Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments III: properties of minimal sequences

TL;DR

This paper investigates properties of minimal sequences of multisegments associated with Bernstein-Zelevinsky derivatives of irreducible representations of GL(n), establishing their commutativity and related properties, and proposing conjectures on module structures.

Contribution

It proves the commutativity and properties of minimal sequences of multisegments in the context of Bernstein-Zelevinsky derivatives, extending previous work on simple quotients.

Findings

Minimal sequences are shown to be commutative.

Properties of minimal sequences are established.

Conjectures on module structures from minimality are proposed.

Abstract

Let $F$ be a non-Archimedean local field. For an irreducible smooth representation $\pi$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, one associates a simple quotient $D_{\mathfrak m}(\pi)$ of a Bernstein-Zelevinsky derivative of $\pi$. In the preceding article, we showed that \[ \mathcal S(\pi, \tau) :=\left\{ \mathfrak m : D_{\mathfrak m}(\pi)\cong \tau \right\} , \] has a unique minimal element under the Zelevinsky ordering, where $\mathfrak m$ runs for all multisegments. The main result of this article includes commutativity and subsequent property of the minimal sequence. At the end of this article, we conjecture some module structure arising from the minimality.