On Character Variety of Anosov Representations

Let $\Gamma$ be the fundamental group of a $k$-punctured, $k \geq 0$, closed connected orientable surface of genus $g \geq 2$. 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 $\Gamma$ into $\SL(n , \C)$, $n \geq 2$, is a complex manifold of complex dimension \hbox{$(2g+k-2)(n^2-1)$}. For $\Gamma=\pi_1(\Sigma_g)$, we also show that these character varieties are holomorphic symplectic manifolds.