Poles of intertwining operators in terms of irreducible components of Lusztig's characteristic variety in type A

TL;DR

This paper proposes a conjectural formula linking the poles of intertwining operators to the geometry of Lusztig's characteristic variety in type A, verified in many cases.

Contribution

It introduces a new conjecture connecting pole orders of intertwining operators with geometric components in Lusztig's variety for type A.

Findings

Conjectural formula relating pole order to Hom-space dimension.

Verification of the conjecture in numerous cases.

Insight into the geometric structure underlying representation theory.

Abstract

In this paper, we propose a conjectural formula for the order of the poles of intertwining operators in the context of the representation theory of general linear groups over $p$-adic fields. More specifically, we conjecturally relate the order of the pole to the dimension of a Hom-space associated with irreducible components of Lusztig's characteristic variety in type $A$. We verify the conjecture in a wide range of cases.