Weyl quantization for semidirect products

Let G be the semidirect product V x K where K is a connected semisimple non-compact Lie group acting linearity on a finite-dimensional real vector space V. Let 0 be a coadjoint orbit of G associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation pi of G. We consider the case when the corresponding little group K-0 is a maximal compact subgroup of K. We realize the representation pi on a Hilbert space of functions on R '' where n = dim(K) - dim(K-0). By dequantizing pi we then construct a symplectomorphism between the orbit 0 and the product R-2n x O' where O' is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on 0 which is adapted to the representation pi in the sense of [B. Cahen, Quantification d'une orbite massive d'un groupe de Poincare gendralise, C. R. Acad. Sci. Paris Serie 1 325 (1997) 803-806]. In particular we recover well-known results for the Poincare group. (c) 2006 Elsevier B. V. All rights reserved.