Observable Actions of Algebraic Groups Lex Renner and Alvaro Rittatore

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ k[X ] such that f | Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if (1) there is a nonempty open subset U ⊆ X consisting of closed orbits, and (2) the field k(X) of G-invariant rational functions on X is equal to the quotient field of k[X ]. In case G is reductive, we conclude that there exists a unique, maximal, G-stable, closed subset Xsoc of X such that G × Xsoc → Xsoc is observable. Furthermore, the canonical map Xsoc//G → X//G is bijective.