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A general Cayley correspondence and higher Teichmüller spaces

Steve Bradlow, Brian Collier, Oscar Garcia-Prada, Peter Gothen, André Oliveira

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 $π_1X$ 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 $Θ$-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.

A general Cayley correspondence and higher Teichmüller spaces

Abstract

We introduce a new class of -triples in a complex simple Lie algebra , which we call magical. Such an -triple canonically defines a real form and various decompositions of . Using this decomposition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface for an associated real Lie group, hence also of the corresponding character variety of representations of in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or with , and also constructs previously unknown components for the quaternionic real forms of , , and . The classification of magical -triples is shown to be in bijection with the set of -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 -positive Anosov representations, which indicate that this conjecture holds.

Paper Structure

This paper contains 36 sections, 67 theorems, 194 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Nilpotents and magical $\mathfrak{sl}_2$-triples
  3. Nilpotents and $\mathfrak{sl}_2$-triples
  4. Magical $\mathfrak{sl}_2$-triples
  5. The canonical real form associated to a magical nilpotent
  6. Real nilpotents and the Sekiguchi correspondence
  7. Classification of magical $\mathfrak{sl}_2$-triples
  8. The classification theorem
  9. The proof
  10. Explicit data and real forms for magical $\mathfrak{sl}_2$-triples
  11. $\mathfrak{sl}_2$-data
  12. The centralizer $\mathfrak{c}$ and its centralizer
  13. The Cayley real form
  14. Lie theory structure for magical nilpotents in exceptional Lie algebras
  15. Higgs bundles and the Cayley map
  16. ...and 21 more sections

Key Result

Proposition 2.2

The subspace $V\subset\mathfrak{g}$ is a subalgebra such that $V_0=W_0$ is a reductive subalgebra and $\bigoplus_{j>0}V_j$ is a nilpotent subalgebra. In addition, for each $j,k$, the subspace $W_j\cap \mathfrak{g}_k \subset\mathfrak{g}$ is preserved by bracketing with $W_0$.

Theorems & Definitions (168)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Conjecture
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 158 more