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A general Cayley correspondence and higher Teichmüller components


We introduce a new class of 𝔰𝔩2-triples in a complex simple Lie algebra 𝔤, which we call magical. Such an 𝔰𝔩2-triple canonically defines a real form and various decompositions of 𝔤. Using this decomposition data, we explicitly parameterize 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 π1X 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≤q, and also constructs previously unknown components for the quaternionic real forms of E6, E7, E8 and F4. The classification of magical 𝔰𝔩2-triples is shown to be in bijection with the set of Θ-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmüller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to Θ-positive Anosov representations, which indicate that this conjecture holds.


Year of publication: 2021


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