On character variety of Anosov representations

Abstract

Let Γ be the fundamental group of a k-punctured, $k\ge 0$, closed connected orientable surface of genus $g\ge 2$. We show that the character variety of the $(Q^{+},Q^{-})$-Anosov irreducible representations, resp. the character variety of the $(P^{+},P^{-})$-Anosov Zariski dense representations of Γ into $\mathrm{SL}(n,C)$, $n\ge 2$, is a complex manifold of complex dimension $(2g+k-2)(n^2-1)$. For $\Gamma ={\pi }_1\left({\Sigma }_g\right)$, we also show that these character varieties are holomorphic symplectic manifolds.