On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces

Let $G$ be a simply connected connected real nilpotent Lie group with Lie algebra $\mathfrak{g}$, $H$ a connected closed subgroup of $G$ with Lie algebra $\mathfrak{h}$ and $\beta\in\mathfrak{h}^{*}$ satisfying $\beta ([\mathfrak{h},\mathfrak{h} ])=\{0\}$. Let $\chi_{\beta}$ be the unitary character of $H$ with differential $2\sqrt{-1}\pi\beta$ at the origin. Let $\tau\equiv$ $Ind_{H}^{G}\chi_{\beta}$ be the unitary representation of $G$ induced from the character $\chi_{\beta}$ of $H$. We consider the algebra $\mathcal{D}(G,H,\beta)$ of differential operators invariant under the action of $G$ on the bundle with basis $H\backslash G$ associated to these data. We consider the question of the equivalence between the commutativity of $\mathcal{D}(G,H,\beta)$ and the finite multiplicities of $\tau$. Corwin and Greenleaf proved that if $\tau$ is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases.