Weyl quantization for semidirect products

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 K0 is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on Rn where n=dim(K)−dim(K0). By dequantizing π we then construct a symplectomorphism between the orbit O and the product R2n×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.