QUANTIZATION OF SYMPLECTIC REDUCTION MICHAEL ANSHELEVICH Summary: QUANTIZATION OF SYMPLECTIC REDUCTION MICHAEL ANSHELEVICH Abstract. Symplectic reduction, also known as Marsden-Weinstein reduction, is an important construction in Poisson geometry. Following N.P. Landsman [22], we propose a quantization of this procedure by means of M. Rieffel's theory of induced representations. Here to an equivariant momentum map there corresponds an operator-valued rigged inner product. We define the operator- algebraic notions that are involved in this construction, and give a number of examples. Acknowledgements: This report is mainly based on a series of papers by N. Landsman ([19, 20, 21, 23] and especially [22]). I also would like to thank Prof. Rieffel for a helpful discussion. And of course many ideas in this report have been touched upon in Prof. Weinstein's course. Added in print: (1) The contemporary name for the structure with the rigged inner product is a Hilbert module. (2) Some related articles are: Survey of Strict Deformation Quantization by Bina Bhattacharyya (written for the same course, http://math.berkeley.edu/~alanw/242papers.html), and Morita Equivalence in Algebra and Geometry by Ralf Meyer, Von Neumann Algebras and Pois- son Manifolds by Dimitri Shlyakhtenko, both written for Prof. Weinstein's course Geometric Models for Noncommutative Algebras, http://math.berkeley.edu/~aweinst/277papers/277papers.html. 1. Introduction Symplectic reduction [1, 12, 28] (also known as Marsden-Weinstein reduction) is one of the basic constructions in symplectic geometry. Given a Lie group G and a hamiltonian action of G on a symplectic manifold S, one gets a momentum mapping J : S g from S to the dual of the Lie Collections: Mathematics