Representations of Abelian Algebraic Groups

There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adèle group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group.