Bernstein-zelevinsky derivatives: A hecke algebra approach

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.