A Hilbert-Mumford criterion for SL_2-actions

Let the special linear group $G:=S{L}_2$ act regularly on a $Q$-factorial variety $X$. Consider a maximal torus $T\subset G$ and its normalizer $N \subset G$\mathrm{.}Weprove:If$U \subset X$isamaximalopen$N$-invariant subset admitting a good quotient $U\to U\mathrm{/}\mathrm{/}N$ with a divisorial quotient space, then the intersection $W(U)$ of all translates $g\dot{U}$ is open in $X$ and admits a good quotient $W(U)\to W(U)\mathrm{/}\mathrm{/}G$ with a divisorial quotient space. Conversely, we obtain that every maximal open $G$-invariant subset $W \subset X$admittingagoodquotient$W \to W // G$ with a divisorial quotient space is of the form $W=W(U)$ for some maximal open $N$-invariant $U$ as above.