Weyl quantization for principal series

Benjamin Cahen

Abstract: Let $G$ be a connected semisimple non-compact Lie group and $\pi$ a principal series representation of $G$. Let $\cal O$ be the coadjoint orbit of $G$ associated by the Kirillov-Kostant 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

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