Heckman, Kostant, and Steinberg formulas for symplectic manifolds

Let M be a compact symplectic manifold on which a compact connected Lie group, K, acts in a Hamiltonian fashion. In Part I we derive a formula for the pushforward of the Liouville measure via the moment map, which generalizes the Heckman formula for co-adjoint orbits. In Part II we derive a “quantum” analogue of this formula which extends to the symplectic setting classical multiplicity formulas of Kostant and Steinberg.