  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.

Group Theory;Differential Geometry;Functional Analysis;Symplectic Geometry