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 .
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