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L'involution de Zelevinski modulo $ \ell$

Authors: Alberto Mínguez and Vincent Sécherre
Journal: Represent. Theory 19 (2015), 236-262
MSC (2010): Primary 22E50, 20G40
Published electronically: October 29, 2015
MathSciNet review: 3416734
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Abstract: 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.

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

Keywords: Modular representations, $p$-adic reductive groups, finite reductive groups, Zelevinski involution, Alvis-Curtis duality, type theory
Received by editor(s): December 17, 2014
Received by editor(s) in revised form: August 28, 2015
Published electronically: October 29, 2015
Article copyright: © Copyright 2015 American Mathematical Society