Title: Algorithms for parabolic inductions and Jacquet modules in $\mathrm{GL}_n$ Show Chinese title

Title: 在$\mathrm{GL}_n$中的抛物诱导和雅克布斯模块算法

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 $\pi$ of $G$ and an essentially square-integrable representation $\sigma$, we explicitly determine the Jacquet module of $\pi$ with respect to $\sigma$ and the socle of the normalized parabolic induction $\pi \times \sigma$. Our result builds on and extends some previous work of M\oe 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. Show Chinese abstract

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 $\pi$ of $G$ and an essentially square-integrable representation $\sigma$, we explicitly determine the Jacquet module of $\pi$ with respect to $\sigma$ and the socle of the normalized parabolic induction $\pi \times \sigma$. 我们的结果建立在M\oe glin-Waldspurger、Jantzen、Mínguez和Lapid-Mínguez的一些先前工作之上，并且还使用了其他方法，如导数序列和一种奇特的对偶性。 作为应用，我们给出了一个计算最高导数多段的简单算法。