Algorithms for parabolic inductions and Jacquet modules in $\mathrm{GL}_n$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911349006663680 |
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| author | Chan, Kei Yuen Pattanayak, Basudev |
| author_facet | Chan, Kei Yuen Pattanayak, Basudev |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00886 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algorithms for parabolic inductions and Jacquet modules in $\mathrm{GL}_n$ Chan, Kei Yuen Pattanayak, Basudev Representation Theory Number Theory 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. |
| title | Algorithms for parabolic inductions and Jacquet modules in $\mathrm{GL}_n$ |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2503.00886 |