Location of reducibility points of induced representations I: A toy example

TL;DR

This paper investigates the reducibility points of induced representations of two Speh representations over p-adic fields by analyzing singularities of intertwining operators, proposing an analytic approach inspired by M{\

Contribution

It introduces a new analytic method to determine reducibility points, potentially extendable to classical groups, based on singularity analysis of intertwining operators.

Findings

Identifies explicit reducibility points for specific induced representations.

Suggests the analytic approach may generalize to classical groups.

Stimulates new questions in the representation theory of p-adic groups.

Abstract

By analyzing the singularity of standard intertwining operators, we provide a new way to understand the explicit location of reducibility points of induced representations of two Speh representations for general linear groups over a p-adic field. Through playing with this toy example, it seems that the analytic approach, in the spirit of M{\oe}glin--Waldspurger, could play a role for analogous reducibility problems in the setting of classical groups. On the other hand, it also stimulates the arising of some interesting questions.