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

Eric Stade

Annales de l'institut Fourier (1994)

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

Abstract

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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 , 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 G L ( 3 , ℝ ) and G L ( 4 , 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.

How to cite

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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

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  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

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