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.

25 pages, preliminary version, comments are welcome!

References in corpus (1)

• SL(n,C)- Representation spaces of Knot Groups

Cited by in corpus (1)

• On GIT quotients and real forms