This paper is about the rigidity of compact group actions in the
Poisson context. The main result 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.