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Generic Properties of Hitchin Representations

Hongtaek Jung

TL;DR

The paper develops a framework to study generic properties of $G$-Hitchin representations for orbifold groups by leveraging the Goldman product formula and a new lifting theorem for Goldman flows. It proves that, for a broad class of split real Lie groups, the Jordan projections along regular elements generically avoid fixed linear constraints, and that generic orbifold Hitchin representations are strongly dense, extending known density results to additional Lie types. The approach unifies eigenvalue-avoidance with flow-based deformations and uses finite covers to transfer results from surfaces to orbifolds. The work provides concrete implications for several Lie types and offers explicit first-variation analyses in the PSL$_3(\mathbb{R})$ case, along with open questions connecting geometry, representation theory, and dynamics.

Abstract

Let $G$ be a split real form of a complex simple adjoint group whose Weyl group contains $-1$, let $λ$ be the Jordan projection of $G$, and let $S$ be a closed orientable surface of genus at least 2. For a $G$-Hitchin representation $ρ$, we define the set $J(ρ):=\{λ(ρ(x))\,|\,x\inπ_1(S)\setminus \{1\}\}$. Choose any hyperplane $H$ in the maximal abelian subalgebra of the Lie algebra of $G$. Our main result shows that, for a generic $G$-Hitchin representation $ρ$, we have $J(ρ)\cap H=\emptyset$. As an application, we prove that generic orbifold Hitchin representations are strongly dense. This extends the result of Long, Reid, and Wolff for the Hitchin representations of surface groups. Our theorem also shows that the split real forms of many simple adjoint Lie groups contain strongly dense orbifold fundamental groups, partially generalizing the work of Breuillard, Guralnick, and Larsen.

Generic Properties of Hitchin Representations

TL;DR

The paper develops a framework to study generic properties of -Hitchin representations for orbifold groups by leveraging the Goldman product formula and a new lifting theorem for Goldman flows. It proves that, for a broad class of split real Lie groups, the Jordan projections along regular elements generically avoid fixed linear constraints, and that generic orbifold Hitchin representations are strongly dense, extending known density results to additional Lie types. The approach unifies eigenvalue-avoidance with flow-based deformations and uses finite covers to transfer results from surfaces to orbifolds. The work provides concrete implications for several Lie types and offers explicit first-variation analyses in the PSL case, along with open questions connecting geometry, representation theory, and dynamics.

Abstract

Let be a split real form of a complex simple adjoint group whose Weyl group contains , let be the Jordan projection of , and let be a closed orientable surface of genus at least 2. For a -Hitchin representation , we define the set . Choose any hyperplane in the maximal abelian subalgebra of the Lie algebra of . Our main result shows that, for a generic -Hitchin representation , we have . As an application, we prove that generic orbifold Hitchin representations are strongly dense. This extends the result of Long, Reid, and Wolff for the Hitchin representations of surface groups. Our theorem also shows that the split real forms of many simple adjoint Lie groups contain strongly dense orbifold fundamental groups, partially generalizing the work of Breuillard, Guralnick, and Larsen.
Paper Structure (9 sections, 16 theorems, 76 equations, 2 figures)

This paper contains 9 sections, 16 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Orbifolds
  4. Lie theory and Hitchin representations
  5. The Goldman Product Formula
  6. Lifting Goldman Flows
  7. Proofs of Main Theorems
  8. Further Discussions and Questions
  9. Analyticity of the Jordan Projection

Key Result

Theorem 1.2

Let $O$ be a closed orientable non-elementary 2-orbifold of negative Euler characteristic. Assume that $O$ is neither $\mathbb{S}^2(2,2,2,2,p)$, $p\ge 2$ nor $\mathbb{S}^2(2,2,p,q)$, $\frac{1}{p}+\frac{1}{q}< 2$. Then for any $i=1,2,\cdots, n$, the following set is generic in $\operatorname{Hit}_n(O)$.

Figures (2)

  • Figure 1: A sphere with six order two cone points. Black dots are order two cone points.
  • Figure 2: The effect of the Dehn twist. Black geodesics are lifts of $y$ and red geodesic is a lift of $x$ in $\mathbb{H}^2$. The red curve in the right hand side is a lift of $\operatorname{tw}^{k}_y(x)$.

Theorems & Definitions (31)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Example 2.2
  • Theorem 3.1: Goldman Product formula
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 21 more