On GIT quotients and real forms

TL;DR

This paper investigates the relationship between complex GIT quotients and their real forms, establishing conditions for lifting real points and applying results to character varieties, especially for SL_3(C).

Contribution

It introduces new links between real points of complex GIT quotients and real GIT quotients, including conditions for lifting and cohomological classifications.

Findings

Some real points of complex GIT quotients can be lifted to real GIT quotients after changing real forms.

The number of possible lifts is related to a cohomology set.

Application to SL_3(C)-character varieties for Z demonstrates the theory.

Abstract

We consider actions of complex algebraic groups $\mathbf{G}$ on complex algebraic varieties $\mathbf{X}$, coming from actions of real forms $G$ of $\mathbf{G}$ and $X$ of $\mathbf{X}$. We explore the links between the real points of the complex GIT quotient $\mathbf{X /\!\!/ G}$ and the real GIT quotient $X /\!\!/ G$ defined by Richardson and Slodowy. We prove that some type of real points of $\mathbf{X /\!\!/ G}$ can be lifted to a quotient of the form $X /\!\!/ G$ maybe after changing the real forms, and we link the number of possible lifts to a co-homology set. We apply then the results to character varieties, and study the particular case of the $\mathrm{SL}_3(\mathbb{C})$-character variety for $\mathbb{Z}$.