Parabolic induction via Hecke algebras and the Zelevinsky dualityconjecture

Abstract

This paper describes parabolic induction, for smooth representations of finite length of the general linear group $\mbox{GL}(N, F)$ of a non-archimedean local field $F$, in terms of a functor between categories of modules over certain affine Hecke algebras, using the 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 involutive automorphism 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 the study of multiplicities of composition factors of induced representations.

1991 Mathematics Subject Classification: 11S37, 22E50.