Resumen de Unicity of types for supercuspidal representations of $ {\rm GL}_N$

Vytautas Paskunas

• Let $F$ be a non-Archimedean local field, with the ring of integers $\mathfrak{o}_F$. Let $G = \mathrm{GL}_N(F)$, $K = \mathrm{GL}_N (\mathfrak{o}_F)$, and $\pi$ be a supercuspidal representation of $G$. We show that there exists 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.