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Représentations banales de ${\rm GL}_{m}({\rm D})$

Part of: Lie groups

Published online by Cambridge University Press:  02 January 2013

Alberto Mínguez  and
Vincent Sécherre
Alberto Mínguez
Affiliation:
Institut de Mathématiques de Jussieu, 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 (email: vincent.secherre@math.uvsq.fr)
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Abstract

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Let ${\rm F}$ be a non-Archimedean locally compact field of residue characteristic $p$, let ${\rm D}$ be a finite-dimensional central division ${\rm F}$-algebra and let ${\rm R}$ be an algebraically closed field of characteristic different from $p$. We define banal irreducible ${\rm R}$-representations of the group ${\rm G}={\rm GL}_{m}({\rm D})$. This notion involves a condition on the cuspidal support of the representation depending on the characteristic of ${\rm R}$. When this characteristic is banal with respect to ${\rm G}$, in particular when ${\rm R}$ is the field of complex numbers, any irreducible ${\rm R}$-representation of ${\rm G}$ is banal. In this article, we give a classification of all banal irreducible ${\rm R}$-representations of ${\rm G}$ in terms of certain multisegments, called banal. When ${\rm R}$ is the field of complex numbers, our method provides a new proof, entirely local, of Tadić’s classification of irreducible complex smooth representations of ${\rm G}$.

Keywords

representations of $p$-adic groupsmodular representationmultisegmentbanal representation

MSC classification

Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 4 , April 2013 , pp. 679 - 704
Copyright
Copyright © 2013 The Author(s)

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