Monatsh. Math. 134, 19±37 (2001) Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces By A. Baklouti 1 and J. Ludwig 2 1 Faculte  des Sciences de Sfax, Tunisia 2 Universite  de Metz, France (Received 15 November 2000) Abstract. Let G exp g be a nilpotent connected and simply connected Lie group, and H exp h an analytic subgroup of G. Let f ; f 2 g , be a unitary character of H and let Ind G H . Suppose that the multiplicities of all the irreducible components of are ®nite. Corwin and Greenleaf conjectured that the algebra D G=H of the differential operators on the Schwartz-space of which commute with is isomorphic to the algebra of H-invariant polynomials on the af®ne space f h ? . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in f h ? . We prove also that if h is an ideal of g, then the ®nite multiplicities of is equivalent to the fact that the algebra D G=H is commutative. 2000 Mathematics Subject Classi®cation: 22 E 27 Key words: Representation, orbit, multiplicity, polarization, operator 1. Introduction Let G be a connected simply connected nilpotent Lie group and let g be the Lie algebra of G. We consider the monomial representation f ;h Ind G H induced by the unitary character of a closed connected subgroup H exph of G, where exp X e ih f ;Xi for X 2 h; f 2 g and f jh is a Lie homomorphism. In [7], Corwin, Greenleaf and Gre Âlaud (and Lipsman in [13]) gave an orbital description of the irreducible representations 2 ^ G appearing in the disintegration of . In particular they showed that ' f h ? =H l dl; 1:1 where H acts on f f ;h f h ? by the coadjoint action, l corresponds to l 2 f by the Kirillov correspondance, and is the push-forward of a ®nite measure equivalent to Lebesgue measure. The decomposition (1.1) leads to a primary decomposition ' ^ G md; 1:2 where is the push-forwards of under the Kirillov map : g ! ^ G; m is the number of Ad H-orbits in f \ , where is the coadjoint orbit associated to .