Unicity of types for supercuspidals

TL;DR

This paper proves the uniqueness of certain types within supercuspidal representations of GL_N over non-Archimedean local fields, establishing a one-to-one correspondence with unramified characters and confirming multiplicity one results.

Contribution

It establishes the unicity of types for supercuspidal representations of GL_N, linking them to unramified characters and providing a form of inertial local Langlands correspondence.

Findings

Unique irreducible smooth representation of K contained in supercuspidals

Supercuspidals contain the type with multiplicity one

Establishment of a local Langlands correspondence variant

Abstract

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.