Geodesic
Weyl quantization for the semidirect product of a compact Lie group and a vector space
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 325-347 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 ${\mathbb C}^{2n}\times {\mathcal O}'$ where ${\mathcal O}'$ 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].
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 ${\mathbb C}^{2n}\times {\mathcal O}'$ where ${\mathcal O}'$ 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].
Classification : 22E46, 22E99, 32M10, 81S10
Keywords: Weyl quantization; Berezin quantization; semidirect product; coadjoint orbits; unitary representations 
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     author = {Cahen, Benjamin},
     title = {Weyl quantization for the semidirect product of a compact {Lie} group and a vector space},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {325--347},
     year = {2009},
     volume = {50},
     number = {3},
     mrnumber = {2573408},
     zbl = {1212.81015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2009_50_3_a1/}
}
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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, Tome 50 (2009) no. 3, pp. 325-347. http://geodesic.mathdoc.fr/item/CMUC_2009_50_3_a1/