Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 48, No. 1, pp. 175190 (2007) 

Weyl quantization for principal seriesBenjamin CahenUniversité de Metz, UFRMIM, Département de mathématiquesLMMAS, ISGMPBât. A, Ile du Saulcy 57045, Metz cedex 01, France email: cahen@univmetz.fr. Abstract: Let $G$ be a connected semisimple noncompact Lie group and $\pi$ a principal series representation of $G$. Let $\cal O$ be the coadjoint orbit of $G$ associated by the KirillovKostant method of orbits to the representation $\pi$. By dequantizing $\pi$ we construct an explicit symplectomorphism between a dense open set of $\cal O$ and a symplectic product ${\mathbb R}^{2n}\times{\cal O}'$ where ${\cal O}' $ is a coadjoint orbit of a compact subgroup of $G$. This allows us to obtain a Weyl correspondence on $\cal O$ which is adapted to the representation $\pi$ in the sense of [6]. Keywords: Principal series representations, coadjoint orbits, Weyl quantization, Berezin quantization, dequantization. Classification (MSC2000): 81S10, 22E46 Full text of the article:
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