From conjugacy classes in the Weyl group to unipotent classes, III

TL;DR

This paper generalizes a previous construction by defining a map from twisted conjugacy classes in the Weyl group to unipotent conjugacy classes in a possibly disconnected affine algebraic group, extending the theory to broader cases.

Contribution

It introduces a new map linking twisted conjugacy classes in the Weyl group to unipotent classes in disconnected groups, broadening the scope of prior work.

Findings

Established a new correspondence between twisted conjugacy classes and unipotent classes.

Extended the theory from connected to disconnected affine algebraic groups.

Provided a framework for analyzing unipotent classes in more general algebraic groups.

Abstract

Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element. In this paper we define a map from the set of "twisted conjugay classes" in W to the set of unipotent G^0-conjugacy classes in D, generalizing an earlier construction which applied when G is connected.