A review on deformation quantization of coadjoint orbits of semisimple Lie groups

In this paper we make a review of the results obtained in previous works by the authors on deformation quantization of coadjoint orbits of semisimple Lie groups. We motivate the problem with a new point of view of the well known Moyal-Weyl deformation quantization. We consider only semisimple orbits. Algebraic and differential deformations are compared.