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A Method for Weight Multiplicity Computation Based on Berezin Quantization
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\Gamma _{\mathrm{hol}}(\mathcal L^\lambda)$ 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 $\Gamma _{\mathrm{hol}}(\mathcal L^\lambda)$ 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.
Keywords: Berezin quantization; representation theory. 
@article{SIGMA_2009_5_a90,
     author = {David Bar-Moshe},
     title = {A~Method for {Weight} {Multiplicity} {Computation} {Based} on {Berezin} {Quantization}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2009},
     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a90/}
}
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David Bar-Moshe. A Method for Weight Multiplicity Computation Based on Berezin Quantization. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a90/

[1] Arnal D., Ben Ammar M., Selmi M., “Le problème de la réduction à un sous-groupe dans la quantification par déformation”, Ann. Fac. Sci. Toulouse Math. (5), 12 (1991), 7–27 | MR | Zbl

[2] Baldoni M. W., Beck M., Cochet C., Vergne M., “Volume computation for polytopes and partition functions of classical root systems”, Discrete Comput. Geom., 35 (2006), 551–595 ; math.CO/0504231 | DOI | MR | Zbl

[3] Bando M., Kuratomo T., Maskawa T., Uehera S., “Structure of nonlinear realization in supersymmetric theories”, Phys. Lett. B, 138 (1984), 94–98 | DOI | MR | Zbl

[4] Bando M., Kuratomo T., Maskawa T., Uehera S., “Nonlinear realizations in supersymmetric theories”, Progr. Theoret. Phys., 72 (1984), 313–349 | DOI | MR

[5] Bar-Moshe D., Marinov M. S., “Realization of compact Lie algebras in Kähler manifolds”, J. Phys. A: Math. Gen., 27 (1994), 6287–6298 ; hep-th/9407092 | DOI | MR | Zbl

[6] Bar-Moshe D., Marinov M. S., “Berezin quantization and unitary representations of Lie groups”, Topics in Statistical and Theoretical Physics, Amer. Math. Soc. Transl. Ser. 2, 177, Amer. Math. Soc., Providence, RI, 1996, 1–21 ; hep-th/9407093 | MR | Zbl

[7] Berezin F. A., “Quantization”, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 1116–1175 | MR | Zbl

[8] Berezin F. A., “Covariant and contravariant symbols of operators”, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134–1167 | MR | Zbl

[9] Billey S., Guillemin V., Rassart E., “A vector partition function for the multiplicities of $\mathfrak{sl}_k(\mathbb C)$”, J. Algebra, 278 (2004), 251–293 ; math.CO/0307227 | DOI | MR | Zbl

[10] Baoua O. B., “Gelfand pairs with coherent states”, Lett. Math. Phys., 46 (1998), 247–263 | DOI | MR | Zbl

[11] Cahen B., “Multiplicities of compact Lie group representations via Berezin quantization”, Mat. Vesnik, 60 (2008), 295–309 | MR | Zbl

[12] Guillemin V., Sternberg S., “Geometric quantization and multiplicities of group representations”, Invent. Math., 67 (1982), 515–538 | DOI | MR | Zbl

[13] Heckman G. J., “Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups”, Invent. Math., 67 (1983), 333–356 | DOI | MR

[14] Hurt N. E., Geometric quantization in action, Mathematics and Its Applications (East European Series), 8, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1983 | MR | Zbl

[15] Kostant B., “A formula for the multiplicity of a weight”, Trans. Amer. Math. Soc., 93 (1959), 53–73 | DOI | MR | Zbl

[16] Littelmann P., “Paths and root operators in representation theory”, Ann of Math. (2), 142 (1995), 499–525 | DOI | MR | Zbl

[17] O'Raifeartaigh L., Group structures of gauge theories, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1986 | MR

[18] Peetre J., “The Berezin transform and Ha-plitz operators”, J. Operator Theory, 24 (1990), 165–168 | MR

[19] Rawnsley J., Cahen M., Gutt S., “Quantization of Kähler manifolds. I. Geometric interpretation of Berezin's quantization”, J. Geom. Phys., 7 (1990), 45–62 | DOI | MR | Zbl

[20] Vergne M., “Multiplicities formula for geometric quantization. I”, Duke Math. J., 82 (1996), 143–179 | DOI | MR | Zbl

[21] Vergne M., “Multiplicities formula for geometric quantization. II”, Duke Math. J., 82 (1996), 181–194 | DOI | MR | Zbl