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 connected closed subgroup of G with Lie algebra b and f a linear form on tl satisfying f([h,h) = {0}. Let chif be the unitary character of H with differential root -1f at the origin. Let tau = Ind(H chif)(G) be the unitary representation 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 tau 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.