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
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.References
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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: minguez@math.jussieu.fr
- 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: vincent.secherre@math.uvsq.fr
- 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: https://doi.org/10.1090/ert/466
- MathSciNet review: 3416734