# From conjugacy classes in the Weyl group to semisimple conjugacy classes

## TL;DR

This paper presents a uniform algorithm to map elliptic conjugacy classes of a Weyl group to semisimple conjugacy classes of a complex semisimple group, including the twisted case, enhancing understanding of their algebraic structure.

## Contribution

It introduces a novel, uniform algorithm for computing the map from elliptic conjugacy classes in Weyl groups to semisimple conjugacy classes in complex semisimple groups, including twisted cases.

## Key findings

- Algorithm effectively computes the conjugacy class mappings.
- Provides a unified approach applicable to both untwisted and twisted cases.
- Enhances the understanding of the relationship between Weyl group classes and group conjugacy classes.

## Abstract

Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple conjugacy classes of $G$. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.