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

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 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 $L$ 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 $G_{\mathrm {split}}$, a split real subgroup of $G$. Our results generalize those by Hitchin, who considered the case 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–Teichmüller components of the moduli space of representations of the fundamental group of $X$ in $G_{\mathrm {split}}$, a split real form of $G$. The construction involves the notion of a maximal split subgroup of a real reductive Lie group and builds on results by Kostant and Rallis.