On Jacobi sums arising from the classical doubling method
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917131379015680 |
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| author | Yost-Wolff, Calvin Zelingher, Elad |
| author_facet | Yost-Wolff, Calvin Zelingher, Elad |
| contents | We define the notion of a non-abelian Jacobi sum $\mathcal{J}^{\mathrm{dbl}}\left(π, χ\right)$ attached to an irreducible representation $π$ of a general linear group or a classical group over a finite field and a character $χ$ of the multiplicative group of the finite field or its quadratic extension. These sums emerge in the study of the doubling method of Piatetski-Shapiro--Rallis and Lapid--Rallis. For general linear groups, we express these non-abelian Jacobi sums in terms of Kondo's non-abelian Gauss sums. For classical groups and for characters that are not conjugate-dual, we give an explicit formula for these non-abelian Jacobi sums in terms of Gauss sums attached to the Deligne--Lusztig data of the representation, and we prove that these Jacobi sums are constant on geometric Lusztig series. Our results rely on a multiplicativity result of non-abelian Jacobi sums obtained by Girsch--Zelingher. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_06588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Jacobi sums arising from the classical doubling method Yost-Wolff, Calvin Zelingher, Elad Number Theory Representation Theory 20C33, 11L05, 11T24 We define the notion of a non-abelian Jacobi sum $\mathcal{J}^{\mathrm{dbl}}\left(π, χ\right)$ attached to an irreducible representation $π$ of a general linear group or a classical group over a finite field and a character $χ$ of the multiplicative group of the finite field or its quadratic extension. These sums emerge in the study of the doubling method of Piatetski-Shapiro--Rallis and Lapid--Rallis. For general linear groups, we express these non-abelian Jacobi sums in terms of Kondo's non-abelian Gauss sums. For classical groups and for characters that are not conjugate-dual, we give an explicit formula for these non-abelian Jacobi sums in terms of Gauss sums attached to the Deligne--Lusztig data of the representation, and we prove that these Jacobi sums are constant on geometric Lusztig series. Our results rely on a multiplicativity result of non-abelian Jacobi sums obtained by Girsch--Zelingher. |
| title | On Jacobi sums arising from the classical doubling method |
| topic | Number Theory Representation Theory 20C33, 11L05, 11T24 |
| url | https://arxiv.org/abs/2512.06588 |