Equivariant deformation quantization and coadjoint orbit method

Our goal here is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive Lie groups. We first prove some general results on the existence of equivariant deformation quantizations of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to the orbit method and construct nontrivial irreducible Harish-Chandra modules for certain nilpotent coadjoint orbits. Our examples include new geometric construction of representations associated to a large class of nilpotent orbits of real exceptional Lie groups.