Observable actions of algebraic groups

Abstract

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 ∈ \(\Bbbk\) [X]G such that f| Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if: (1) the action is stable, that is there exists a nonempty open subset U ⊆ X consisting of closed orbits; and (2) the field \(\Bbbk\)(X)G of G-invariant rational functions on X is equal to the quotient field of \(\Bbbk\)[X]G. In case G is reductive, we conclude that there exists a unique, maximal, G-stable, closed subset X soc of X such that G × X soc → X soc is observable. Furthermore, the canonical map X soc//G → X//G is bijective.