Weyl quantization for semidirect products

Abstract: Let G be the semidirect product where K is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space V. Let 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 is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on where . By dequantizing π we then construct a symplectomorphism between the orbit and the product where is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on 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. [Copyright &y& Elsevier]