Homotopy type of moduli spaces of G-Higgs bundles and reducibility of the nilpotent cone

Abstract

Let G be a real reductive Lie group, and HC the complexification of its maximal compact subgroup H⊂G. We consider classes of semistable G-Higgs bundles over a Riemann surface X of genus g⩾2 whose underlying HC-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of G-Higgs bundles over X to the moduli space of HC-bundles over X, in contrast with the situation when g=1, and to show reducibility of the nilpotent cone of the moduli space of G-Higgs bundles, for G complex.