Characteristic classes of star products on Marsden-Weinstein reduced symplectic manifolds

In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map but in the Cartan model of equivariant cohomology. As a byproduct we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.