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 $O_{0}$-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.[-]

The automorphism group of a compact Kähler manifold satisfies Tits alternative: any subgroup either admits a solvable subgroup of finite index or contains a free non-abelian group of two generators (Campana-WangZhang). In the first case, this group cannot be too big. Some algebraic (rational) manifolds with special automorphisms admit infinitely many nonequivalent real forms. This talk is based on my (old and recent) works with F. Hu, H.-Y. Lin, V.-A. Nguyen, K. Oguiso, N. Sibony, X. Yu, D.-Q. Zhang.[-]