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 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-Teichmüller 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.

50 pages, we have made minor corrections to version 1 and suppressed the last section

References in corpus (2)

• Higgs bundles, the Toledo invariant and the Cayley correspondence
• Fibration de Hitchin et endoscopie

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• Gradings on the real form ${\mathfrak{e}}_{6,-14}$
• Higgs bundles over elliptic curves for real groups
• Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system
• Generalized Punctual Hilbert Schemes and $\mathfrak{g}$-complex structures
• Multiplicative Higgs bundles and involutions