Local newforms for the general linear groups over a non-archimedean local field

TL;DR

This paper extends the theory of local newforms for irreducible generic representations of p-adic general linear groups by defining new compact open subgroups and introducing Rankin--Selberg integrals for Speh representations.

Contribution

It introduces a new family of compact open subgroups and extends local newform theory to all irreducible representations, including Speh representations.

Findings

Extended local newform theory to all irreducible representations

Defined new compact open subgroups indexed by tuples

Introduced Rankin--Selberg integrals for Speh representations

Abstract

In [12], Jacquet--Piatetskii-Shapiro--Shalika defined a family of compact open subgroups of $p$-adic general linear groups indexed by non-negative integers, and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of non-negative integers. For the proof, we introduce the Rankin--Selberg integrals for Speh representations.