On the Homotopy of Symplectomorphism Groups of Homogeneous Spaces

Let $\mathcal{O}$ be a quantizable coadjoint orbit of a semisimple Lie group $G$. Under certain hypotheses we prove that $#(\pi_1($\text{Ham}$({\cal O})))\geq #(Z(G))$,where$$\text{Ham}$({\cal O})$ is the group of Hamiltonian symplectomorphisms of $\mathcal{O}$.