Invariant differential operators on certain nilpotent homogeneous spaces

Let G = exp g be a nilpotent connected and simply connected Lie group, and H = exp h an analytic subgroup of G. Let chi = chi (f), f is an element of g*, be a unitary character of H and let tau = Ind(H)(G)chi. Suppose that the multiplicities of all the irreducible components of tau are finite. Corwin and Greenleaf conjectured that the algebra D-tau (G/H) of the differential operators on the Schwartz-space of tau which commute with tau is isomorphic to the algebra of H-invariant polynomials on the affine space f + h(perpendicular to). We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in f + h(perpendicular to). We prove also that if h is an ideal of g, then the finite multiplicities of tau is equivalent to the fact that the algebra D-tau(G/H) is commutative.