Kontsevich quantization and invariant distributions on lie groups

We study Kontsevich's deformation quantization for the dual of a finite-dimensional real Lie algebra (or superalgebra) g. In this case the Kontsevich -product defines a new convolution on S(g), regarded as the space of distributions supported at 0 ∈ g. For p ∈ S(g), we show that the convolution operator f → p f is a differential operator with analytic germ. We use this fact to prove a conjecture of Kashiwara and Vergne on invariant distributions on a Lie group G. This implies local solvability of bi-invariant differential operators on a Lie supergroup. In the special case of Lie groups, we get a new proof Duflo's theorem.