Non-Abelian symplectic cuts and the geometric quantization of noncompact manifolds - Dedicated to the memory of Moshe Flato

Let (M, omega) be a Hamiltonian U(n)-space with proper moment map. In the c ase where n = 1, Lerman constructed a one-parameter family of Hamiltonian U (1)-spaces M-xi called the symplectic cuts of M. We generalize this constru ction to Hamiltonian U(n) spaces. Motivated by recent theorems that show th at 'quantization commutes with reduction,' we next give a definition of geo metric 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) tha t such quantizations exist in general.