JGP Vol.3, n. 3, 1986 Deformations on coadjoint orbits D. ARNAL et M. CAHEN (4) Université de Nancy S. GUTT (4) (*4) Université de Metz Abstract. A deformation of the polynomial algebra S(~) on ~ when S(~1) is a free I(~)module (I(ç~) = algebra of invariant polinomials). This deformation restricts nicely to a large class of orbits. We also give an example to show that deformations of S(~) restricting to orbits may not always be defined by bidifferen- tial operators. 0. INTRODUCTION Deformations (and * products which form a particular class of deformations) have been introduced by M. Flato, C. Fronsdal and A. Lichnerowicz (see for instance [1]) in order to give a formulation of quantum mechanics without ope- rators in the general framework of a Poisson manifold; this geometrical approach to quantum theory is of a different nature from what one calls usually geometric quantization [2, 3]. Geometric quantization has been from the start deeply related to the method of orbits in the representation theory of Lie groups. In this context one of the important results has been the description of unitary irreducible representations of solvable groups [4] and of certains classes of representations of more general (*) On leave of absence from Université Libre de Bruxelles. (**) Chercheur qualifle au F.N.R.S. Key-words: Deformations; orbits in the dual of the Lie algebra. Universal enveloping algebra. 1980 Mathematics Subject Classification: 8] CXX C 99.