  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.
