  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.
