Rigidity of Hamiltonian actions on Poisson manifolds

TL;DR

This paper establishes the rigidity of Hamiltonian actions of compact semisimple groups on Poisson manifolds using a Nash-Moser normal form theorem, extending classical results to the Poisson and symplectic contexts.

Contribution

It introduces a Nash-Moser normal form theorem for closed subgroups of SCI-type, demonstrating the rigidity of Hamiltonian actions of compact semisimple groups on Poisson manifolds.

Findings

Hamiltonian actions of compact semisimple groups are rigid

Develops a Nash-Moser normal form theorem for SCI-type subgroups

Extends classical rigidity results to Poisson and symplectic settings

Abstract

This paper is about the rigidity of compact group actions in the Poisson context. The main resut is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash-Moser normal form theorem for closed subgroups of SCI-type. This Nash-Moser normal form has other applications to stability results that we will explore in a future paper. We also review some classical rigidity results for differentiable actions of compact Lie groups and export it to the case of symplectic actions of compact Lie groups on symplectic manifolds.