Geometric quantization, reduction and decomposition of group representations

We consider a Hamiltonian action of a connected group G on a symplectic manifold (P, ω) with an equivariant momentum map and its quantization in terms of a K?hler polarization which gives rise to a unitary representation of G on a Hilbert space . If O is a co-adjoint orbit of G quantizable with respect to a K?hler polarization, we describe geometric quantization of algebraic reduction of J ⁻¹(O). We show that the space of normalizable states of quantization of algebraic reduction of J ⁻¹(O) gives rise to a projection operator onto a closed subspace of on which is unitarily equivalent to a multiple of the irreducible unitary representation of G corresponding to O. This is a generalization of the results of Guillemin and Sternberg obtained under the assumptions that G and P are compact and that the action of G on P is free. None of these assumptions are needed here. Dedicated to Vladimir Igorevich Arnold