The Wigner function for general Lie groups and the wavelet transform

We build Wigner maps, functions and operators on general phase spaces arisi ng from a class of Lie groups, including non-unimodular groups (such as the affine group). The phase spaces are coadjoint orbits in the dual of the Li e algebra of these groups and they come equipped with natural symplectic st ructures and Liouville-type invariant measures. When the group admits squar e-integrable representations, we present a very general construction of a W igner function which enjoys all the desirable properties, including full co variance and reconstruction formulae. We study in detail the case of the af fine group on the line. In particular, we put into focus the close connecti on between the well-known wavelet transform and the Wigner function on such groups.