We show that every orbital measure, μx, on a compact exceptional Lie group or algebra has the property that for every positive integer either μxK ∈ L2 and the support of μxK has non-empty interior, or μ xK is singular to Haar measure and the support of μxK 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.
Bibliographical noteFunding Information:
This research was supported in part by NSERC and by the Chinese University of Hong Kong.
- Adjoint orbit
- Conjugacy class
- Exceptional Lie group/algebra
- Orbital measure
- Root system