Unicity of types for supercuspidals

Let $F$ be a non-Archimedean local field, with the ring of integers ${\mathfrak{o}}_F\mathrm{.}Let$G=GL_N(F)$,$K=GL_N(\mathfrak{o}_F)$and$\pi$ a supercuspidal representation of $G$. We show that there exist a unique irreducible smooth representation $\tau$ of $K$, such that the restriction to $K$ of a smooth irreducible representation ${\pi }^{\mathrm{\prime }}$ of $G$ contains $\tau$ if and only if $p{i}^{\mathrm{\prime }}$ is isomorphic to $\pi \otimes \chi \circ \det$, where $\chi$ is an unramified quasicharacter of $F^{\times }$. Moreover, we show that $\pi$ contains $\tau$ with the multiplicity 1. As a corollary we obtain a kind of inertial local Langlands correspondence.