A method for weight multiplicity computation based on Berezin quantization. (English) Zbl 1188.22009

Bar-Moshe, David

SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 091, 12 p. (2009).

Summary: 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 \(\Gamma _{\text{hol}}(L^{\lambda })\) be the reproducing kernel Hilbert space of holomorphic sections of the homogeneous line bundle \(L^{\lambda }\) over \(G/T\) associated with the highest weight \(\lambda \) of the irreducible representation \(\pi _{\lambda }\) of \(G\). The multiplicity of a weight \(m\) in \(\pi _{\lambda }\) is computed from the functional analytical structure of the Berezin symbol of the projector in \(\Gamma _{\text{hol}}(L^{\lambda })\) onto a 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.

MSC:

22E46	Semisimple Lie groups and their representations
32M05	Complex Lie groups, group actions on complex spaces
32M10	Homogeneous complex manifolds
53D50	Geometric quantization
81Q70	Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory

Keywords:

Berezin quantization; representation theory