A general Cayley correspondence and higher rank Teichmuller spaces

We introduce a new class of sl2-triples in a complex simple Lie algebra g,which we call magical. Such an sl2-triple canonically de_nes 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 _1X.in the associated real Lie group. This recovers known components when.the real group is split, Hermitian of tube type, or SOp;q with 1 < p 6 q,and also constructs previously unknown components for the quaternionic.real forms of E6, E7, E8 and F4. The classi_cation of magical sl2-triples is.shown to be in bijection with the set of _-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 _-positive Anosov representations,which indicate that this conjecture holds.