Centralizers of maximal regular subgroups in simple Lie groups and relative congruence classes of representations

TL;DR

This paper provides a comprehensive description of the structure of centralizers of maximal regular subgroups in compact simple Lie groups, including explicit formulas for their action on irreducible representations.

Contribution

It introduces a uniform framework for describing these centralizers across all types and ranks of simple Lie groups, with explicit formulas for their actions.

Findings

Centralizers are classified as finite cyclic, rank 1 continuous, or products thereof.

Explicit formulas for the action of centralizers on irreducible representations are provided.

The description applies uniformly across all types and ranks of compact simple Lie groups.

Abstract

In the paper we present a new, uniform and comprehensive description of centralizers of the maximal regular subgroups in compact simple Lie groups of all types and ranks. The centralizer is either a direct product of finite cyclic groups, a continuous group of rank 1, or a product, not necessarily direct, of a continuous group of rank 1 with a finite cyclic group. Explicit formulas for the action of such centralizers on irreducible representations of the simple Lie algebras are given.