  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.
