Unitarity in "quantization commutes with reduction"

Abstract

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.