CHARACTER VARIETIES OF VIRTUALLY NILPOTENT KAHLER GROUPS AND G-HIGGS BUNDLES

Let G be a connected complex reductive affine algebraic group, and let K subset of G be a maximal compact subgroup. Let X be a compact connected Kahler manifold whose fundamental group Gamma is virtually nilpotent. We prove that the character variety Hom(Gamma, G)//G admits a natural strong deformation retraction to the subset Hom(Gamma, K)/K subset of Hom(Gamma, G)//G. The natural action of C* on the moduli space of G-Higgs bundles over X extends to an action of C. This produces the above mentioned deformation retraction.