Singular unitarity in “quantization commutes with reduction”

Let MM be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group GG. Let M//G=ϕ−1(0)/G=M0M//G=ϕ−1(0)/G=M0 be the symplectic quotient at value 0 of the moment map ϕϕ. The space M0M0 may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over M0M0 and the GG-invariant subspace of the quantum Hilbert space over MM. In this paper, without any regularity assumption on the quotient M0M0, 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.