Algorithms for parabolic inductions and Jacquet modules in ${\mathrm{GL}}_n$

Abstract

In this article, we present algorithms for computing parabolic inductions and Jacquet modules for the general linear group $G$ over a non-Archimedean local field. Given the Zelevinsky data or Langlands data of an irreducible smooth representation $\ddot{I}€$ of $G$ and an essentially square-integrable representation $\ddot{I}ƒ$, we explicitly determine the Jacquet module of $\ddot{I}€$ with respect to $\ddot{I}ƒ$ and the socle of the normalized parabolic induction $\ddot{I}€\times \ddot{I}ƒ$. Our result builds on and extends some previous work of Mœglin-Waldspurger, Jantzen, Mínguez, and Lapid-Mínguez, and also uses other methods such as sequences of derivatives and an exotic duality. As an application, we give a simple algorithm for computing the highest derivative multisegment and an algorithm for computing the Langlands parameter of the highest Bernstein-Zelevinsky derivatives.

v1: 46pages, v2: minor editions