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Représentations banales de
${\rm GL}_{m}({\rm D})$
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 4 / April 2013
- Published online by Cambridge University Press:
- 02 January 2013, pp. 679-704
- Print publication:
- April 2013
-
- Article
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- You have access
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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 
${\rm G}$.
