Algorithms for parabolic inductions and Jacquet modules in GLₙ

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 × σ. Our result builds on and extends some previous work of M glin-Waldspurger, Jantzen, M\'inguez, and Lapid-M\'inguez, 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.