On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces
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- by Hidénori Fujiwara, Gérard Lion and Salah Mehdi
- Trans. Amer. Math. Soc. 353 (2001), 4203-4217
- DOI: https://doi.org/10.1090/S0002-9947-01-02850-1
- Published electronically: June 6, 2001
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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.References
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Bibliographic 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
- MR Author ID: 609901
- Email: smehdi@math.okstate.edu
- Received by editor(s): March 17, 2000
- Published electronically: June 6, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 4203-4217
- MSC (2000): Primary 43A85, 22E27, 22E30
- DOI: https://doi.org/10.1090/S0002-9947-01-02850-1
- MathSciNet review: 1837228