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 $π$ of $G$ and an essentially square-integrable representation $σ$, we explicitly determine the Jacquet module of $π$ with respect to $σ$ and the socle of the normalized parabolic induction $π\times σ$. 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.