Hamiltonian Carleman approximation and the density property for coadjoint orbits

TL;DR

This paper proves that Hamiltonian automorphisms of certain coadjoint orbits can be approximated by invariant symplectic automorphisms, establishing a density property and Hamiltonian density property for these orbits.

Contribution

It introduces a Carleman approximation result for Hamiltonian automorphisms on coadjoint orbits and proves the Hamiltonian density property for all closed coadjoint orbits of complex Lie groups.

Findings

Hamiltonian automorphisms can be approximated by holomorphic invariant symplectic automorphisms.

Established the Hamiltonian density property for closed coadjoint orbits.

Proved approximation results under the condition of simply connected components.

Abstract

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.