Jordan classes and Lusztig strata in disconnected reductive groups

TL;DR

This paper studies Jordan classes and Lusztig strata in disconnected reductive groups, providing descriptions of their closures and proving local closedness of Lusztig strata.

Contribution

It offers a new description of the regular part of Jordan class closures and establishes local closedness of Lusztig strata in non-connected groups.

Findings

Description of the regular part of Jordan class closures via induction.

Lusztig strata are shown to be locally closed.

Provides structural insights into disconnected reductive groups.

Abstract

Let $G$ be a non-connected reductive algebraic group over an algebraically closed field $\mathbb{K}$ and let $D$ be a connected component of $G$. We investigate Jordan classes of $D$ and we obtain a description of the regular part of the closure of a Jordan class in terms of induction of $G^{\circ}$-orbits. We use this result to show that Lusztig strata in a non-connected reductive algebraic group are locally closed.