Generic Smooth Representations

Abstract

Let $F$ be a non-archimedean local field. In this paper we explore genericity of irreducible smooth representations of $G{L}_n(F)$ by restriction to a maximal compact subgroup $K$ of $G{L}_n(F)$. 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 $G{L}_n(F)$ 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.