  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(Z^{2d},K)/K is a strong deformation retract of the space Hom(Z^{2d},G)/G of
equivalence classes of representations of Z^{2d} 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.
