Notes on partial conjugation

TL;DR

This paper explores the method of partial conjugation in Coxeter groups, its applications to Lusztig's G-stable pieces, and recent developments in affine Deligne-Lusztig varieties, providing insights into conjugacy classes and group actions.

Contribution

It offers an exposition of partial conjugation actions, extending to applications in affine groups and recent research on conjugacy classes in loop groups.

Findings

Analysis of partial conjugation actions in Coxeter groups

Applications to Lusztig's G-stable pieces and affine generalizations

Discussion of recent work on conjugacy classes in loop groups

Abstract

In this notes, we will give an exposition of some results on the method of partial conjugation action. We first discuss the partial conjugation action of a parabolic subgroup of a Coxeter group. We then discuss some applications to Lusztig's $G$-stable pieces and its affine generalization. We also discuss some recent work on the $\s$-conjugacy classes of loop groups and affine Deligne-Lusztig varieties.