On measures derived from orbital integrals

Abstract

The present work develops a framework to derive piecewise polynomial measures arising from invariant measures on adjoint orbits in the context of compact and semisimple Lie groups. These measures are computed from orbital integrals via transformations on spaces of polynomials endowed with the apolar inner product. In the case of the unitary group, we obtain a formula for the moments of the projection of an orbital measure.