Let G=exp( g) be a nilpotent Lie group, H=exp( h) a closed subgroup of G .Let = f be a unitary character of H and = ind G H the induced representation of G .Let us suppose that the multiplicities of the irreducible representations occuring in the disintegration of are finite.We prove in this note the conjecture of Corwin-Greenleaf, which says that the algebra D (G/H) of the differential operators which commute with is isomorhic to the algebra C H under the condition that the H -orbits in are of dimension 1 .
H. Fujiwara (Wed,) studied this question.