  Let $M$ be a connected compact quantizable K\"ahler manifold equipped with a
Hamiltonian action of a connected compact Lie group $G$. Let
$M//G=\phi^{-1}(0)/G=M_0$ be the symplectic quotient at value 0 of the moment
map $\phi$. The space $M_0$ may in general not be smooth. It is known that, as
vector spaces, there is a natural isomorphism between the quantum Hilbert space
over $M_0$ and the $G$-invariant subspace of the quantum Hilbert space over
$M$. In this paper, without any regularity assumption on the quotient $M_0$, we
discuss the relation between the inner products of these two quantum Hilbert
spaces under the above natural isomorphism; we establish asymptotic unitarity
to leading order in Planck's constant of a modified map of the above
isomorphism under a ``metaplectic correction'' of the two quantum Hilbert
spaces.

Symplectic Geometry