Affine representations of the fundamental group

Unitary representations of the fundamental group of a compact Kahler manifo ld correspond to polystable vector bundles (with vanishing Chern classes). Semisimple linear representations correspond to polystable Higgs bundles. I n this paper we find the objects corresponding to affine representations: t he linear part gives a Higgs bundle and the translation part corresponds to an element of a generalized de Rham cohomology. We use the same techniques to study representations into a parabolic subgroup of G1(n)(C).