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Quantification géométrique et réduction symplectique. (Geometric quantization and symplectic reduction). (French) Zbl 1037.53062
Bourbaki seminar. Volume 2000/2001. Exposés 880-893. Paris: Société Mathématique de France (ISBN 2-85629-130-9/pbk). Astérisque 282, 249-278, Exp. No. 888 (2002).
The author and many other researchers (J. J. Duistermaat, J. Fogarty, V. Guillemin, L. C. Jeffrey, F. C. Kirwan, B. Kostant, E. Lerman, J. Marsden, E. Meinrenken, D. Mumford, P.-E. Paradan, S. Sternberg, R. Sjamaar, C. Teleman, Y. Tian, A. Weinstein, E. Witten, S. Wu, W. P. Zhang, et.al.) contributed to geometric quantization and symplectic reduction. This paper is a survey on many related topics. The geometry and topology of moduli spaces, via the connections between symplectic reduction and Mumford’s geometric invariant theory are presented. The author gives a nice overview of recent developments concerning symplectic cutting, singular reductions, the cohomology ring of symplectic reductions, the Guillemin-Sternberg conjecture (quantization commutes with reduction) stating that a part of the equivariant index of the manifold, which is invariant under the action of the Lie group, is equal to the Riemann-Roch number of the symplectic quotient of the manifold, provided that the quotient is nonsingular, and the Meinrenken-Sjamaar theorems generalizing the Guillemin-Sternberg theorem.
For the entire collection see [Zbl 1007.00024].

MSC:
53D50 Geometric quantization
53D20 Momentum maps; symplectic reduction
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
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