ar X iv : 0 81 0 . 49 33 v 1 [ m at h . SG ] 2 7 O ct 2 00 8 Higher Asymptotics of Unitarity in “ Quantization Commutes with Reduction ”

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In [Invent. Math. 67 (1982), no. 3, 515–538], Guillemin and Sternberg showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M/G. This map, though, is not in general unitary, even to leading order in ħ. In [Comm. Math. Phys. 275 (2007), no. 2, 401–422], Hall and the author showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed ħ, becomes unitary in the semiclassical limit ħ → 0. The unitarity of the classical Guillemin–Sternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M/G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as ħ→ 0.