Hamiltonian Carleman approximation and the density property for coadjoint orbits

For a complex Lie group $G$ with a real form $G_0\subset G$, we prove that any Hamiltionian automorphism $\phi$ of a coadjoint orbit $\mathcal O_0$ of $G_0$ whose connected components are simply connected, may be approximated by holomorphic $\mathcal O_0$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal O$ is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.