Generic smooth representations

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 $\mathrm{\Omega }$. The work of Schneider–Zink gives an irreducible $K$-representation ${\sigma }_{min}(\lambda )$, which appears with multiplicity one in ${\mathrm{I}\mathrm{n}\mathrm{d}}_J^K\lambda$. Let $\pi$ be an irreducible smooth representation of $G{L}_n(F)$ in $\mathrm{\Omega }$. We will prove that $\pi$ is generic if and only if ${\sigma }_{min}(\lambda )$ is contained in $\pi$ with multiplicity one.