  Consider a compact prequantizable symplectic manifold M on which a compact
Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with
reduction'' theorem asserts that the G-invariant part of the equivariant index
of M is equal to the Riemann-Roch number of the symplectic quotient of M,
provided the quotient is nonsingular. We extend this result to singular
symplectic quotients, using partial desingularizations of the symplectic
quotient to define its Riemann-Roch number. By similar methods we also compute
multiplicities for the equivariant index of the dual of a prequantum bundle,
and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is
invariant under symplectic reduction.
