Doubling constructions and tensor product L-functions: coverings of the symplectic group
In this work, we develop an integral representation for the partial L-function of a pair $\pi \times \tau $ of genuine irreducible cuspidal automorphic representations, $\pi $ of the m-fold covering of Matsumoto of the symplectic group $\operatorname {\mathrm {Sp}}_{2n}$ and...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942400063X/type/journal_article |
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| author | Eyal Kaplan |
| author_facet | Eyal Kaplan |
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| description | In this work, we develop an integral representation for the partial L-function of a pair
$\pi \times \tau $
of genuine irreducible cuspidal automorphic representations,
$\pi $
of the m-fold covering of Matsumoto of the symplectic group
$\operatorname {\mathrm {Sp}}_{2n}$
and
$\tau $
of a certain covering group of
$\operatorname {\mathrm {GL}}_k$
, with arbitrary m, n and k. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-
$1$
twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Our global results are subject to certain conjectures, but when
$k=1$
they are unconditional for all m. One possible future application of this work will be a Shimura-type lift of representations from covering groups to general linear groups. In a recent work, we used the present results in order to provide an analytic definition of local factors for representations of the m-fold covering of
$\operatorname {\mathrm {Sp}}_{2n}$
. |
| format | Article |
| id | doaj-art-b1ad7367fb594ff082d101735fabad36 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-b1ad7367fb594ff082d101735fabad362025-02-10T12:03:38ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.63Doubling constructions and tensor product L-functions: coverings of the symplectic groupEyal Kaplan0https://orcid.org/0000-0002-0727-8529Department of Mathematics, Bar-Ilan University, Ramat Gan, 5290002, IsraelIn this work, we develop an integral representation for the partial L-function of a pair $\pi \times \tau $ of genuine irreducible cuspidal automorphic representations, $\pi $ of the m-fold covering of Matsumoto of the symplectic group $\operatorname {\mathrm {Sp}}_{2n}$ and $\tau $ of a certain covering group of $\operatorname {\mathrm {GL}}_k$ , with arbitrary m, n and k. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank- $1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Our global results are subject to certain conjectures, but when $k=1$ they are unconditional for all m. One possible future application of this work will be a Shimura-type lift of representations from covering groups to general linear groups. In a recent work, we used the present results in order to provide an analytic definition of local factors for representations of the m-fold covering of $\operatorname {\mathrm {Sp}}_{2n}$ .https://www.cambridge.org/core/product/identifier/S205050942400063X/type/journal_article11F7011F5511F6622E5022E55 |
| spellingShingle | Eyal Kaplan Doubling constructions and tensor product L-functions: coverings of the symplectic group Forum of Mathematics, Sigma 11F70 11F55 11F66 22E50 22E55 |
| title | Doubling constructions and tensor product L-functions: coverings of the symplectic group |
| title_full | Doubling constructions and tensor product L-functions: coverings of the symplectic group |
| title_fullStr | Doubling constructions and tensor product L-functions: coverings of the symplectic group |
| title_full_unstemmed | Doubling constructions and tensor product L-functions: coverings of the symplectic group |
| title_short | Doubling constructions and tensor product L-functions: coverings of the symplectic group |
| title_sort | doubling constructions and tensor product l functions coverings of the symplectic group |
| topic | 11F70 11F55 11F66 22E50 22E55 |
| url | https://www.cambridge.org/core/product/identifier/S205050942400063X/type/journal_article |
| work_keys_str_mv | AT eyalkaplan doublingconstructionsandtensorproductlfunctionscoveringsofthesymplecticgroup |