Higher Asymptotics of Unitarity in ``Quantization Commutes with Reduction''

William Kirwin

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Submission date: 30. Oct. 2008
Pages: 14
Bibtex
MSC-Numbers: 53D50, 53D20, 41A60
Keywords and phrases: geometric quantization, symplectic reduction, semiclassical limit
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Abstract:
Let M be a compact Kaehler 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 h-bar. 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 h-bar, becomes unitary in the semiclassical limit as h-bar approaches 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 h-bar goes to 0.