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 ) / K is a strong deformation retract of the GIT quotient space Hom ( 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 ) ⫽ 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.