On Symplectic Linearizable Actions

We prove that linearizable actions are also symplectically linearizable (either smoothly or analytically) in a neighborhood of a fixed point. Specifically, the fundamental vector fields associated with the action can be simultaneously linearized in Darboux coordinates. This result extends equivariant symplectic local normal forms to non-compact group actions. In both formal and analytic frameworks, the existence of linearizing coordinates is tied to a cohomological equation, which admits a solution for semisimple actions [9, 8]. Consequently, an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates, enabling the simultaneous analytic linearization of Hamiltonian vector fields near a shared zero. However, in the smooth setting, this result is restricted to semisimple Lie algebras of compact type. We construct an explicit example of a smooth, non-linearizable Hamiltonian action with a semisimple linear part, thereby answering in the negative a question posed by Eliasson [5].