Symplectic aspects of polar actions

Abstract

An isometric compact group action $G\times (M,g)\to (M,g)$ is called polar if there exists a closed embedded submanifold $\Sigma \subseteq M$ which meets all orbits orthogonally. Let $\Pi$ be the associated generalized Weyl group. We study the properties of the lifting action $G$ on the cotangent bundle $T^{\ast }M$. In particular, we show that the restriction map $(C^{\infty }(T^{\ast }M){)}^G\to (C^{\infty }(T^{\ast }\Sigma ){)}^{\Pi }$ is a surjective homomorphism of Poisson algebras. As a corollary, the singular symplectic reductions $T^{\ast }M//G$ and $T^{\ast }\Sigma //\Pi$ are isomorphic as stratified symplectic spaces, which gives a partial answer to a conjecture of Lerman, Montgomery and Sjamaar.