Deformation quantization and invariant distributions

We study Kontsevich's deformation quantization for the dual of a finite-dim ensional Lie algebra g. Regarding elements of S(g) as distributions on g, w e show that the *-multiplication operator (r bar right arrow r * p) is a di fferential operator with analytic germ at 0. We use this to establish a con jecture of Kashiwara and Vergne which, in turn, gives a new proof of Duflo' s result on the local solvability of bi-invariant differential operators on a Lie group. (C) 2000 Academie des sciences/Editions scientifiques et medi cales Elsevier SAS.