Commuting elements in reductive groups and Higgs bundles on abelian varieties

TL;DR

This paper investigates the topology of representation spaces of abelian groups into reductive groups and shows that the moduli space of principal G-Higgs bundles on abelian varieties has a homotopy type determined solely by a maximal compact subgroup.

Contribution

It proves that the conjugation orbit space of representations into a maximal compact subgroup is a strong deformation retract of the space of all representations into the reductive group, linking Higgs bundle moduli to compact subgroups.

Findings

Hom(Z^{2d},K)/K is a strong deformation retract of Hom(Z^{2d},G)/G

The homotopy type of the Higgs bundle moduli space depends only on K

Representation spaces have topological structures determined by maximal compact subgroups

Abstract

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.