A commutativity criterion for the algebra of invariant differential operators on a nilpotent homogeneous space

Let G be a connected simply connected real nilpotent Lie group, H a connect ed closed subgroup of G with Lie algebra b and f a linear form on tl satisf ying f([h,h) = {0}. Let chif be the unitary character of H with differentia l root -1f at the origin. Let tau = Ind(H chif)(G) be the unitary represent ation of G induced from the character chi (f) of H. Let D-G,D-H,D-f be the algebra of G-invariant differential operators on the bundle with basis G/H associated to these data. Corwin and Greenleaf have shown in 1992 that if t au is of finite multiplicities, this algebra is commutative. We prove here the converse part. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.