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

TL;DR

This paper constructs a natural section of the Hitchin map for moduli spaces of G-Higgs bundles over Riemann surfaces, generalizing Hitchin's work to real groups and arbitrary line bundles, advancing the understanding of these moduli spaces.

Contribution

It introduces a new construction of a Hitchin map section for real reductive Lie groups, extending Hitchin's original complex case to real groups and arbitrary line bundles.

Findings

Constructed a Hitchin map section for real groups

Generalized Hitchin's section beyond complex groups

Connected the section to Hitchin-Teichmüller components

Abstract

We consider the moduli space of polystable $L$-twisted $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a real reductive Lie group, and $L$ is a holomorphic line bundle over $X$. Evaluating the Higgs field at a basis of the ring of polynomial invariants of the isotropy representation, one defines the Hitchin map. This is a map to an affine space, whose dimension is determined by $L$ and the degrees of the polynomials in the basis. Building up on the work of Kostant-Rallis and Hitchin, in this paper, as a first step in the study of the Hitchin map, we construct a section of this map. This generalizes the section constructed by Hitchin when $L$ is the canonical line bundle of $X$ and $G$ is complex. In this case the image of the section is related to the Hitchin-Teichm\"uller components of the moduli space of representations of the fundamental group of $X$ in $G_{\mathrm{Split}}$, a split real form of $G$. In fact, our construction is very natural in that we can start with the moduli space for $G_{\mathrm{Split}}$, instead of $G$, and construct the section for the Hitchin map for $G_{\mathrm{Split}}$ directly. The construction involves the notion of maximal split subgroup of a real reductive Lie group.