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TWISTED DOUBLING INTEGRALS FOR BRYLINSKI–DELIGNE EXTENSIONS OF CLASSICAL GROUPS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 December 2021

Yuanqing Cai Open the ORCID record for Yuanqing Cai [Opens in a new window]
Yuanqing Cai*
Affiliation:
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa, 920-1192, Japan
*
cai@se.kanazawa-u.ac.jp
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Abstract

We explain how to develop the twisted doubling integrals for Brylinski–Deligne extensions of connected classical groups. This gives a family of global integrals which represent Euler products for this class of nonlinear extensions.

Keywords

Brylinski–Deligne extensionstwisted doubling integralstensor product L-functions

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 4 , July 2023 , pp. 1931 - 1985
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Brylinski, J.-L. and Deligne, P., Central extensions of reductive groups by ${\mathbf{K}}_2$ , Publ. Math. Inst. Hautes Études Sci. 94 (2001), 5–85.10.1007/s10240-001-8192-2CrossRefGoogle Scholar
Bump, D. and Ginzburg, D., Symmetric square $L$ -functions on $\mathrm{GL}(r)$ , Ann. Math. (2) 136(1) (1992), 137–205.10.2307/2946548CrossRefGoogle Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376.Google Scholar
Bump, D., Automorphic Forms and Representations , Vol. 55 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1997).Google Scholar
Cai, Y., Twisted doubling integrals for classical groups, Math. Z. 297(3–4) (2021), 1075–1104.10.1007/s00209-020-02547-zCrossRefGoogle Scholar
Cai, Y., Friedberg, S., Ginzburg, D. and Kaplan, E., Doubling constructions for covering groups and tensor product L-functions, Preprint, 2016, arXiv:1601.08240.Google Scholar
Cai, Y., Friedberg, S., Ginzburg, D. and Kaplan, E., Doubling constructions and tensor product $L$ -functions: the linear case, Invent. Math. 217(3) (2019), 985–1068.10.1007/s00222-019-00883-4CrossRefGoogle Scholar
Friedberg, S. and Ginzburg, D., On the genericity of Eisenstein series and their residues for covers of $G{L}_m$ , Int. Math. Res. Not. IMRN 4 (2017), 1000–1012.Google Scholar
Gan, W. T., Doubling zeta integrals and local factors for metaplectic groups, Nagoya Math. J. 208 (2012), 67–95.10.1017/S002776300001059XCrossRefGoogle Scholar
Gao, F., The Langlands-Shahidi $L$ -functions for Brylinski-Deligne extensions, Amer. J. Math. 140(1) (2018), 83–137.10.1353/ajm.2018.0001CrossRefGoogle Scholar
Gan, W. T. and Gao, F., The Langlands-Weissman program for Brylinski-Deligne extensions, Astérisque 398 (2018), 187–275.Google Scholar
Gomez, R., Gourevitch, D. and Sahi, S., Generalized and degenerate Whittaker models, Compos. Math. 153(2) (2017), 223–256.10.1112/S0010437X16007788CrossRefGoogle Scholar
Gan, W. T., Gao, F. and Weissman, M. H., L-groups and the Langlands program for covering groups: a historical introduction, Astérisque 398 (2018), 1–31.Google Scholar
Ginzburg, D., Tensor product $L$ -functions on metaplectic covering groups of ${\mathrm{GL}}_r$ , Preprint, 2019, arXiv:1908.07720.Google Scholar
Gao, F., Shahidi, F. and Szpruch, D., Local coefficients and gamma factors for principal series of covering groups, Mem. Amer. Math. Soc. to appear, arXiv:1902.02686.Google Scholar
Kaplan, E., Doubling constructions and tensor product $L$ -functions: coverings of the symplectic group, Reprint, 2019, arXiv:1902.00880.Google Scholar
Kazhdan, D. A. and Patterson, S. J., Metaplectic forms, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35–142.10.1007/BF02698770CrossRefGoogle Scholar
Li, W.-W., La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin, J. Reine Angew. Math. 686 (2014), 37–109.Google Scholar
Mezo, P., Metaplectic tensor products for irreducible representations, Pacific J. Math. 215(1) (2004), 85–96.10.2140/pjm.2004.215.85CrossRefGoogle Scholar
Moore, C. C., Group extensions of $p$ -adic and adelic linear groups, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 157–222.10.1007/BF02698923CrossRefGoogle Scholar
Mœglin, C. and Waldspurger, J.-L., Modèles de Whittaker dégénérés pour des groupes $p$ -adiques, Math. Z. 196(3) (1987), 427–452.10.1007/BF01200363CrossRefGoogle Scholar
Mœglin, C. and Waldspurger, J.-L., Spectral Decomposition and Eisenstein Series , Vol. 113 of Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1995). Une paraphrase de l’Écriture [A paraphrase of Scripture].Google Scholar
Piatetski-Shapiro, I. and Rallis, S., $L$ -Functions for the Classical Groups, Vol. 1254 of Lecture Notes in Math. (Springer, New York, 1987).Google Scholar
Suzuki, T., Distinguished representations of metaplectic groups, Amer. J. Math. 120(4) (1998), 723–755.10.1353/ajm.1998.0032CrossRefGoogle Scholar
Takeda, S., Metaplectic tensor products for automorphic representation of $\widetilde{\mathrm{GL}}(r)$ , Canad. J. Math. 68(1) (2016), 179–240.10.4153/CJM-2014-046-2CrossRefGoogle Scholar
Takeda, S., Remarks on metaplectic tensor products for covers of ${\mathrm{GL}}_r$ , Pacific J. Math. 290(1) (2017), 199–230.10.2140/pjm.2017.290.199CrossRefGoogle Scholar
Weissman, M. H., Managing metaplectiphobia: covering $p$ -adic groups, in Harmonic Analysis on Reductive, $p$ -Adic Groups, Vol. 543 of Contemp. Math. (American Mathematical Society, Providence, RI, 2011), 237–277.Google Scholar
Weissman, M. H., Covering groups and their integral models, Trans. Amer. Math. Soc. 368(5) (2016), 3695–3725.10.1090/tran/6598CrossRefGoogle Scholar
Yamana, S., L-functions and theta correspondence for classical groups, Invent. Math. 196(3) (2014), 651–732.10.1007/s00222-013-0476-xCrossRefGoogle Scholar