Structure of Twisted Jacquet Modules of principal series representations of $Sp_{4}(F)$
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arXiv
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| Format: | Preprint |
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2024
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| author | Pandey, Sanjeev Kumar Venketasubramanian, C G |
| author_facet | Pandey, Sanjeev Kumar Venketasubramanian, C G |
| contents | Let $F$ be a non-archimedean local field. For the symplectic group $Sp_{4}(F),$ let $P$ and $Q$ denote respectively its Siegel and Klingen parabolic subgroups with respective Levi decompositions $P=MN$ and $Q=LU.$ For a non-trivial character $ψ$ of the unipotent radical $N$ of $P,$ let $M_ψ$ denote the stabilizer of the character $ψ$ in $M$ under the conjugation action of $M$ on characters of $N.$ For an irreducible representation of the Levi subgroups $M$ or $L,$ let $π$ denote the respective representation of $Sp_{4}(F)$ parabolically induced either from $P$ or from $Q.$ Let $ψ$ be a character of the group $N$ given by a rank one quadratic form. In this article, we determine the structure of the twisted Jacquet module $r_{N,ψ}(π)$ as an $M_ψ$-module. We also deduce the analogous results in the case where $F$ is a finite field of order $q.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure of Twisted Jacquet Modules of principal series representations of $Sp_{4}(F)$ Pandey, Sanjeev Kumar Venketasubramanian, C G Representation Theory 11F70(Primary), 22E50(Secondary) Let $F$ be a non-archimedean local field. For the symplectic group $Sp_{4}(F),$ let $P$ and $Q$ denote respectively its Siegel and Klingen parabolic subgroups with respective Levi decompositions $P=MN$ and $Q=LU.$ For a non-trivial character $ψ$ of the unipotent radical $N$ of $P,$ let $M_ψ$ denote the stabilizer of the character $ψ$ in $M$ under the conjugation action of $M$ on characters of $N.$ For an irreducible representation of the Levi subgroups $M$ or $L,$ let $π$ denote the respective representation of $Sp_{4}(F)$ parabolically induced either from $P$ or from $Q.$ Let $ψ$ be a character of the group $N$ given by a rank one quadratic form. In this article, we determine the structure of the twisted Jacquet module $r_{N,ψ}(π)$ as an $M_ψ$-module. We also deduce the analogous results in the case where $F$ is a finite field of order $q.$ |
| title | Structure of Twisted Jacquet Modules of principal series representations of $Sp_{4}(F)$ |
| topic | Representation Theory 11F70(Primary), 22E50(Secondary) |
| url | https://arxiv.org/abs/2401.16933 |