  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.

Symplectic Geometry;Algebraic Geometry;Differential Geometry