I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold have non-zero Euler characteristic. They prove it by quoting a theorem by Konstant, that implies that the manifold is symplectomorphic to a coadjoint orbit of a semisimple group, and then saying that compact coadjoint orbits of semisimple groups have non-zero Euler characteristic.
I was looking for a more direct proof of that fact. Do you know some?