Abstract:

Consider a connected compact quantizable Kaehler manifold equipped with a Hamiltonian action of a connected compact Lie group. It is known that, as vector spaces, there is a natural isomorphism between the invariant subspace of the quantum Hilbert space over the manifold and the quantum Hilbert space over the symplectic quotient at value zero of the moment map. Without assuming that the symplectic quotient is a manifold, we discuss the inner products of the two quantum Hilbert spaces under the above natural isomorphism, and establish asymptotic unitarity of a modified isomorphism under a "metaplectic correction'' of the two quantum Hilbert spaces.