A general Hilbert-Mumford Criterion

We provide a Hilbert-Mumford Criterion for actions of reductive groups $G$ on $Q$-factorial complex varieties. The result allows to construct open subsets admitting a good quotient by $G$ from certain maximal open subsets admitting a good quotient by a maximal torus of $G$. As an application, we show how to obtain all invariant open subsets with good quotient for a given $G$-action on a complete $Q$-factorial toric variety.