“量子化与约化交换”中么正性的高阶渐近性
Higher Asymptotics of Unitarity in "Quantization Commutes with Reduction"
摘要：
设M是紧Kaehler流形，紧李群G具有Hamilton作用.在[发明。Math.67（1982），no.~ 3，515- 538]，Guillemin和Sternberg证明了M上的G-不变量子希尔伯特空间与辛商M//G上的量子希尔伯特空间之间存在几何上的自然同构.然而，这个映射在一般情况下不是酉的，甚至在h-巴中也不是首序的。在[Comm.Math.Phys.275（2007），no. 2，401- 422]，霍尔和作者表明，当包括的metaplectic修正，一个没有得到一个地图，而不是在一般酉任何固定的h-酒吧，成为酉的半经典极限，h-酒吧走向零。（参见：Ma和Zhang在[C. R.数学学院Sci. Paris 341（2005），no.~ 5，297- 302]和[Astérisque No. 318（2008），viii+154 pp.]）。经典Guillemin-Sternberg映射及其相应的映射的么正性是通过辛商M//G上的某些函数来度量的。在本文中，我们给出了这些函数的精确表达式，并计算了它们的完全渐近展开的h-杆趋于零。
英文摘要：
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 that h-bar goes to zero. (cf. the work of Ma and Zhang in [C. R. Math. Acad. Sci. Paris 341 (2005), no.~5, 297--302], and [Astérisque No. 318 (2008), viii+154 pp.]). 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 zero.
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