  Let M be a compact Kahler manifold equipped with a Hamiltonian action of a
compact Lie group G. In this paper, we study the geometric quantization of the
symplectic quotient M//G. Guillemin and Sternberg [Invent. Math. 67 (1982),
515--538] have shown, under suitable regularity assumptions, that there is a
natural invertible map between the quantum Hilbert space over M//G and the
G-invariant subspace of the quantum Hilbert space over M.
  We prove that in general the natural map of Guillemin and Sternberg is not
unitary, \textit{even to leading order in Planck's constant}. We then modify
the quantization procedure by the "metaplectic correction" and show that in
this setting there is still a natural invertible map between the Hilbert space
over M//G and the G-invariant subspace of the Hilbert space over M. We then
prove that this modified Guillemin--Sternberg map is asymptotically unitary to
leading order in Planck's constant.
