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Representation Theory

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

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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L’involution de Zelevinski modulo $\ell$
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by Alberto Mínguez and Vincent Sécherre PDF
Represent. Theory 19 (2015), 236-262 Request permission


Let $\mathrm {F}$ be a non-Archimedean locally compact field with residual characteristic $p$, let $\mathrm {G}$ be an inner form of $\mathrm {GL}_n(\mathrm {F})$, $n\geqslant 1$ and let $\mathrm {R}$ be an algebraically closed field of characteristic different from $p$. When $\mathrm {R}$ has characteristic $\ell >0$, the image of an irreducible smooth $\mathrm {R}$-representation $\pi$ of $\mathrm {G}$ by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of $\mathrm {G}$) contains a unique irreducible term $\pi ^\star$ with the same cuspidal support as $\pi$. This defines an involution $\pi \mapsto \pi ^\star$ on the set of isomorphism classes of irreducible $\mathrm {R}$-representations of $\mathrm {G}$, that coincides with the Zelevinski involution when $\mathrm {R}$ is the field of complex numbers. The method we use also works for $\mathrm {F}$ a finite field of characteristic $p$, in which case we get a similar result.
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Additional Information
  • Alberto Mínguez
  • Affiliation: Institut de Mathématiques de Jussieu, Université Paris 6, 4 place Jussieu, 75005, Paris, France
  • Address at time of publication: Institut de Mathématiques de Jussieu – Paris Rive Gauche, Université Pierre et Marie Curie, 4 place Jussieu, 75005, Paris, France.
  • Email:
  • Vincent Sécherre
  • Affiliation: Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, 45 avenue des Etats-Unis, 78035 Versailles cedex, France
  • Address at time of publication: Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France
  • MR Author ID: 741262
  • Email:
  • Received by editor(s): December 17, 2014
  • Received by editor(s) in revised form: August 28, 2015
  • Published electronically: October 29, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Represent. Theory 19 (2015), 236-262
  • MSC (2010): Primary 22E50, 20G40
  • DOI:
  • MathSciNet review: 3416734