Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 091, 12 pages      arXiv:math-ph/0306056      https://doi.org/10.3842/SIGMA.2009.091

A Method for Weight Multiplicity Computation Based on Berezin Quantization

David Bar-Moshe
Dune Medical Devices Ltd., P.O. Box 3131, Caesarea Industrial Park, Israel

Received July 26, 2009, in final form September 16, 2009; Published online September 25, 2009

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 Γhol(Lλ) be the reproducing kernel Hilbert space of holomorphic sections of the homogeneous line bundle Lλ over G/T associated with the highest weight λ of the irreducible representation πλ of G. The multiplicity of a weight m in πλ is computed from functional analytical structure of the Berezin symbol of the projector in Γhol(Lλ) 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.

Key words: Berezin quantization; representation theory.

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