Structure of twisted Jacquet modules of principal series representations of $GL_{2n}(F)$
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| Format: | Preprint |
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2025
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| author | Harshitha, C. Venketasubramanian, C. G. |
| author_facet | Harshitha, C. Venketasubramanian, C. G. |
| contents | Let $F$ be a non-archimedean local field or a finite field. Let $π$ be a principal series representation of $GL_{2n}(F)$ induced from any of its maximal standard parabolic subgroups. Let $N$ be the unipotent radical of the maximal parabolic subgroup $P$ of $GL_{2n}(F)$ corresponding to the partition $(n,n).$ In this article, we describe the structure of the twisted Jacquet module $π_{N,ψ}$ of $π$ with respect to $N$ and a non-degenerate character $ψ$ of $N.$ We also provide a necessary and sufficient condition for $π_{N,ψ}$ to be non-zero and show that the twisted Jacquet module is non-zero under certain assumptions on the inducing data. As an application of our results, we obtain the structure of twisted Jacquet modules of certain non-generic irreducible representations of $GL_{2n}(F)$ and establish the existence of their Shalika model in the non-archimedean case. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of $GL_{2n}(F)$ with a non-zero twisted Jacquet module. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24737 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure of twisted Jacquet modules of principal series representations of $GL_{2n}(F)$ Harshitha, C. Venketasubramanian, C. G. Representation Theory 11F70, 22E50 Let $F$ be a non-archimedean local field or a finite field. Let $π$ be a principal series representation of $GL_{2n}(F)$ induced from any of its maximal standard parabolic subgroups. Let $N$ be the unipotent radical of the maximal parabolic subgroup $P$ of $GL_{2n}(F)$ corresponding to the partition $(n,n).$ In this article, we describe the structure of the twisted Jacquet module $π_{N,ψ}$ of $π$ with respect to $N$ and a non-degenerate character $ψ$ of $N.$ We also provide a necessary and sufficient condition for $π_{N,ψ}$ to be non-zero and show that the twisted Jacquet module is non-zero under certain assumptions on the inducing data. As an application of our results, we obtain the structure of twisted Jacquet modules of certain non-generic irreducible representations of $GL_{2n}(F)$ and establish the existence of their Shalika model in the non-archimedean case. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of $GL_{2n}(F)$ with a non-zero twisted Jacquet module. |
| title | Structure of twisted Jacquet modules of principal series representations of $GL_{2n}(F)$ |
| topic | Representation Theory 11F70, 22E50 |
| url | https://arxiv.org/abs/2512.24737 |