Marsden-Weinstein reduction for symplectic connections

We propose a reduction procedure for symplectic connections with symmetry. This is applied to coadjoint orbits whose isotropy is reductive. Key-words: Marsden-Weinstein reduction, symplectic connections, Hamiltonian group actions MSC 2000: 53C15, 53D20 1 e-mail: [email protected] 2 Research supported by the Marie Curie Fellowship Nr. HPMF-CT-1999-00062 3 e-mail: [email protected] 4 Research supported by an ARC of the “Communauté française de Belgique” 2 marsden-weinstein reduction of symplectic connections 0. The aim of this paper is to show that under very mild conditions, MarsdenWeinstein reduction is “compatible” with a symplectic connection. This means that if a symplectic manifold (M,ω) is endowed with a strongly Hamiltonian action of a connected Lie group G and with a G-invariant symplectic connection ∇, there is a natural way to construct a symplectic connection ∇ on a reduced manifold (M , ω). The construction always works when G is compact, and in many non-compact cases as well. The interest of the construction if two-fold. First it leads to interesting examples of symplectic connections when (M,ω) is a very simple symplectic manifold and G is, for example, one-dimensional or multidimensional but abelian (see [2]). Secondly, it may be a useful tool in dealing with the general problem of commutation of quantization and reduction in the framework of deformation quantization. The paper is organized as follows. We first recall some classical results about strongly Hamiltonian actions. In the second paragraph we show how to construct a reduced connection with a technical assumption and we prove that this is always possible in the compact case. The third paragraph collects several examples where this construction gives interesting results. We finally indicate some possible further developments. 1. Let (M,ω) be a symplectic manifold and let σ:G × M → M be a strongly Hamiltonian action of a connected Lie group G, (g, x) 7→ g · x, which we will assume to be effective. If g is the Lie algebra of G, we denote by J :M → g the corresponding G-equivariant momentum map: i(X)ω = d(JX), ∀X ∈ g (1) where X is the infinitesimal generator of the action corresponding to X : X x = d dt exp(−tX) · x ∣