The Fourier Transform of Semi-Simple Coadjoint Orbits

LetMbe a closed coadjoint orbit of a real, connected, semi-simple Lie groupG, and letFM∈C−∞(g)Gbe its Fourier transform. In this paper we compute the restriction ofFMto the Lie algebra k of a maximal compact subgroupKofG. Using our technique of localization in equivariant cohomology developed recently, we extend previous results by M. Duflo, G. Heckman, M. Vergne, and I. Sengupta.