Berezin quantization and unitary representations of Lie groups

In 1974, Berezin proposed a quantum theory for dynamical systems having a K\"{a}hler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous K\"{a}hler manifold, the Lie algebra of its group of motions may be represented either by holomorphic differential operators (``quantum theory"), or by functions on the manifold with Poisson brackets, generated by the K\"{a}hler structure (``classical theory"). The K\"{a}hler potentials and the corresponding Lie algebras are constructed now explicitly for all unitary representations of any compact simple Lie group. The quantum dynamics can be represented in terms of a phase-space path integral, and the action principle appears in the semi-classical approximation.