The diagonal distribution for the invariant measure of a unitary type symmetric space

Let Θ denote an involution for a simply connected compact Lie group U let K denote the fixed point set and let μ denote the U-invariant probability measure on U/K. Consider the geodesic embedding φ :U/K → U:u → uu ^-Θ of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for φ*μ relative to a compatible triangular decomposition of G the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation involving a Poisson structure on the dual symmetric space G₀/K discovered by Evens and Lu.