Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments II: Minimal sequences

TL;DR

This paper investigates the structure of certain quotient representations of Bernstein-Zelevinsky derivatives of GL(n) representations, establishing the existence of a unique minimal multisegment within a poset framework using new ordering concepts.

Contribution

It introduces the concepts of fine chain orderings and local minimizability to analyze the poset of multisegments associated with derivatives, extending previous work on simple quotients.

Findings

The poset of multisegments has a unique minimal element.

Introduction of fine chain orderings for multisegment analysis.

Development of local minimizability as a new tool.

Abstract

Let $F$ be a non-Archimedean local field. For any irreducible smooth representation $\pi$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, we have an operation $D_{\mathfrak m}(\pi)$ to construct a simple quotient $\tau$ of a Bernstein-Zelevinsky derivative of $\pi$. This article continues the previous one to study the following poset \[ \mathcal S(\pi, \tau) :=\left\{ \mathfrak n : D_{\mathfrak n}(\pi)\cong \tau \right\} , \] where $\mathfrak n$ runs for all the multisegments. Here the partial ordering on $\mathcal S(\pi, \tau)$ comes from the Zelevinsky ordering. We show that the poset has a unique minimal multisegment. Along the way, we introduce two new ingredients: fine chain orderings and local minimizability.