## Benjamin Cahen

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

Comment.Math.Univ.Carolinae 50,3 (2009) 325-347.**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 ${\mathbb C}^{2n}\times {\mathcal O}'$ 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, {\it Quantification d'une orbite massive d'un groupe de Poincaré généralisé\/}, C.R. Acad. Sci. Paris t. 325, série I (1997), 803--806].

**Keywords:** Weyl quantization, Berezin quantization, semidirect product, coadjoint orbits, unitary representations

**AMS Subject Classification:** 81S10 22E46 22E99 32M10

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