Stability with respect to actions of real reductive Lie groups

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of noncompact real reductive Lie groups on topological spaces that admit functions similar to the Kempf–Ness function. The point of this construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight. We apply this setting to the actions of a real noncompact reductive Lie group G on a real compact submanifold M of a Kähler manifold Z and to the action of G on measures of M.