An inductive Ext non-vanishing theorem for the $p$-adic general linear group

TL;DR

This paper investigates homological properties of parabolic induction in the $p$-adic general linear group, establishing an Ext-group embedding theorem and a variation of the Gan-Gross-Prasad conjecture related to homological branching laws.

Contribution

It introduces a new embedding theorem for Ext-groups in the context of parabolic induction and applies it to prove a variation of the Gan-Gross-Prasad conjecture for $p$-adic groups.

Findings

Established an embedding theorem for Ext-groups

Proved a variation of the Gan-Gross-Prasad conjecture

Enhanced understanding of homological properties in $p$-adic representation theory

Abstract

We study some homological properties of the parabolic induction functor for the $p$-adic general linear group. We obtain an embedding theorem of Ext-groups in the context of parabolic induction. As an application, we establish and prove a variation of the non-tempered Gan-Gross-Prasad conjecture in homological branching laws for $p$-adic general linear groups.