Abstract
We present a Rankin–Selberg integral on the exceptional group \(G_2\) which represents the L-function for generic cuspidal representations of \(\widetilde{\mathrm {SL}}_2\times {\mathrm {GL}}_2\). As an application, we show that certain Fourier–Jacobi type periods on \(G_2\) are non-vanishing.
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References
Bump, D., Friedberg, S., Hoffstein, J.: \(p\)-adic Whittaker functions on metaplectic groups. Duke Math. J. 63, 379–397 (1991)
Gan,W.T.: The Shimura Correspondence, À la Waldspurger, preprint
Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups (English, with English and French summaries). Astł’erisque 346, 1–10 (2012)
Ginzburg, D.: A Rankin–Selberg integral for the adjoint representation of \({\rm GL}_3\). Invent. Math. 105(3), 571–588 (1991)
Ginzburg, D.: On the standard \(L\)-function for \(G_2\). Duke Math. J. 69, 315–333 (1993)
Ginzburg, D.: On the symmetric fourth power \(L\)-function of \({\rm GL}_2\). Isr. J. Math. 92, 157–184 (1995)
Ginzburg, D., Rallis, S., Soudry, D.: Periods, poles of L-functions and symplectic-orthogonal theta lifts. J. Reine Angew. Math. 487, 85–114 (1997)
Gingzburg, D., Rallis, S., Soudry, D.: L-Functions for symplectic groups. Bull. Soc. Math. Fr. 126, 181–244 (1998)
Jacquet, H., Shalika, J.: Exterior square L-functions, in “Automorphic forms, Shimura varieties, and L-functions, Vol. II” (Ann Arbor, MI, 1988), 143–226, Perspect. Math., 11, Academic Press, Boston, MA (1990)
Kim, H.: The residue spectrum of \(G_2\). Can. J. Math. 48(6), 1245–1272 (1996)
Kudla, S.: Notes on the local theta correspondence, preprint. http://www.math.toronto.edu/skudla/castle.pdf
Ree, R.: A family of simple groups associated with the simple Lie algebra of type \(G_2\). Am. J. Math. 83, 432–462 (1961)
Shahidi, F.: Functional equations satisfied by certain \(L\)-functions. Compos. Math. 37, 171–208 (1978)
Steiberg, R.: Lectures on Chevalley Groups. Yale University Press, New Haven (1967)
Waldspurger, J.P.: Correspondance de Shimura. J. Math. Pures Appl. 59(1), 1–132 (1980)
Zampera, S.: The residue spectrum of the group of type \(G_2\). J. Math. Pures. Appl. 76, 805–835 (1997)
Acknowledgements
I would like to thank D. Ginzburg for helpful communications and pointing out the reference [6]. The debt of this paper to Ginzburg’s papers [5, 6] should be evident for the readers. I also would like to thank Joseph Hundley and Baiying Liu for useful discussions. I appreciate Jim Cogdell and Clifton Cunningham for encouragement and support. I also would like to thank the anonymous referee for his/her careful reading and useful suggestions. This work is supported by a fellowship from Pacific Institute for Mathematical Sciences (PIMS) and NSFC Grant 11801577.
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Zhang, Q. On a Rankin–Selberg integral of the L-function for \(\widetilde{\mathrm {SL}}_2\times {\mathrm {GL}}_2\). Math. Z. 298, 307–326 (2021). https://doi.org/10.1007/s00209-020-02611-8
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DOI: https://doi.org/10.1007/s00209-020-02611-8