Abstract
In this paper, we recreate the Whittaker-Shintani functions for general linear groups over non-archimedean fields given by Kato et al. Those generalized spherical functions naturally arise from global zeta integrals of automorphic L-functions. More explicitly, this formula plays a fundamental role in the local calculation over the split places of tensor product L-functions defined by Jiang and Zhang (2014) and the twisted automorphic descents introduced by Jiang and Zhang (2015).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aizenbud A, Gourevitch D, Rallis S, et al. Multiplicity one theorems. Ann of Math (2), 2010, 172: 1407–1434
Banks W D. A corollary to Bernstein’s theorem and Whittaker functionals on the metaplectic group. Math Res Lett, 1998, 5: 781–790
Casselman W. The unramified principal series of p-adic groups, I: The spherical function. Compos Math, 1980, 40: 387–406
Ginzburg D, Rallis S, Soudry D. The Descent Map from Automorphic Representations of GL(n) to Classical Groups. Hackensack: World Scientific, 2011
Gross B H, Prasad D, Gan W T. Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Asterisque, 2012, 346: 1–109
Jiang D, Sun B, Zhu C B. Uniqueness of Bessel models: the Archimedean case. Geom Funct Anal, 2010, 20: 690–709
Jiang D, Zhang L. A product of tensor product l-functions of quasi-split classical groups of hermitian type. Geom Funct Anal, 2014, 24: 552–609
Jiang D, Zhang L. Arthur parameters and cuspidal automorphic modules of classical groups. ArXiv:1508.03205v2, 2015
Kato S-I, Murase A, Sugano T. Shintani functions for GLn: An explicit formula. Unpublished
Kato S-I, Murase A, Sugano T. Whittaker-Shintani functions for orthogonal groups. Tohoku Math J (2), 2003, 55: 1–64
Murase A, Sugano T. Whittaker-Shintani functions on the symplectic group of Fourier-Jacobi type. Compos Math, 1991, 79: 321–349
Murase A, Sugano T. Shintani functions and automorphic L-functions for GL(n). Tohoku Math J (2), 1996, 48: 165–202
Sakellaridis Y. On the unramified spectrum of spherical varieties over p-adic fields. Compos Math, 2008, 144: 978–1016
Shen X. The Whittaker-Shintani functions for symplectic groups. Int Math Res Not IMRN, 2014, 21: 5769–5831
Sun B Y, Zhu C B. Multiplicity one theorems: The Archimedean case. Ann of Math (2), 2012, 175: 23–44
Acknowledgements
This work was supported by the National University of Singapore’s Start-Up Grant. The author thanks Dihua Jiang, Yifeng Liu and Xin Shen for helpful conversations and thanks the referees for very useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, L. Whittaker-Shintani functions for general linear groups over p-adic fields. Sci. China Math. 61, 95–110 (2018). https://doi.org/10.1007/s11425-016-9064-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-9064-6