Unicity of types for supercuspidals

Let $F$ be a non-Archimedean local field, with the ring of integers $\mathfrak{o}_F. Let$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'$of$G$contains$\tau$if and only if$pi'$is isomorphic to$\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.