Generic Properties of Hitchin Representations

Let $G$ be a split real form of a complex simple adjoint group whose Weyl group contains $-1$, let $\lambda$ be the Jordan projection of $G$, and let $S$ be a closed orientable surface of genus at least 2. For a $G$-Hitchin representation $\rho$, we define the set $J(\rho):=\{\lambda(\rho(x))\,|\,x\in\pi_1(S)\setminus \{1\}\}$. 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 $\rho$, we have $J(\rho)\cap H=\emptyset$. 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.