Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-06-28T06:49:30.743Z Has data issue: false hasContentIssue false

Whittaker Functions on Real Semisimple Lie Groups of Rank Two

Published online by Cambridge University Press:  20 November 2018

Taku Ishii*
Affiliation:
Faculty of Science and Technology, Seikei University, Japan, e-mail: ishii@st.seikei.ac.j
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Bailey, W. N., Generalized Hypergeometric Series. Cambridge University Press, Cambridge, 1935.Google Scholar
[2] Bump, D., Automorphic forms on GL(3, R). Lecture Notes in Mathematics 1083. Springer-Verlag, Berlin, 1984.Google Scholar
[3] Bump, D., The Rankin-Selberg method: a survey. In: Number Theory, Trace Formulas, and Discrete Groups. Academic Press, Boston, 1989, pp 49–109.Google Scholar
[4] Bump, D., Friedberg, S. and Ginzburg, D., Rankin-Selberg integrals in two complex variables. Math. Ann. 313(1999), no. 4, 731–761. doi:10.1007/s002080050280 Google Scholar
[5] Gelbart, S., Piatetski-Shapiro, I., and Rallis, S., Explicit constructions of automorphic L-functions. Lecture Notes in Mathematics 1254. Springer-Verlag, Berlin, 1987.Google Scholar
[6] Gradshteyn, I. S. and Ryzhik, I. M., Tables of integrals, series, and products. Sixth edition. Academic Press, Sandiego, CA, 2000.Google Scholar
[7] Hashizume, M., Whittaker functions on semisimple Lie groups. Hiroshima Math. J. 12(1982), no. 2, 259–293.Google Scholar
[8] Hirano, M., Ishii, T., and Oda, T., Whittaker functions for PJ-principal series representations of Sp(3, R). Adv. Math. 215(2007), no. 2, 734–765. doi:10.1016/j.aim.2007.04.015 Google Scholar
[9] Ishii, T., On principal series Whittaker functions on Sp(2, R). J. Funct. Anal. 225(2005), no. 1, 1–32. doi:10.1016/j.jfa.2005.03.023 Google Scholar
[10] Ishii, T., Class one Whittaker functions on real semisimple Lie groups. In: Automorphic representations, L-functions, and periods. Surikaisekikenkyusho Kokyuroku No. 1523 (2006), 70–78. http://www.kurims.kyoto-u.ac.jp/-kyodo/kokyuroku/contents/1523.html Google Scholar
[11] Ishii, T. and Moriyama, T., Spinor L-functions for generic cusp forms on GSp(2) belonging to principal series representations. Trans. Amer. Math. Soc. 360(2008), no. 11, 5683–5709. doi:10.1090/S0002-9947-08-04724-7 Google Scholar
[12] Ishii, T. and Stade, E., New formulas for Whittaker functions on GL(n, R). J. Funct. Anal. 244(2007), no. 1, 289–314. doi:10.1016/j.jfa.2006.12.004 Google Scholar
[13] Jacquet, H., Fonctions de Whittaker associées aux groupes de Chevalley. Bull. Soc. Math. France 95(1967), 243–309.Google Scholar
[14] Niwa, S., Commutation relations of differential operators and Whittaker functions on Sp2(R). Proc. Japan Acad. Ser. A Math. Sci. 71(1995), no. 8, 189–191. doi:10.3792/pjaa.71.189 Google Scholar
[15] Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series. Vol. 3, Gordon and Breach Science Publishers, New York, 1990.Google Scholar
[16] Stade, E., Poincaré series for GL(3, R)-Whittaker functions. Duke Math. J. 58(1989), no. 3, 695–729. doi:10.1215/S0012-7094-89-05833-X Google Scholar
[17] Stade, E., Hypergeometric series and Euler factors at infinity for L-functions on GL(3, R) × GL(3, R). Amer. J. Math. 115(1993), no. 2, 371–387. doi:10.2307/2374862 Google Scholar
[18] Stade, E., Archimedean L-factors on GL(n) × GL(n) and generalized Barnes integrals. Israel J. Math. 127(2002), 201–219. doi:10.1007/BF02784531 Google Scholar
[19] Vinogradov, I. and Tahtadžjan, L., Theory of the Eisenstein series for the group SL(3, R) and its application to a binary problem. I. Fourier expansion of the highest Eisenstein series. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LO MI) 76(1978), 5–52, 216, translated in J. of Soviet Math. vol. 18(1982), no. 3, 293–324.Google Scholar
[20] Wallach, N., Asymptotic expansions of generalized matrix entries of representations of real reductive groups. Lecture Notes in Mathematics 1024. Springer, Berlin, 1983, pp. 287–369.Google Scholar