Geometric quantization, reduction and decomposition of group representations

Abstract

We consider a Hamiltonian action of a connected group G on a symplectic manifold (P, ω) with an equivariant momentum map \(J : P \rightarrow {\mathfrak{g}^{*}}\) and its quantization in terms of a Kahler polarization which gives rise to a unitary representation \({\mathcal{U}}\) of G on a Hilbert space \({\mathcal{H}}\). If O is a co-adjoint orbit of G quantizable with respect to a Kahler 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 \({\mathcal{H}}\) on which \({\mathcal{U}}\) 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.