Character varieties for real forms of classical complex groups

Abstract: Let $\Gamma$ be a finitely generated group, $G_\mathbb{C}$ be a classical complex group and $G_\mathbb{R}$ a real form of $G_\mathbb{C}$. We propose a definition of the $G_\mathbb{R}$-character variety of $\Gamma$ as a subset $\mathcal{X}{G\mathbb{R}}(\Gamma)$ of the $G_\mathbb{C}$-character variety $\mathcal{X}{G\mathbb{C}}(\Gamma)$. We prove that these subsets cover the set of irreducible $G_\mathbb{C}$-characters fixed by an anti-holomorphic involution $\Phi$ of $\mathcal{X}{G\mathbb{C}}(\Gamma)$. Whenever $G_\mathbb{R}$ is compact, we prove that $\mathcal{X}{G\mathbb{R}}(\Gamma)$ is homeomorphic to the topological quotient $\mathrm{Hom}(\Gamma,G_\mathbb{R})/G_\mathbb{R}$. Finally, we identify the reducible points of $\mathcal{X}_{\mathrm{GL}(n,\mathbb{C})}(\Gamma)$ fixed by an anti-holomorphic involution $\Phi$ as coming from direct sums of representations with values in real groups.