The L²-singular dichotomy for exceptional lie groups and algebras

We show that every orbital measure, μ_x, on a compact exceptional Lie group or algebra has the property that for every positive integer either μ_x^K ∈ L² and the support of μ_x^K has non-empty interior, or μ _x^K is singular to Haar measure and the support of μ_x^K has Haar measure zero. We also determine the index k where the change occurs; it depends on properties of the set of annihilating roots of x. This result was previously established for the classical Lie groups and algebras. To prove this dichotomy result we combinatorially characterize the subroot systems that are kernels of certain homomorphisms.