# Symplectic aspects of polar actions

## TL;DR

This paper explores the symplectic geometry of polar group actions, demonstrating how the cotangent bundle's invariants relate to a reduced space, thereby advancing understanding of stratified symplectic structures and confirming a conjecture.

## Contribution

It establishes a surjective Poisson algebra homomorphism between invariants on the cotangent bundle and a submanifold, linking their symplectic reductions and confirming a conjecture.

## Key findings

- The restriction map between invariant functions is surjective.
- Singular symplectic reductions are isomorphic as stratified spaces.
- Provides partial confirmation of a conjecture by Lerman, Montgomery, and Sjamaar.

## Abstract

An isometric compact group action $G \times (M,g) \rightarrow (M,g)$ 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 $T^*M$. In particular, we show that the restriction map $(C^{\infty}(T^*M))^G \rightarrow (C^{\infty}(T^* \Sigma))^{\Pi}$ is a surjective homomorphism of Poisson algebras. As a corollary, the singular symplectic reductions $T^*M // G $ and $T^* \Sigma // \Pi$ are isomorphic as stratified symplectic spaces, which gives a partial answer to a conjecture of Lerman, Montgomery and Sjamaar.