Sums of adjoint orbits and L2-singular dichotomy for SU(m)

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 L2-singular dichotomy: The convolution of invariant measures on adjoint orbits, or conjugacy classes, is either singular to Haar measure or in L2.