Generic smooth representations

TL;DR

This paper characterizes the genericity of irreducible smooth representations of $GL_n(F)$ over a non-archimedean local field by examining their restriction to a maximal compact subgroup, linking it to specific $K$-representations.

Contribution

It establishes a precise criterion for genericity of representations based on their restriction to a maximal compact subgroup, connecting Bushnell--Kutzko types and Schneider--Zink results.

Findings

A representation is generic if and only if a specific $K$-representation appears with multiplicity one.

Provides a new characterization of genericity in terms of restriction to compact subgroups.

Links the theory of types with the concept of genericity in smooth representations.

Abstract

Let $F$ be a non-archimedean local field. In this paper we explore genericity of irreducible smooth representations of $GL_n(F)$ by restriction to a maximal compact subgroup $K$ of $GL_n(F)$. Let $(J, \lambda)$ be a Bushnell--Kutzko type for a Bernstein component $\Omega$. The work of Schneider--Zink gives an irreducible $K$-representation $\sigma_{min}(\lambda)$, which appears with multiplicity one in $\mathrm{Ind}_J^K \lambda$. Let $\pi$ be an irreducible smooth representation of $GL_n(F)$ in $\Omega$. We will prove that $\pi$ is generic if and only if $\sigma_{min}(\lambda)$ is contained in $\pi$ with multiplicity one.