Geodesic
Generic Smooth Representations
Documenta mathematica, Tome 25 (2020), pp. 2473-2485.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $F$ be a non-archimedean local field. In this paper we explore genericity of irreducible smooth representations of $GL_n(F)$ by restriction to a maximal compact subgroup $K$ of $GL_n(F)$. Let $(J, \lambda)$ be a Bushnell-Kutzko type for a Bernstein component $\Omega $. The work of Schneider-Zink gives an irreducible $K$-representation $\sigma_{min}(\lambda)$, which appears with multiplicity one in $\text{Ind}_J^K \lambda $. Let $\pi$ be an irreducible smooth representation of $GL_n(F)$ in $\Omega $. We will prove that $\pi$ is generic if and only if $\sigma_{min}(\lambda)$ is contained in $\pi $, in which case it occurs with multiplicity one.
Classification : 22E50, 11F70, 11F85
Keywords: representations, \(p\)-adic groups 
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     author = {Pyvovarov, Alexandre},
     title = {Generic {Smooth} {Representations}},
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Pyvovarov, Alexandre. Generic Smooth Representations. Documenta mathematica, Tome 25 (2020), pp. 2473-2485. http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a3/