Higgs bundles for real groups and the Hitchin–Kostant–Rallis section

Abstract

We consider the moduli space of polystable L L -twisted G G -Higgs bundles over a compact Riemann surface X X , where G G is a real reductive Lie group and L L is a holomorphic line bundle over X X . Evaluating the Higgs field on a basis of the ring of polynomial invariants of the isotropy representation defines the Hitchin map. This is a map to an affine space whose dimension is determined by L L and the degrees of the polynomials in the basis. In this paper, we construct a section of this map and identify the connected components of the moduli space containing the image. This section factors through the moduli space for G s p l i t G_{\mathrm {split}} , a split real subgroup of G G . Our results generalize those by Hitchin, who considered the case when L L is the canonical line bundle of X X and G G is complex. In this case, the image of the section is related to the Hitchin–Teichmüller components of the moduli space of representations of the fundamental group of X X in G s p l i t G_{\mathrm {split}} , a split real form of G G . The construction involves the notion of a maximal split subgroup of a real reductive Lie group and builds on results by Kostant and Rallis.