Moment maps, their quantization and reduction

Given a Lie group acting on a Poisson manifold and a moment map, Marsden-Weinstein reduction allows us to reduce the manifold and the Poisson structure. This reduction procedure extends to invariant star products and quantum moment maps via the BRST approach. On the other hand, invariant Poisson structures (resp. star products) together with corresponding moment maps (resp. quantum moment maps) can be described as (curved) Maurer Cartan elements of certain DGLA's. The main aim of this talk is to relate these DGLA's via an $L_\infty$-morphism and explain that, under mild assumptions on the group action, it is a quasi-isomorphism. This implies that we can identify the Maurer Cartan elements, up to gauge equivalence in both DGLA's and hence we classify invariant star products admitting quantum moment maps. In the last part of this talk, I try to connect the two equivariant DGLA's to the BRST approach of Marsden-Weinstein reduction. This is a work in progress with Chiara Esposito, Niek de Kleijn and Andreas Kraft.