FIRST STEPS TOWARDS [FORMALITY, REDUCTION]=0?

One open question in deformation quantization is its compatibility with reduction in the case of Poisson manifolds. In this talk, we propose a way to study this compatibility by investigating the commutativity of a diagram of certain L-infinity-morphisms. On the classical side, one considers the curved DGLA of equivariant polyvector fields whose Maurer-Cartan elements are formal Poisson structures with formal momentum maps. The quantum analogues are the curved equivariant polydifferential operators with equivariant star products as Maurer-Cartan elements. The quantization is then described by an equivariant formality, i.e. an L-infinity-morphism between these curved DGLAs, and by the Kontsevich formality at the level of the Marsden-Weinstein reduced manifold. Concerning the classical and quantum reduction, we construct two reduction morphisms from the equivariant DGLAs into the polyvector fields and the polydifferential operators, respectively, on the reduced manifold. This is joint work with Chiara Esposito and Jonas Schnitzer.