Un critère de commutativité pour l'algèbre des opérateurs différentiels invariants sur un espace homogène nilpotent

Abstract

Let G be a connected simply connected real nilpotent Lie group, H a connected closed subgroup of G with Lie algebra h and f a linear form on h satisfying f([ h, h])={0} . Let χ f be the unitary character of H with differential −1 f at the origin. Let τ≡ Ind H Gχ f be the unitary representation of G induced from the character χ f of H. Let D G,H,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 τ is of finite multiplicities, this algebra is commutative. We prove here the converse part.