Character varieties for real forms of classical complex groups

Let $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ as a subset ${\mathcal{X}}_{G_{\mathbb{R}}}(\mathrm{\Gamma })$ of the $G_{\mathbb{C}}$-character variety ${\mathcal{X}}_{G_{\mathbb{C}}}(\mathrm{\Gamma })$. We prove that these subsets cover the set of irreducible $G_{\mathbb{C}}$-characters fixed by an anti-holomorphic involution $\mathrm{\Phi }$ of ${\mathcal{X}}_{G_{\mathbb{C}}}(\mathrm{\Gamma })$. Whenever $G_{\mathbb{R}}$ is compact, we prove that ${\mathcal{X}}_{G_{\mathbb{R}}}(\mathrm{\Gamma })$ is homeomorphic to the topological quotient $\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{\Gamma },G_{\mathbb{R}})\mathrm{/}{G}_{\mathbb{R}}$. Finally, we identify the reducible points of ${\mathcal{X}}_{\mathrm{G}\mathrm{L}(n,\mathbb{C})}(\mathrm{\Gamma })$ fixed by an anti-holomorphic involution $\mathrm{\Phi }$ as coming from direct sums of representations with values in real groups.