This paper describes parabolic induction,for smooth representations of finite length of the general linear group $\mbox{GL}(N, F)$ of a non-archimedeanlocal field $F$, in terms of a functor between categories of modules over certain affine Hecke algebras, usingthe categorical equivalences developed in the Bushnell-Kutzko classification of the admissible dual of$\mbox{GL}(N, F)$. This result is used to prove that the Zelevinsky automorphism, which is an involutiveautomorphism on the representation ring of the category of smooth representations of finite length of$\mbox{GL}(N, F)$, preserves the irreducible representations. The result also implies a simplification of thestudy of multiplicities of composition factors of induced representations.

1991 Mathematics SubjectClassification: 11S37, 22E50.