Character varieties for real forms

Abstract

Let $\Gamma$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $\Gamma$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $\Gamma$. We consider two anti-holomorphic involutions of the $\mathrm{SL}_n(\mathbb{C})$ character variety and show that an irreducible representation with character fixed by one of them is conjugate to a representation taking values in a real form of $\mathrm{SL}_n(\mathbb{C})$. We study in detail an example: the $\mathrm{SL}_n(\mathbb{C})$, $\mathrm{SU}(2,1)$ and $\mathrm{SU}(3)$ character varieties of the free product $\mathbb{Z}/3\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$.