Strata of a disconnected reductive group

TL;DR

This paper introduces a new partition of a connected component of a possibly disconnected reductive group into finitely many strata, each comprising conjugacy classes of fixed dimension, generalizing known partitions for connected groups.

Contribution

It extends the concept of stratification to disconnected reductive groups, providing a new framework for understanding their conjugacy class structure.

Findings

Partition of $D$ into finitely many strata is established.

Each stratum is a union of conjugacy classes of fixed dimension.

Recovers known partitions when $D=G^0$.

Abstract

Let $D$ be a connected component of a possibly disconnected reductive group $G$ over an algebraic closed field. We define a partition of $D$ into finitely many Strata each of which is a union of $G^0$-conjugacy classes of fixed dimension. In the case where $D=G^0$ this recovers a known partition.