Abstract
We consider the residues at the poles in the half plane \(Re(s)\ge 0\) of Eisenstein series, on symplectic groups, or their double covers, induced from Speh representations. We show that for each such pole, there is a unique maximal nilpotent orbit, attached to Fourier coefficients admitted by the corresponding residual representation. We find this orbit in each case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cai, Y., Friedberg, S., Ginzburg, D., Kaplan, E.: Doubling constructions and tensor product \(L\)-functions: the linear case. Invent. Math. 217(3), 985–1068 (2019)
Cai, Y., Friedberg, S., Kaplan, E.: Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations. arXiv:1802.02637
Collingwood, D., McGovern, W.: Nilpotent Orbits in Semisimple Lie Algebras: An Introduction. Van Nostrand Reinhold, New York (1993)
Friedberg, S., Ginzburg, D.: Criteria for existence of cuspidal theta representations. Res. Number Theory 2, 16 (2016)
Ginzburg, D.: On construction of CAP representations in classical groups. IMRN 20, 1123–1140 (2003)
Ginzburg, D.: Certain conjectures relating unipotent orbits to automorphic representations. Israel J. Math. 151, 323–355 (2006)
Ginzburg, D., Soudry, D.: Two identities relating Eisenstein series on classical groups. J. Number Theory 221, 1–108 (2021)
Ginzburg, D., Soudry, D.: Double descent in classical groups. J. Number Theory (2021). https://doi.org/10.1016/jnt.2021.08.012
Ginzburg, D., Rallis, S., Soudry, D.: On Fourier coefficients of automorphic forms on symplectic groups. Manuscr. Math. 111, 1–16 (2003)
Ginzburg, D., Rallis, S., Soudry, D.: The Descent Map from Automorphic Representation of \({{\rm GL}}(n)\) to Classical Groups. World Scientific, Singapore (2011)
Jiang, D.: Automorphic integral transforms for classical groups I: endoscopy correspondences. Automorphic forms and related geometry: assessing the legacy of I.I. Piatetski–Shapiro. Comtemp. Math. 614, 179–242 (2014)
Jiang, D., Liu, B.: On Fourier coefficients of certain residual representations of symplectic groups. Pac. J. Math. 281(2), 421–466 (2016)
Jiang, D., Liu, B., Zhang, L.: Poles of certain residual Eisenstein series of classical groups. Pac. J. Math. 264(1), 83–123 (2013)
Kudla, S., Rallis, S.: A regularized Siegel–Weil formula: The first term identity. Ann. Math. 140, 1–80 (1994)
Liu, B.: On extension of Ginzburg–Jiang–Soudry correspondence to certain automorphic forms on \(Sp_{4mn}({\mathbb{A}})\) and \({\widetilde{Sp}}_{4mn\pm 2n}({\mathbb{A}})\). arXiv:1309.6240
Moeglin, C., Waldspurger, J.-L.: Le spectre résiduel de \(GL(n)\). Ann. Sci. École Norm. Sup. (4) 22(4), 605–674 (1989)
Ranga Rao, R.: On some explicit formulas in the theory of Weil representations. Pac. J. Math. 157, 335–371 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was supported by the ISRAEL SCIENCE FOUNDATION (Grant No. 461/18).
Rights and permissions
About this article
Cite this article
Ginzburg, D., Soudry, D. Top Fourier coefficients of residual Eisenstein series on symplectic or metaplectic groups, induced from Speh representations. Res. number theory 8, 10 (2022). https://doi.org/10.1007/s40993-021-00306-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-021-00306-5