A general Cayley correspondence and higher rank Teichmuller spaces

Abstract

We introduce a new class of sl 2-triples in a complex simple Lie algebra g , which we call magical. Such an sl 2-triple canonically defines a real form and various decompositions of g . Using this decomposition data, we explicitly parametrize special connected components of the moduli space of Higgs bundles on a compact Riemann surface X for an associated real Lie group, hence also of the corresponding character variety of representations of pi 1X in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or SO p,q with 1 < p < q, and also constructs previously unknown components for the quaternionic real forms of E-6, E-7, E(8 )and F-4. The classification of magical sl 2-triples is shown to be in bijection with the set of Theta-positive structures in the sense of Guichard-Wienhard, thus the mentioned parametrization conjecturally detects all examples of higher rank Teichmuller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to Theta-positive Anosov representations, which indicate that this conjecture holds.