On E-functions of Semisimple Lie Groups

TL;DR

This paper develops continuous and discrete E-transform methods for class functions on compact semisimple Lie groups, focusing on their expansions into special functions invariant under even Weyl group actions, especially in rank 3 cases.

Contribution

It introduces new E-transform techniques for semisimple Lie groups, extending previous work to higher dimensions and detailed rank 3 cases.

Findings

E-transforms for semisimple Lie groups of rank 3 are explicitly described.

Two types of even Weyl groups are distinguished and analyzed.

Transform methods are applicable to both simple and non-simple compact semisimple Lie groups.

Abstract

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of $G$. We distinguish two cases of even Weyl groups -- one is the direct product of even Weyl groups of simple components of $G$, the second is the full even Weyl group of $G$. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two -- we describe in detail $E-$transforms of semisimple Lie groups of rank 3.