On parabolic induction on inner forms of the general linear group over a non-archimedean local field

TL;DR

This paper establishes new criteria for the irreducibility of parabolic induction on inner forms of the general linear group over non-archimedean fields, simplifying the classification of the unitary dual.

Contribution

It provides a necessary and sufficient condition for irreducibility when the inducing data involves ladder representations, advancing understanding of representation theory.

Findings

New irreducibility criteria for parabolic induction

Simplified proof of unitary dual classification

Characterization for ladder representation induction

Abstract

We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form $\pi\otimes\sigma$ where $\pi$ is a ladder representation and $\sigma$ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.