Representation Theory and Quantization on K\"{a}hler Manifolds

Abstract

Homogeneous K\"{a}hler manifold is considered as a phase space of a dynamical system. Berezin introduced the Hilbert space of holomorphic wave functions, which is a basis for the quantum theory. The Lie algebra of the transformation group is realized by differential operators, or their symbols, i.e. functions in the phase space. Commutation relations for the canonical operators are in the direct correspondence to the Poisson brackets on the manifold. A complete and explicit solution is obtained for all compact Lie groups.