MOYAL PRODUCT AND REPRESENTATIONS OF SOLVABLE LIE-GROUPS

Using a parametrization for the universal covering O-0 of any coadjoint orbit O of a solvable (connected and simply connected) Lie group G, we prove that the Moyal product on O-0 gives a realization of the unitary representations canonically associated to the orbit O in Pukanszky's theory by deformation of the algebra of C-infinity functions on O-0. By restriction to the standard covering O, we describe a structure of algebra on L(2)(O) and an adapted Fourier transform for type I group G. (C) 1995 Academic Press, Inc.