Properties of gradient maps associated with action of real reductive group

Let (Z,ω) be a Kähler manifold and let U be a compact connected Lie group with Lie algebra acting on Z and preserving ω. We assume that the U-action extends holomorphically to an action of the complexified group Uc and the U-action on Z is Hamiltonian. Then there exists a U-equivariant momentum map μ: Z → . If G Uc is a closed subgroup such that the Cartan decomposition Uc = Uexp(i) induces a Cartan decomposition G = Kexp(), where K = U ∩ G, = ∩ i and g = p ⊕ p is the Lie algebra of G, there is a corresponding gradient map μ: Z → . If X is a G-invariant compact and connected real submanifold of Z, we may consider μ as a mapping μ: X → . Given an Ad(K)-invariant scalar product on, we obtain a Morse like function f = 1/2 ||μp||2 on X. We point out that, without the assumption that X is a real analytic manifold, the Lojasiewicz gradient inequality holds for f. Therefore, the limit of the negative gradient flow of f exists and it is unique. Moreover, we prove that any G-orbit collapses to a single K-orbit and two critical points of f which are in the same G-orbit belong to the same K-orbit. We also investigate convexity properties of the gradient map μ in the Abelian case. In particular, we study two-orbit variety X and we investigate topological and cohomological properties of X.