Weyl quantization for the semidirect product of a compact Lie group and a vector space

Benjamin Cahen

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 3, page 325-347
  • ISSN: 0010-2628

Abstract

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Let G be the semidirect product V ⋊ K where K is a semisimple compact connected Lie group acting linearly on a finite-dimensional real vector space V . Let 𝒪 be a coadjoint orbit of G associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation π of G . We consider the case when the corresponding little group H is the centralizer of a torus of K . By dequantizing a suitable realization of π on a Hilbert space of functions on ℂ n where n = dim ( K / H ) , we construct a symplectomorphism between a dense open subset of 𝒪 and the symplectic product ℂ 2 n × 𝒪 ' where 𝒪 ' is a coadjoint orbit of H . This allows us to obtain a Weyl correspondence on 𝒪 which is adapted to the representation π in the sense of [B. Cahen, Quantification d’une orbite massive d’un groupe de Poincaré généralisé, C.R. Acad. Sci. Paris t. 325, série I (1997), 803–806].

How to cite

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Cahen, Benjamin. "Weyl quantization for the semidirect product of a compact Lie group and a vector space." Commentationes Mathematicae Universitatis Carolinae 50.3 (2009): 325-347. <http://eudml.org/doc/33318>.

@article{Cahen2009,
abstract = {Let $G$ be the semidirect product $V\rtimes K$ where $K$ is a semisimple compact connected Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\mathcal \{O\}$ be a coadjoint orbit of $G$ associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation $\pi $ of $G$. We consider the case when the corresponding little group $H$ is the centralizer of a torus of $K$. By dequantizing a suitable realization of $\pi $ on a Hilbert space of functions on $\{\mathbb \{C\}\}^n$ where $n=\dim (K/H)$, we construct a symplectomorphism between a dense open subset of $\{\mathcal \{O\}\}$ and the symplectic product $\{\mathbb \{C\}\}^\{2n\}\times \{\mathcal \{O\}\}^\{\prime \}$ where $\{\mathcal \{O\}\}^\{\prime \}$ is a coadjoint orbit of $H$. This allows us to obtain a Weyl correspondence on $\{\mathcal \{O\}\}$ which is adapted to the representation $\pi $ in the sense of [B. Cahen, Quantification d’une orbite massive d’un groupe de Poincaré généralisé, C.R. Acad. Sci. Paris t. 325, série I (1997), 803–806].},
author = {Cahen, Benjamin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Weyl quantization; Berezin quantization; semidirect product; coadjoint orbits; unitary representations; Weyl quantization; Berezin quantization; semidirect product; coadjoint orbit; unitary representation},
language = {eng},
number = {3},
pages = {325-347},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Weyl quantization for the semidirect product of a compact Lie group and a vector space},
url = {http://eudml.org/doc/33318},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Cahen, Benjamin
TI - Weyl quantization for the semidirect product of a compact Lie group and a vector space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 3
SP - 325
EP - 347
AB - Let $G$ be the semidirect product $V\rtimes K$ where $K$ is a semisimple compact connected Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\mathcal {O}$ be a coadjoint orbit of $G$ associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation $\pi $ of $G$. We consider the case when the corresponding little group $H$ is the centralizer of a torus of $K$. By dequantizing a suitable realization of $\pi $ on a Hilbert space of functions on ${\mathbb {C}}^n$ where $n=\dim (K/H)$, we construct a symplectomorphism between a dense open subset of ${\mathcal {O}}$ and the symplectic product ${\mathbb {C}}^{2n}\times {\mathcal {O}}^{\prime }$ where ${\mathcal {O}}^{\prime }$ is a coadjoint orbit of $H$. This allows us to obtain a Weyl correspondence on ${\mathcal {O}}$ which is adapted to the representation $\pi $ in the sense of [B. Cahen, Quantification d’une orbite massive d’un groupe de Poincaré généralisé, C.R. Acad. Sci. Paris t. 325, série I (1997), 803–806].
LA - eng
KW - Weyl quantization; Berezin quantization; semidirect product; coadjoint orbits; unitary representations; Weyl quantization; Berezin quantization; semidirect product; coadjoint orbit; unitary representation
UR - http://eudml.org/doc/33318
ER -

References

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