Sums of Adjoint Orbits

TL;DR

This paper characterizes when sums of adjoint orbits in compact Lie algebras and products of conjugacy classes in compact Lie groups contain open sets, using root data and eigenvalue multiplicities, with specific results for su(m) and SU(m).

Contribution

It provides a complete characterization for su(m) and SU(m) regarding when sums or products contain open sets, based on eigenvalue multiplicities, and establishes an L^2--singular dichotomy for convolutions.

Findings

Exact criteria for su(m) and SU(m) for open set containment.

Determination of when convolutions are in L^2 or singular.

Identification of eigenvalue multiplicity conditions.

Abstract

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g, or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su(m) and products of conjugacy classes in SU(m) contain an open set, in terms of the highest multiplicities of eigenvalues. For su(m) and SU(m) we show L^2--singular dichotomy: The convolution of invariant measures on adjoint orbits, or conjugacy classes, is either singular to Haar measure or in L^2.