Commuting elements in reductive groups and Higgs bundles on abelian varieties

Let G be a connected real reductive algebraic group, and let K be a maximal compact subgroup of G. We prove that the conjugation orbit space Hom(Z2d,K)/K is a strong deformation retract of the space Hom(Z2d,G)//G of equivalence classes of representations of Z2d into G. This is proved by showing that the homotopy type of the moduli space of principal G-Higgs bundles of vanishing rational characteristic classes on a complex abelian variety of dimension d depends only on K.