Reduction principles for proper actions

Abstract

We study the core of a proper action by a Lie group $G$ on a smooth manifold $M$, extending the construction for $G$ compact by Skjelbred and Straume. Moreover, we show that many properties of a proper $G$-action on $M$ are determined by the action of a group $G'$ on the corresponding core $_cM$. We say that such properties admit a reduction principle. In particular, we prove that a proper isometric $G$-action on $M$ is polar (resp. hyperpolar) if and only if the $G'$-action on $_cM$ is polar (resp. hyperpolar). In the case of a proper action by syplectomorphisms on a symplectic manifold, we show that a reduction principle holds for coisotropic and infinitesimally almost homogeneous actions. We further study the coisotropic condition for the case of a proper Hamiltonian action and its relation with the symplectic stratification described by Lermann and Bates. In particular, we obtain several characterizations for coisotropic actions, some of which extend known results for the action of a compact group of holomorphic automorphisms on a compact K\"ahler manifold obtained by Huckleberry and Wurzbacher. Finally, we study some applications of the core construction for the action of a compact Lie group on a K\"ahler manifold by holomorphic isometries.