Non-Abelian Symplectic Cuts and the Geometric Quantization of Noncompact Manifolds

Let (M, ω) be a Hamiltonian U(n)-space with proper moment map. In the case where n = 1, Lerman constructed a one-parameter family of Hamiltonian U(1)-spaces Mξ called the symplectic cuts of M. We generalize this construction to Hamiltonian U(n) spaces. Motivated by recent theorems that show that 'quantization commutes with reduction,' we next give a definition of geometric quantization for noncompact Hamiltonian G-spaces with proper moment map, and use our cutting technique to illustrate the proof of existence of such quantizations in the case of U(n) spaces. We then show (Theorem 1) that such quantizations exist in general.