A compact connected semisimple Lie group G acts in a Hamiltonian fashion on its coadjoint orbits, i.e. G maps into the group of Hamiltonian transformations of the coadjoint orbit. McDuff and Tolman showed that the induced map on fundamental groups is injective, answering a question of A. Weinstein. In this talk we will show that this is not quite a symplectic phenomenon, but a topological one, since the (finite) fundamental group of G already injects in the fundamental group of the group of diffeomorphisms. This is joint work with I. Mundet.