Heckman, Kostant, and Steinberg formulas for symplectic manifolds

Abstract

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 J∗(β M) which generalizes the Heckman formula for co-adjoint orbits. (Here J is the moment map associated with the action of K on M and β M the Liouville measure on M.) In Part II we derive a “quantum” analogue of this formula which extends to the symplectic setting classical multiplicity formulas of Kostant and Steinberg.