The Berezin quantization on a simply connected homogeneous Kähler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finitedimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the Kähler structure. The Kähler potentials are constructed for the manifolds related to all compact semi-simple Lie groups. The complex coordinates are introduced by means of the Borel method. The Kähler structure is obtained explicitly for any unitary group representation. The cocycle functions for the Lie algebra and the Killing vector fields on the manifold are also obtained.
In 1974, Berezin proposed a quantum theory for dynamical systems having a Kähler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous Kähler 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 Kähler structure ("classical theory"). The Kähler 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.
Abstract. Let G be a compact semisimple Lie group and T be a maximal torus of G. We describe a method for weight multiplicity computation in unitary irreducible representations of G, based on the theory of Berezin quantization on G/T . Let Γ hol (L λ ) be the reproducing kernel Hilbert space of holomorphic sections of the homogeneous line bundle L λ over G/T associated with the highest weight λ of the irreducible representation π λ of G. The multiplicity of a weight m in π λ is computed from functional analytical structure of the Berezin symbol of the projector in Γ hol (L λ ) onto subspace of weight m. We describe a method of the construction of this symbol and the evaluation of the weight multiplicity as a rank of a Hermitian form. The application of this method is described in a number of examples.
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