Abstract
Let k be a local field. Let Iv and \(I_{v^{\prime}}\) be smooth principal series representations of GLn(k) and GLn-−1(k), respectively. The Rankin-Selberg integrals yield a continuous bilinear map \(I_{v}\times I_{v^{\prime}}\rightarrow\mathbb{C}\) with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map \(I_{v}\times I_{v^{\prime}}\rightarrow\mathbb{C}\) with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for GLn(k) × GLn(k).
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References
Aizenbud A, Gourevitch D. Schwartz functions on Nash manifolds. Int Math Res Not IMRN, 2008, 2008: rnm155
Coates J. On p-adic L-functions. Astérisque, 1989, 1989: 33–59
Coates J. On p-adic L-functions attached to motives over \(\mathbb{Q}\) II. Bull Braz Math Soc (NS), 1989, 20: 101–112
Coates J, Perrin-Riou B. On p-adic L-functions attached to motives over \(\mathbb{Q}\). In: Algebraic Number Theory. Advanced Studies in Pure Mathematics, vol. 17. Boston: Academic Press, 1989, 23–54
Deligne P. Valeurs de fonctions L et périodes d’intégrales. In: Automorphic Forms, Representations and L-Functions. Proceedings of Symposia in Pure Mathematics, vol. 33. Providence: Amer Math Soc, 1979, 313–346
Dixmier J, Malliavin P. Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull Sci Math, 1978, 102: 305–330
Hirano M, Ishii T, Miyazaki T. Archimedean zeta integrals for GL(3) × GL(2). Mem Amer Math Soc, 2022, 278: 1366
Ishii T, Miyazaki T. Calculus of archimedean Rankin-Selberg integrals with recurrence relations. Represent Theory, 2022, 26: 714–763
Jacquet H. Principal L-functions of the linear group. In: Automorphic Forms, Representations and L-Functions. Proceedings of Symposia in Pure Mathematics, vol. 33. Providence: Amer Math Soc, 1979, 63–86
Jacquet H. Archimedean Rankin-Selberg integrals. In: Automorphic Forms and L-Functions II. Local Aspects. Contemporary Mathematics, vol. 489. Providence: Amer Math Soc, 2009, 57–172
Jacquet H, Piatetski-Shapiro I I, Shalika J A. Rankin-Selberg convolutions. Amer J Math, 1983, 105: 367–464
Kudla S S. Tate’s thesis. In: Bernstein J, Gelbart S, eds. An Introduction to the Langlands Program. Boston: Birkhäuser, 2004, 109–131
Li J-S, Liu D W, Sun B Y. Period relations for Rankin-Selberg convolutions for GL(n) × GL(n − 1). arXiv:2109.05273, 2021.
Tate J. Fourier analysis in number fields and Hecke’s zeta-functions. In: Algebraic Number Theory. Washington: Thompson, 1967, 305–347
Tate J. Number theoretic background. In: Automorphic Forms, Representations and L-functions. Proceedings of Symposia in Pure Mathematics, vol. 33. Providence: Amer Math Soc, 1979, 3–26
Vogan D A. The algebraic structure of the representation of semisimple Lie groups I. Ann of Math (2), 1979, 109: 1–60
Wallach N R. Real Reductive Groups II. Pure and Applied Mathematics, vol. 132. Boston: Academic Press, 1992
Acknowledgements
Dongwen Liu was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LZ22A010006) and National Natural Science Foundation of China (Grant No. 12171421). Feng Su was supported by National Natural Science Foundation of China (Grant No. 11901466) and the Qinglan Project of Jiangsu Province. Binyong Sun was supported by the National Key R&D Program of China (Grant No. 2020YFA0712600). The authors thank the referees for the careful reading and comments.
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Li, JS., Liu, D., Su, F. et al. Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n−1) for principal series representations. Sci. China Math. 66, 2203–2218 (2023). https://doi.org/10.1007/s11425-022-2050-5
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DOI: https://doi.org/10.1007/s11425-022-2050-5