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On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces
Authors:Hidé  nori Fujiwara   Gé  rard Lion   Salah Mehdi
Affiliation:Faculté de Technologie à Kyushu, Université de Kinki, Iizuka 820-8555, Japon ; Equipe Modal'X, Université Paris X, 200 Avenue de la République, 92001 Nanterre, France - Equipe de Théorie des Groupes, Représentations et Applications, Institut de Mathé- matiques de Jussieu, Université Paris VII, 2 Place Jussieu, 75251 Paris Cedex 05, France ; Equipe Modal'X, Université Paris X, 200 Avenue de la République, 92001 Nanterre, France - Equipe de Théorie des Groupes, Représentations et Applications, Institut de Mathé- matiques de Jussieu, Université Paris VII, 2 Place Jussieu, 75251 Paris Cedex 05, France
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 $betainmathfrak{h}^{*}$ satisfying $beta ([mathfrak{h},mathfrak{h} ])={0}$. Let $chi_{beta}$ be the unitary character of $H$ with differential $2sqrt{-1}pibeta$ at the origin. Let $tauequiv$ $Ind_{H}^{G}chi_{beta}$ 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 $Hbackslash G$ 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.

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