Weyl quantization for semidirect products

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Abstract

Let G be the semidirect product V ⋊ K where K is a connected semisimple non-compact Lie group acting linearily 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 K0is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on Rnwhere n = dim (K) - dim (K0). By dequantizing π we then construct a symplectomorphism between the orbit O and the product R2 n× O′where O′is a little group coadjoint orbit. 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 Série I 325 (1997) 803-806]. In particular we recover well-known results for the Poincaré group. © 2006 Elsevier B.V. All rights reserved.

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Cahen, B. (2007). Weyl quantization for semidirect products. Differential Geometry and Its Application, 25(2), 177–190. https://doi.org/10.1016/j.difgeo.2006.08.005

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