A general Cayley correspondence and higher rank Teichmüller spaces

Abstract

We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$. 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 $\pi_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $\mathrm{SO}_{p,q}$ with $1<p\leq q$, and also constructs previously unknown components for the quaternionic real forms of $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{F}_4$. The classification of magical $\mathfrak{sl}_2$-triples is shown to be in bijection with the set of $\Theta$-positive structures in the sense of Guichard–Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher rank Teichmüller 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.

Authors

Steven Bradlow

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Brian Collier

Department of Mathematics, University of California Riverside, Riverside, CA 92521, USA

Oscar García-Prada

Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Carbrera, 13--15, 28049 Madrid, Spain

Peter B. Gothen

Centro de Matemática da Universidade do Porto and Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

André Oliveira

Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, Escola de Ciências e Tecnologia, Quinta de Prados, 5001 - 801 Vila Real, Portugal

Current address:

Centro de Matemática da Universidade do Porto and Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal