Symplectic and Hamiltonian properties of holomorphic coadjoint orbits

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant K\"ahlerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G with finite center, such that G/K is a Hermitian symmetric space, where K is a maximal compact subgroup of G. Holomorphic coadjoint orbits are a generalization of the Hermitian symmetric space G/K. In this thesis, we prove that the McDuff's symplectomorphism on Hermitian symmetric spaces, has an analogous for holomorphic coadjoint orbits. Then, using this symplectomorphism and recent GIT arguments from Ressayre, we compute the equations of the projection of the O, relatively to the maximal compact subgroup K.