2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67724Let $F$ 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 $π$ a supercuspidal representation of $G$. We show that there exist a unique irreducible smooth representation $τ$ of $K$, such that the restriction to $K$ of a smooth irreducible representation $π'$ of $G$ contains $τ$ if and only if $pi'$ is isomorphic to $π\otimesχ\circ\det$, where $χ$ is an unramified quasicharacter of $F^{\times}$. Moreover, we show that $π$ contains $τ$ with the multiplicity 1. As a corollary we obtain a kind of inertial local Langlands correspondence.42 pagesNumber TheoryRepresentation Theory22E50text