A Method for Weight Multiplicity Computation Based on Berezin Quantization

TL;DR

This paper introduces a novel method for computing weight multiplicities in irreducible representations of compact semisimple Lie groups using Berezin quantization on homogeneous spaces, providing a functional analytical approach.

Contribution

It develops a new technique based on Berezin quantization and reproducing kernel Hilbert spaces to calculate weight multiplicities, linking geometric quantization with representation theory.

Findings

Method successfully computes weight multiplicities in examples.

Provides explicit construction of Berezin symbols for projectors.

Demonstrates the approach's effectiveness through multiple case studies.

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 $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ be the reproducing kernel Hilbert space of holomorphic sections of the homogeneous line bundle $\mathcal{L}^{\lambda}$ 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 functional analytical structure of the Berezin symbol of the projector in $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ 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.