Holomorphic Discrete Models of Semisimple Lie Groups and their Symplectic Constructions

Let G be a connected real semisimple Lie group which contains a compact Cartan subgroup such that it has non-empty discrete series. A holomorphic discrete model of G is a unitary G-representation consisting of all its holomorphic discrete series with multiplicity one. We perform geometric quantization to a class of G-invariant pseudo-Kنhler manifolds and construct a holomorphic discrete model. The construction of discrete series which are not holomorphic is also discussed.