Let M be a closed coadjoint orbit of a real, connected, semi-simple Lie group G, and let F(M) be its Fourier transform. In this paper we compute the restriction of F(M) to the Lie algebra of a maximal compact subgroup K of G. Using our technique of localization in equivariant cohomology developed recently, we extend previous results by M. Duflo, G. Heckman, M. Vergne, and I. Sengupta.