Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments I: reduction to combinatorics

TL;DR

This paper introduces a combinatorial framework using highest derivative multisegments to analyze simple quotients of Bernstein-Zelevinsky derivatives of irreducible representations of GL_n(F), with a focus on reduction to combinatorics.

Contribution

It defines highest derivative multisegments and develops a combinatorial approach to study simple quotients, including a double derivative result.

Findings

Introduces highest derivative multisegments concept.

Provides a combinatorial method for analyzing derivatives.

Proves a double derivative theorem.

Abstract

Let $F$ be a local non-Archimedean field. A sequence of derivatives of generalized Steinberg representations can be used to construct simple quotients of Bernstein-Zelevinsky derivatives of irreducible representations of $\mathrm{GL}_n(F)$. In the first of a series of articles, we introduce a notion of a highest derivative multisegment, which in turn gives a combinatorial approach to study problems about those simple quotients. We also prove a double derivative result along the way.