On GIT quotients and real forms

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 $\mathbb{Z}$.