Stability of measures on Kähler manifolds

Let (M,ω) be a Kähler manifold and let K be a compact group that acts on M in a Hamiltonian fashion. We study the action of the complexification of K on probability measures on M. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystability. Next we apply this setting to the action of the complexification of K on measures. We get various stability criteria for measures on Kähler manifolds. The same circle of ideas gives a very general surjectivity result for a map originally studied by Hersch and Bourguignon–Li–Yau.