Abstract
This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by Waldspurger. Ifn=2, they were considered by Novodvorsky and Piatetski-Shapiro, who called them Bessel models. In the works of these authors, uniqueness of the models is established; in this paper functional equations and explicit formulas for them are obtained. As a global application, the Bessel period of the Eisenstein series on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n) is computed—it is shown to be a product of L-series. This generalizes work of Böcherer and Mizumoto forn=2 and base field ℚ, and puts it in a representation-theoretic context. In an appendix by M. Furusawa, a new Rankin-Selberg integral is given for the standardL-function on SO(2n+1)×GL(n). The local analysis of the integral is carried out using the formulas of the paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Andrianov,Euler products corresponding to Siegel modular forms of genus 2, Russian Mathematical Surveys29 (1974), 45–116.
W. Banks,The Casselman-Shalika formula for a distinguished model, Proceedings of the American Mathematical Society123 (1995), 681–692.
J. Bernstein,Letter to Piatetski-Shapiro, Fall 1985, to appear in a book by J. Cogdell and I. Piatetski-Shapiro.
S. Böcherer,Über gewisse Siegelsche Modulformen zweiten Grades, Mathematische Annalen261 (1982), 23–41.
D. Bump, S. Friedberg and D. Ginzburg,Whittaker-orthogonal models, functoriality, and the Rankin-Selberg method, Inventiones mathematicae109 (1992), 55–96.
D. Bump, S. Friedberg and J. Hoffstein,Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives, Annals of Mathematics131 (1990), 53–127.
D. Bump, S. Friedberg and J. Hoffstein,p-adic Whittaker functions on the metaplectic group, Duke Mathematical Journal63 (1991), 379–397.
D. Bump and D. Ginzburg,Spin L-functions on the symplectic group, International Mathematical Research Notices8 (1992), 153–160.
W. Casselman,The unramified principal series of p-adic groups I: the spherical function, Compositio Mathematica40 (1980), 387–406.
W. Casselman,Introduction to the theory of admissible representations of p-adic reductive groups, manuscript.
W. Casselman and J. Shalika,The unramified principal series of p-adic groups II: the Whittaker function, Compositio Mathematica41 (1980), 207–231.
M. Furusawa,On L-functions for GSp(4) × GL(2)and their special values, Journal für die reine und angewandte Mathematik438 (1993), 187–218.
M. Furusawa,On the theta lift from SO2n+1 to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatuuDJXwAK1uy0Hwmaerbfv3ySLgzG0uy0Hgip5% wzamXvP5wqonvsaeHbfv3ySLgzaeXatLxBI9gBamXvP5wqSXMqHnxA% Jn0BKvguHDwzZbqehqvATv2CG4uz3bIuV1wyUbqehm0B1jxALjhiov% 2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qq% aqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8% qqQ8frFve9Fve9Ff0dmeaaciGacmaadaWabiqaeaqbaqaagaaakeaa% daWfGaqaaGabaiaa-nfacaWFWbaaleqakeaacWaGGVipIOdaamaaBa% aaleaacqWGUbGBaeqaaaaa!4A7D!\[\mathop {Sp}\limits^ \sim _n \], Journal für die reine und angewandte Mathematik466 (1995), 87–110.
S. Gelbart and I. Piatetski-Shapiro,L-functions for G × GL(n), inExplicit Constructions of L-functions, Springer Lecture Notes in Mathematics1254, Springer-Verlag, Berlin, 1987.
D. Ginzburg,Fax to Daniel Bump, 1994.
D. Ginzburg, I. Piatetski-Shapiro and S. Rallis,L-functions for the orthogonal group, Memoirs of the American Mathematical Society, to appear.
H. Jacquet,Fonctions de Whittaker associeés aux groupes de Chevalley, Bulletin de la Société Mathématique de France95 (1967), 243–309.
S. Mizumoto,Fourier coefficients of generalized Eisenstein series of degree two, I., Inventiones mathematicae65 (1981), 115–135.
M. Novodvorsky,New unique models of representations of unitary groups, Compositio Mathematica33 (1976), 289–295.
M. Novodvorskii and I. Pjateckii-Šapiro,Generalized Bessel models for a symplectic group of rank 2, Mathematics of the USSR-Sbornik19 (1973), 243–255.
I. Piatetski-Shapiro and D. Soudry,On a correspondence of automorphic forms on orthogonal groups of order five, Journal de Mathématiques Pures et Appliquées (9)66 (1987), 407–436.
T. Sugano,On holomorphic cusp forms on quaternion unitary groups of degree 2, Journal of the Faculty of Science of the University of Tokyo, Section IA, Mathematics31 (1984), 521–568.
J.-L. Waldspurger,Correspondance de Shimura, Journal de Mathématiques Pures et Appliquées59 (1980), 1–133.
J.-L. Waldspurger,Sur les valeurs de certaines fonctions L automorphes en leur centre de symetrie, Compositio Mathematica54 (1985), 173–242.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by National Science Foundation grants DMS 9023441 (Bump) and DMS-9123845 (Friedberg), by the AMS Centennial Research Fellowship (Bump), by National Security Agency grant MDA904-95-H-1053 (Friedberg) and by NSF Postdoctoral Research Fellowship DMS 9206242 (Furusawa).
Rights and permissions
About this article
Cite this article
Bump, D., Friedberg, S. & Furusawa, M. Explicit formulas for the waldspurger and bessel models. Isr. J. Math. 102, 125–177 (1997). https://doi.org/10.1007/BF02773797
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02773797