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

Abstract

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.