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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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 PDF
Trans. Amer. Math. Soc. 353 (2001), 4203-4217 Request permission

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
  • 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