Character varieties for real forms

TL;DR

This paper defines and explores the properties of $G$-character varieties for real forms of $ ext{SL}_n( ext{C})$, analyzing involutions and specific examples like free products of cyclic groups.

Contribution

It introduces a new definition for $G$-character varieties within the complex $ ext{SL}_n( ext{C})$-character variety framework and studies their properties under involutions.

Findings

Irreducible representations fixed by involutions are conjugate to real form representations.

Detailed analysis of character varieties for free product groups.

Identification of fixed points under anti-holomorphic involutions.

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 $\mathbb{Z}/3\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$.