Generic smooth representations

Abstract

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 $\mathrm{Ind}_J^K \lambda$. 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 $\pi$ with multiplicity one.