Singular unitarity in “quantization commutes with reduction”

Abstract

Let M be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group G . Let M / / G = ϕ − 1 ( 0 ) / G = M 0 be the symplectic quotient at value 0 of the moment map ϕ . 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.