Abstract:Quantization of a Hamiltonian system with symmetry gives rise to a unitary representation U of the symmetry group G. It can be decomposed into a direct sum/integral of irreducible unitary representations of G. On the other hand, classical reduction describes the structure on the space of orbits of G contained in the phase space of the system.
In 1983 Guillemin and Sternberg proved, for a Kahler quantization of a compact symplectic manifold with a free action of a compact group G, that the multiplicity of the irreducible unitary representation of G corresponding to a coadjoint orbit O in the decomposition of U is determined by the Marsden-Weinstein reduction at O. I will discuss a program of obtaining similar results for strongly admissible polarizations without assuming that G or P are compact or that the action of G on P is proper.