A Hilbert-Mumford-Criterion for SL2-Actions

Let the special linear group G := SL2 act regularly on a Q-factorial variety X. Consider a maximal torus T subset G and its normalizer N subset G. We prove: If U subset X is a maximal open N-invariant subset admitting a good quotient U -> U // N with a divisorial quotient space, then the intersection W(U) of all translates g U is open in X and admits a good quotient W(U) -> W(U) // G with a divisorial quotient space. Conversely, we obtain that every maximal open G-invariant subset W subset X admitting a good quotient W -> W // G with a divisorial quotient space is of the form W = W(U) for some maximal open N-invariant U as above