On Character Variety of Anosov Representations

Abstract

Let $\Gamma$ be the free group $F_n$ of $n$ generators, resp. the fundamental group $\pi_1(\Sigma_g)$ of a closed, connnected, orientatble surface of genus $g \geq 2$. We show that the charater variety of irreducible, resp. Zariski dense, Anosov representations of $\Gamma$ into $\SL(n, \C)$ is a complex manifold of (complex) dimension $(n-1)(n^2-1)$, resp. $(2g-2) (n^2-1)$. For $\Gamma=\pi_1(\Sigma_g)$, we also show that these character varieties are holomorphic symplectic manifolds.