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 $\mathbb{ Z}$.
Categories: math.AG math.GT math.RT