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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Hidénori Fujiwara, Gérard Lion and Salah Mehdi
Journal: Trans. Amer. Math. Soc. 353 (2001), 4203-4217
MSC (2000): Primary 43A85, 22E27, 22E30
Published electronically: June 6, 2001
MathSciNet review: 1837228
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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.


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Additional Information

Hidénori Fujiwara
Affiliation: Faculté de Technologie à Kyushu, Université de Kinki, Iizuka 820-8555, Japon
Email: fujiwara@fuk.kindai.ac.jp

Gérard Lion
Affiliation: 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
Email: glion@math.jussieu.fr

Salah Mehdi
Affiliation: 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
Address at time of publication: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: smehdi@math.okstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02850-1
PII: S 0002-9947(01)02850-1
Received by editor(s): March 17, 2000
Published electronically: June 6, 2001
Article copyright: © Copyright 2001 American Mathematical Society