The diagonal distribution for the invariant measure of a unitary type   symmetric space
  Let U denote a simply connected compact Lie group, let K denote the fixed point set for an involutive automorphism of U, and let m denote the U-invariant probability measure on the symmetric space U/K. Consider the geodesic embedding U/K into U of Cartan. In this paper we compute the diagonal distribution for m, 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 G_0/K discovered by Evens and Lu. 
Doug Pickrell