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Higher asymptotics of unitarity in “quantization commutes with reduction”

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Abstract

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. Guillemin and Sternberg (Invent Math 67:515–538, 1982, no. 3), 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 \({\hslash}\). Hall and Kirwin (Commun Math Phys 275:401–422, 2007, no. 2), showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed \({\hslash}\), becomes unitary in the semiclassical limit \({\hslash\rightarrow0}\) (cf. the work of Ma and Zhang (C R Math Acad Sci Paris 341:297–302, 2005, no. 5), and (Astérisque No. 318:viii+154, 2008). 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 \({\hslash\rightarrow0}\).

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Correspondence to William D. Kirwin.

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Kirwin, W.D. Higher asymptotics of unitarity in “quantization commutes with reduction”. Math. Z. 269, 647–662 (2011). https://doi.org/10.1007/s00209-010-0755-9

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  • DOI: https://doi.org/10.1007/s00209-010-0755-9

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