Description
Higher Teichmüller theory is the study of connected components in the $G^R$-character variety consisting entirely of discrete and faithful surface group representations. Under the nonabelian Hodge correspondence, various problems in higher Teichmüller theory can be approached by using topological methods on the $G^R$-Higgs bundle moduli space.
Following recent developments of Theta-positivity on the character variety side, it has been proposed by Bradlow, Collier, Garcia-Prada, Gothen and Oliveira that the corresponding components on the Higgs bundle moduli space are characterized by Slodowy slices of magical $\mathfrak{sl}_2$-triples. In this talk, I will explain how the above framework can be extended to include the case of maximal components of a nontube type Hermitian Lie group by introducing the notion of odd magical triples, further supporting the expectation that all higher Teichmüller components arise via a Cayley correspondence.