Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-09T01:31:52.481Z Has data issue: false hasContentIssue false
  • English
  • Français

Mellin Transforms of Whittaker Functions

Published online by Cambridge University Press:  20 November 2018

Anton Deitmar
Anton Deitmar*
Affiliation:
University of Exeter Department of Mathematics Exeter EX4 4QE Devon United Kingdom
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.

In this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers.

Keywords

11F3022E3011F7022E45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 3 , 01 September 2002 , pp. 364 - 377
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Cogdell, J., Li, J. and Pyatetskii-Shapiro, I., The meromorphic continuation of Kloosterman-Selberg Zeta functions. Springer Lecture Notes 1422 (1990), 1422–1990.Google Scholar
[2] Cogdell, J., Li, J., Pyatetskii-Shapiro, I. and Sarnak, P., Poincaré series for SO(n, 1). Acta Math. 167, 229-285 (1991).Google Scholar
[3] Bump, D., Automorphic Forms on GL(3; R). Lecture Notes in Math. 1083, Springer-Verlag, 1984.Google Scholar
[4] Bump, D., Friedberg, S., Goldfeld, D., Poincaré series and Kloosterman sums for SL(3; Z). Acta Arith. 50 (1988), 50–1988.Google Scholar
[5] Bump, D. and Friedberg, S., On Mellin transforms of unramified Whittaker functions on GL(3, C). J. Math. Anal. Appl. 139 (1989), 139–1989.Google Scholar
[6] Casselman, W., Canonical extensions of Harish-Chandra modules to representations of G. Canad. J. Math. 41 (1989), 41–1989.Google Scholar
[7] Dixmier, J. and Malliavin, P., Factorisations de fonctions et de vecteurs indefiniment differentiables. Bull. Sci. Math. (2) 102 (1978), 102–1978.Google Scholar
[8] Friedberg, S. and Goldfeld, D., Mellin transforms of Whittaker functions. Bull. Soc. Math. France 121 (1993), 121–1993.Google Scholar
[9] Goldfeld, D., Kloosterman Zeta Functions for GL(n, Z). Proceedings of the ICM, Berkeley, CA, 1986.Google Scholar
[10] Humphreys, J., Introduction to Lie Algebras and Representation Theory. Springer-Verlag, 1972.Google Scholar
[11] Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic representations. I. Amer. J. Math. 103 (1981), 103–1981.Google Scholar
[12] Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic forms. II. Amer. J. Math. 103 (1981), 103–1981.Google Scholar
[13] Raghunathan, M., Lattices in Lie Groups. Springer-Verlag, New York, 1972.Google Scholar
[14] Selberg, A., On the estimation of Fourier coefficients of modular forms. Proc. Sympos. Pure Math. VIII, Amer. Math. Soc., Providence, R.I. 1965, 1–15.Google Scholar
[15] Stade, E., Mellin transforms of GL(n, R) Whittaker functions. Amer. J. Math, to appear.Google Scholar
[16] Wallach, N., Asymptotic expansions of generalized matrix coefficients of real reductive groups. Lie Group Representations I. SLNM 1024 (1983), 1024–1983.Google Scholar
[17] Wallach, N., Real Reductive Groups II. Academic Press, 1993.Google Scholar