Characters of Elements of Finite Order in Lie Groups

Abstract

In this paper we use the theory of elements of finite order (EFO) as a new and very effective tool for discrete methods in simple Lie groups. The EFO provide a systematic way of discretely approximating the group. Their character values allow us to systematically determine information about Lie groups and their representations for groups well beyond the range of standard methods. We discuss the theory of EFO, the use of algebraic number fields to single out finite classes of them, and methods of explicitly determining such classes. We introduce an algorithm for effectively computing their character values which utilizes double coset decompositions in the Weyl group and a fast algorithm for determining weight space multiplicities which we developed earlier. The methods are uniform for all simple Lie groups. We briefly discuss a number of applications of this work and finish with a number of tables (including some for $E_6 $) of EFO and their character values.