Oscar García-Prada (ICMAT-CSIC), Steven Bradlow (University of Illinois at Urbana-Champaign), Brian Collier (University of California Riverside), Peter B. Gothen (University of Porto) and André Oliveira (University of Trás-os-Montes e Alto Douro) publish their results in the journal Annals of Mathematics.Title: A general Cayley correspondence and higher rank Teichmüller spaces

It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2, R). Higher rank Teichmüller spaces are generalizations of this that exist in the moduli space of representations of the fundamental group into certain simple real non-compact Lie groups of higher rank. As with the usual Teichmüller space, these spaces consist entirely of discrete and faithful representations.

The first instances of such higher rank Teichmüller spaces were provided by Hitchin for split real groups in his seminal paper “Lie groups and Teichmüller space”, using the theory of Higgs bundles that he had introduced earlier in 1987. A subsequent class of higher rank Teichmüller spaces was given by the maximal Toledo invariant representations in groups of Hermitian type. The Higgs bundle construction of these for general groups of Hermitian type was given by Biquard–García-Prada–Rubio in the paper “Higgs bundles, Toledo invariant and the Cayley correspondence”, after several constructions by Bradlow, García-Prada, Gothen, and Mundet i Riera for classical groups. The combination of the theory of Higgs bundles with work by Labourie on Anosov representations and Burger–Iozzi–Wienhard provided the main tools to conclude that all representations in these components were discrete and faithful.

For some time, it was believed that split real groups and groups of Hermitian type were the only two classes of groups for which higher rank Teichmüller spaces existed. However, again the theory of Higgs bundles, and work by Guichard–Wienhard on positive structures on Lie groups, generalizing Lusztig positivity for split groups, indicated that this was not the case. A Higgs bundle construction for the family of groups SO(p, q) was given in the paper by M. Aparicio-Arroyo, S.B. Bradlow, B. Collier, O. García-Prada, P.B. Gothen, and A. Oliveira, SO(p, q)-Higgs bundles and higher Teichmüller components, Inventiones Mathematicae, 218 (2019), 197–299.

In a new paper, Oscar García-Prada (ICMAT-CSIC), Steven Bradlow (University of Illinois at Urbana-Champaign), Brian Collier (University of California Riverside), Peter B. Gothen (University of Porto) and André Oliveira (University of Trás-os-Montes e Alto Douro) give a general Higgs bundle classification and unified construction of higher rank Teichmüller spaces. In addition to recovering the cases mentioned above, they identify new groups for which higher rank Teichmüller spaces exist. The groups in question are quaternionic real forms of the exceptional groups  E6, E7, E8, and F4. An important tool for this construction is the introduction and classification of a special class of sl2-triples of a simple complex Lie algebra, that they call magical. Each situation appears by means of a Cayley-type correspondence for a different choice of magical triple. The classification of magical triples matches completely the classification of positive structures by Guichard–Wienhard, which plays a central role in the characterization of the higher rank Teichmüller spaces.