The reciprocal of the beta function and G L ( n , ) Whittaker functions

Eric Stade

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 1, page 93-108
  • ISSN: 0373-0956

Abstract

top
In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of G L ( n , ) Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental” Whittaker functions for G L ( 3 , ) and G L ( 4 , ) . The integrals obtained are seen to resemble very closely known integrals for the “class-one” Whittaker functions on these groups. It is expected that this correspondence between expressions for different types of Whittaker functions will carry over to groups of arbitrary rank.

How to cite

top

Stade, Eric. "The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions." Annales de l'institut Fourier 44.1 (1994): 93-108. <http://eudml.org/doc/75062>.

@article{Stade1994,
abstract = {In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of $GL(n, \{\Bbb R\})$ Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental” Whittaker functions for $GL(3,\{\Bbb R\})$ and $GL(4, \{\Bbb R\})$. The integrals obtained are seen to resemble very closely known integrals for the “class-one” Whittaker functions on these groups. It is expected that this correspondence between expressions for different types of Whittaker functions will carry over to groups of arbitrary rank.},
author = {Stade, Eric},
journal = {Annales de l'institut Fourier},
keywords = {automorphic forms; beta function; Whittaker functions},
language = {eng},
number = {1},
pages = {93-108},
publisher = {Association des Annales de l'Institut Fourier},
title = {The reciprocal of the beta function and $GL(n, \{\mathbb \{R\}\})$ Whittaker functions},
url = {http://eudml.org/doc/75062},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Stade, Eric
TI - The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 1
SP - 93
EP - 108
AB - In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of $GL(n, {\Bbb R})$ Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental” Whittaker functions for $GL(3,{\Bbb R})$ and $GL(4, {\Bbb R})$. The integrals obtained are seen to resemble very closely known integrals for the “class-one” Whittaker functions on these groups. It is expected that this correspondence between expressions for different types of Whittaker functions will carry over to groups of arbitrary rank.
LA - eng
KW - automorphic forms; beta function; Whittaker functions
UR - http://eudml.org/doc/75062
ER -

References

top
  1. [1] T. BROMWICH, Introduction to the Theory of Infinite Series, Macmillan & Co. Ltd., 1908. Zbl39.0306.02JFM39.0306.02
  2. [2] D. BUMP, Automorphic Forms on GL (3, R), Springer Lecture Notes in Mathematics, n°1083 (1984). Zbl0543.22005MR86g:11028
  3. [3] D. BUMP and J. HUNTLEY, Unramified Whittaker functions for GL (3, R) to appear. Zbl0839.22016
  4. [4] R. GODEMENT and H. JACQUET, Zeta Functions of Simple Algebras, Springer Lecture Notes in Mathematics, n°260 (1972). Zbl0244.12011MR49 #7241
  5. [5] M. HASHIZUME, Whittaker functions on semisimple Lie groups, Hiroshima Math. J., 12 (1982), 259-293. Zbl0524.43005MR84d:22018
  6. [6] H. JACQUET, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France, 95 (1967), 243-309. Zbl0155.05901MR42 #6158
  7. [7] B. KOSTANT, On Whittaker vectors and representation theory, Inventiones Math., 48 (1978), 101-184. Zbl0405.22013MR80b:22020
  8. [8] H. NEUNHÖFFER, Uber die analytische Fortsetzung von Poincaréreihen, Sitz. Heidelberger Akad. Wiss., 2 (1973), 33-90. Zbl0272.10015
  9. [9] D. NIEBUR, A class of nonanalytic automorphic functions, Nagoya Math. J., 52 (1973), 133-145. Zbl0288.10010MR49 #2557
  10. [10] I.I. PIATETSKI-SHAPIRO, Euler subgroups, Lie Groups and their Representations, John Wiley and Sons, 1975. Zbl0329.20028MR53 #10720
  11. [11] A. SELBERG, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (956), 47-87. Zbl0072.08201MR19,531g
  12. [12] J. SHALIKA, The multiplicity one theorem for GL(n), Annals of Math., 100 (1974), 171-193. Zbl0316.12010MR50 #545
  13. [13] L. SLATER, Generalized Hypergeometric Functionsl Cambridge University Press, 1966. Zbl0135.28101
  14. [14] E. STADE, Poincaré series for GL(3,R)-Whittaker functions, Duke Mathematical Journal, 58-3 (1989), pages 131-165. Zbl0699.10041MR90i:22022
  15. [15] E. STADE, On explicit integral formulas for GL(n, R)-Whittaker functions, Duke Mathematical Journal, 60-2 (1990), 313-362. Zbl0731.11027
  16. [16] E. STADE, GL(4, R)-Whittaker functions and 4F3(1) hypergeometric series, Trans. Amer. Math. Soc., 336-1 (1993), 253-264. Zbl0786.11027MR93e:22018
  17. [17] N. WALLACH, Asymptotic expansions of generalized matrix coefficients of representations of real reductive groups, Springer Lecture Notes in Mathematics, n°1024 (1983). Zbl0553.22005MR85g:22029
  18. [18] E. WHITTAKER and G. WATSON, A Course of Modern Analysis, Cambridge University Press, 1902. Zbl45.0433.02JFM45.0433.02

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.