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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


A proof of the polynomial conjecture for restrictions of nilpotent lie groups representations
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by Ali Baklouti, Hidenori Fujiwara and Jean Ludwig
Represent. Theory 26 (2022), 616-634
Published electronically: June 2, 2022


Let $G$ be a connected and simply connected nilpotent Lie group, $K$ an analytic subgroup of $G$ and $\pi$ an irreducible unitary representation of $G$ whose coadjoint orbit of $G$ is denoted by $\Omega (\pi )$. Let $\mathscr U(\mathfrak g)$ be the enveloping algebra of ${\mathfrak g}_{\mathbb C}$, $\mathfrak g$ designating the Lie algebra of $G$. We consider the algebra $D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K$ of the $K$-invariant elements of $\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )$. It turns out that this algebra is commutative if and only if the restriction $\pi |_K$ of $\pi$ to $K$ has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that $D_{\pi }(G)^K$ is isomorphic to the algebra $\mathbb C[\Omega (\pi )]^K$ of $K$-invariant polynomial functions on $\Omega (\pi )$. The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341).
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Bibliographic Information
  • Ali Baklouti
  • Affiliation: Laboratory LAMHA, Faculty of Sciences of Sfax, University of Sfax, Route de Soukra, BP 1171, Sfax 3038, Tunisia
  • MR Author ID: 347205
  • Email:
  • Hidenori Fujiwara
  • Affiliation: Faculté de Science et Technologie pour l’Humanité, Université de Kinki, Iizuka 820-8555, Japan
  • Email:
  • Jean Ludwig
  • Affiliation: Institut Elie Cartan de Lorraine, Université de Lorraine, Site de Metz, 3, rue Augustin Fresnel, 57000 Metz, Technopole Metz France
  • MR Author ID: 224117
  • Email:
  • Received by editor(s): February 22, 2021
  • Received by editor(s) in revised form: September 8, 2021
  • Published electronically: June 2, 2022
  • Additional Notes: This research has been partially supported by the DGRSRT through the Research Laboratory LR 11ES52

  • Dedicated: This work is dedicated to the memory of Takaaki Nomura
  • © Copyright 2022 American Mathematical Society
  • Journal: Represent. Theory 26 (2022), 616-634
  • MSC (2020): Primary 22E27
  • DOI:
  • MathSciNet review: 4433082