On the commutativity of the algebra of invariant differential operators oncertain nilpotent homogeneous spaces

Let G be a simply connected connected real nilpotent Lie group with Lie alg ebra g, H a connected closed subgroup of G with Lie algebra h and beta is a n element of h* satisfying beta([h, h]) = {0}. Let chi (beta) be the unitar y character of H with differential 2 root -1 pi beta at the origin. Let tau equivalent to Ind(H)(G)chi (beta) be the unitary representation of G induc ed from the character chi (beta) of H. We consider the algebra D (G; H, bet a) of differential operators invariant under the action of G on the bundle with basis H\G associated to these data. We consider the question of the eq uivalence between the commutativity of D (G, H, beta) and the finite multip licities of tau. Corwin and Greenleaf proved that if tau is of finite multi plicities, this algebra is commutative. We show that the converse is true i n many cases.