Abstract
We consider an Hamiltonian action of a connected Lie group G on a symplectic manifold P with an equivariant momentum map J from P to the dual of the Lie algebra of G and its quantization in terms of a Kahler polarization which gives rise to a unitary representation U of G on a Hilbert space H. If O is a co-adjoint orbit of G quantizable with respect to a Kahler polarization, we describe geometric quantization of the pre-image K of O under the momentum map J. We show that the space of normalizable states of quantization of algebraic reduction of K gives rise to a projection operator on a closed subspace of H on which U is unitarily equivalent to a multiple of the irreducible unitary representation of G corresponding to O. This is a generalization of the results by Guillemin and Sternberg obtained the under assumptions that G and P are compact and that the action of G on P is free. None of these assumptions are needed here.