# Bernstein-Zelevinsky derivatives: a Hecke algebra approach

## TL;DR

This paper connects Bernstein-Zelevinsky derivatives of smooth representations of general linear groups over p-adic fields with Hecke algebra modules, providing explicit descriptions and a comprehensive theoretical framework.

## Contribution

It introduces a Hecke algebra approach to Bernstein-Zelevinsky derivatives, offering explicit module descriptions and a unified theory for these derivatives.

## Key findings

- Explicit Hecke algebra modules for Bernstein components of Gelfand-Graev representations
- Translation of Bernstein-Zelevinsky derivatives into Hecke algebra language
- Development of a comprehensive theory of derivatives via Hecke algebra representations

## Abstract

Let $G$ be a general linear group over a $p$-adic field. It is well known that Bernstein components of the category of smooth representations of $G$ are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of $G$ by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.