  In this paper I construct a geometric transformation for generalized
1-motives which extends the Fourier-Mukai transformation for O-Modules on
abelian varieties, the geometric Fourier transformation for D-Modules on vector
spaces and the geometric Mellin transformation for D-Modules on tori. In
particular, I construct an equivalence of triangulated categories between the
derived category of quasi-coherent D-Modules on an abelian variety and the
derived category of quasi-coherent O-Modules on the universal extension of the
dual abelian variety. This equivalence has also been obtained by Mitchell
Rothstein.
