Kac Diagrams for Elliptic Weyl Group Elements

TL;DR

This paper establishes an injective correspondence between elliptic conjugacy classes in Weyl groups and Kac diagrams for semisimple Lie algebras, providing explicit descriptions of this relationship.

Contribution

It proves the injectivity of the map from elliptic conjugacy classes to Kac diagrams and explicitly characterizes its image for semisimple Lie algebras.

Findings

The map from elliptic conjugacy classes to Kac diagrams is injective.

Explicit description of the image of this map.

Results hold in both standard and twisted settings.

Abstract

Suppose $\mathfrak{g}$ is a semisimple complex Lie algebra and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$. To the pair $(\mathfrak{g},\mathfrak{h})$ one can associate both a Weyl group and a set of Kac diagrams. There is a natural map from the set of elliptic conjugacy classes in the Weyl group to the set of Kac diagrams. In both this setting and the twisted setting, this paper (a) shows that this map is injective and (b) explicitly describes this map's image.