Character varieties for real forms of classical complex groups

TL;DR

This paper defines and studies the properties of real form character varieties within complex classical groups, revealing their structure and relation to fixed points under anti-holomorphic involutions, with implications for representation theory.

Contribution

It introduces a new definition of real form character varieties and analyzes their topological and geometric properties, including their relation to fixed points and reducible representations.

Findings

Real form character varieties cover fixed points of anti-holomorphic involutions.

When compact, these varieties are homeomorphic to certain quotient spaces.

Reducible points correspond to direct sums of real group representations.

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.