Dominant weight associated with actions of real reductive groups

We study the action of a real reductive Lie group G on a real submanifold X of a Kähler manifold Z. Suppose the action of a compact Lie group U with Lie algebra u extends holomorphically to an action of the complexified group UC and that the U-action on Z is Hamiltonian. If G⊂UC is compatible, there is a corresponding G-gradient map μp:X→p, where g=k⊕p is a Cartan decomposition of the Lie algebra of G, k is the Lie algebra of K=U∩G, and p=iu∩g. One can associate a G-invariant maximal weight function to this action. To each x∈X, there is a dominant weight that minimizes the maximal weight function. We study the dominant weight's existence and uniqueness under various stability conditions.