not contained in any standard parabolic subgroup of $W$, the intersection of the Bruhat cell $B w B$ with any regular conjugacy class of $G$ is always irreducible. We then prove that the associated Lusztig varieties are irreducible. This extends the previous work of Kim \cite{kim2020homology} on the regular semisimple and regular unipotent elements. The irreducibilitiy result of Lusztig varieties will be used in an upcoming work in the study of affine Lusztig varieties.