Topology of character varieties of Abelian groups

Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let Γ   be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(Γ,K)/KHom(Γ,K)/K is a strong deformation retract of the GIT quotient space Hom(Γ,G)⫽GHom(Γ,G)⫽G. Moreover, this result remains true when G   is replaced by its locus of real points. As a corollary, we determine necessary and sufficient conditions for the character variety Hom(Γ,G)⫽GHom(Γ,G)⫽G to be irreducible when G is connected and semisimple. For a general connected reductive G, analogous conditions are found to be sufficient for irreducibility, when Γ is free Abelian.