Moyal Product and Representations of Solvable Lie Groups

Using a parametrization for the universal covering O0 of any coadjoint orbit O of a solvable (connected and simply connected) Lie group G, we prove that the Moyal product on O0 gives a realization of the unitary representations canonically associated to the orbit O in Pukanszky′s theory by deformation of the algebra of C∞ functions on O0. By restriction to the standard covering Ô, we describe a structure of algebra on L2(Ô) and an adapted Fourier transform for type I group G.