Momentum Maps, Dual Pairs and Reduction in Deformation Quantization∗

This paper is a brief survey of momentum maps, dual pairs and reduction in deformation quantization. We recall the classical theory of momentum maps in Poisson geometry and present its quantum counterpart. We also discuss quantization of momentum maps and applications of quantum momentum maps to quantum versions of Marsden-Weinstein reduction. This paper is organized as follows. We recall the classical notions of momentum map, hamiltonian action and symplectic dual pair in Section 1. In Section 2.2 we discuss quantum momentum maps and show how they produce examples of quantum dual pairs in Section 2.4. The problem of quantizing momentum maps is briefly discussed in Section 2.5. We mention in Section 2.6 how quantum momentum maps play a role in quantum versions of the classical Marsden-Weinstein reduction procedure. 1 Momentum Maps, Symplectic Dual Pairs and Reduction Let (M,ω) be a symplectic manifold and G a Lie group. We will assume for simplicity that both G and M are connected. Let ψ : G×M −→M be an action of G on M satisfying ψ∗ gω = ω. Such an action is called symplectic. We will refer to the triple (M,ω, ψ) as a symplectic G-space. A momentum map for a symplectic action is a C∞ map J :M −→ g∗ (1.1) such that ivMω = dJ , (1.2) where vM denotes the infinitesimal generator of the action corresponding to v ∈ g and Jv ∈ C∞(M) is defined by Jv(x) = 〈J(x), v〉. Note that J naturally defines a linear map J : g −→ C∞(M) by J(v) = Jv, called a comomentum map. Conversely, any linear map J : g −→ C∞(M) ∗This is a survey paper for the Berkeley Math277 course Topics in Differential Geometry: Momentum mappings, taught by Alan Weinstein in Fall 2000. I would like to thank Prof. Alan Weinstein for many useful comments.