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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942400063X/type/journal_article |
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| Summary: | 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
$\operatorname {\mathrm {Sp}}_{2n}$
. |
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| ISSN: | 2050-5094 |