 
Summary: QUANTIZATION OF SYMPLECTIC REDUCTION
MICHAEL ANSHELEVICH
Abstract. Symplectic reduction, also known as MarsdenWeinstein 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 operatorvalued 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 MarsdenWeinstein 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
