Spherical gradient manifolds

Abstract

We study the action of a real-reductive group $G=K\exp(\lie{p})$ on real-analytic submanifold $X$ of a K\"ahler manifold $Z$. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^\mbb{C}$ such that the action of a maximal Hamiltonian subgroup is Hamiltonian. The moment map $\mu$ induces a gradient map $\mu_\lie{p}\colon X\to\lie{p}$. We show that $\mu_\lie{p}$ almost separates the $K$--orbits if and only if a minimal parabolic subgroup of $G$ has an open orbit. This generalizes Brion's characterization of spherical K\"ahler manifolds with moment maps.