The reciprocal of the beta function and GL(n,) Whittaker functions
Annales de l'Institut Fourier, Volume 44 (1994) no. 1, p. 93-108

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,) 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,) and GL(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.

Dans cet article nous obtenons, à l’aide du théorème de la sommation de Gauss pour les séries hypergéométriques, une expression intégrale très simple pour l’inverse de la fonction beta d’Euler. Cette expression est semblable dans sa forme à plusieurs intégrales bien connues de la fonction beta proprement dite.

Nous appliquons ensuite notre nouvelle formule aux fonctions de Whittaker pour les groupes GL(n,). Ces dernières sont des fonctions spéciales que l’on retrouve dans la théorie de Fourier appliquée aux formes automorphes du groupe linéaire général. Plus précisément, nous déduisons des représentations intégrales explicites pour les fonctions “fondamentales” de Whittaker pour les cas GL(3,) et GL(4,). Les intégrales ainsi obtenues sont très semblables aux intégrales connues des fonctions “classe-première” de Whittaker sur ces groupes. Nous croyons que cette correspondance entre les expressions des différents types de fonctions de Whittaker, pourra s’étendre aux groupes de rang arbitraire.

@article{AIF_1994__44_1_93_0,
     author = {Stade, Eric},
     title = {The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {1},
     year = {1994},
     pages = {93-108},
     doi = {10.5802/aif.1390},
     zbl = {0801.33001},
     mrnumber = {96h:11046},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_1_93_0}
}
Stade, Eric. The reciprocal of the beta function and $GL(n, {\mathbb {R}})$ Whittaker functions. Annales de l'Institut Fourier, Volume 44 (1994) no. 1, pp. 93-108. doi : 10.5802/aif.1390. http://www.numdam.org/item/AIF_1994__44_1_93_0/

[1] T. Bromwich, Introduction to the Theory of Infinite Series, Macmillan & Co. Ltd., 1908. | JFM 39.0306.02

[2] D. Bump, Automorphic Forms on GL (3, R), Springer Lecture Notes in Mathematics, n°1083 (1984). | MR 86g:11028 | Zbl 0543.22005

[3] D. Bump and J. Huntley, Unramified Whittaker functions for GL (3, R) to appear. | Zbl 0839.22016

[4] R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Springer Lecture Notes in Mathematics, n°260 (1972). | MR 49 #7241 | Zbl 0244.12011

[5] M. Hashizume, Whittaker functions on semisimple Lie groups, Hiroshima Math. J., 12 (1982), 259-293. | MR 84d:22018 | Zbl 0524.43005

[6] H. Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France, 95 (1967), 243-309. | Numdam | MR 42 #6158 | Zbl 0155.05901

[7] B. Kostant, On Whittaker vectors and representation theory, Inventiones Math., 48 (1978), 101-184. | MR 80b:22020 | Zbl 0405.22013

[8] H. Neunhöffer, Uber die analytische Fortsetzung von Poincaréreihen, Sitz. Heidelberger Akad. Wiss., 2 (1973), 33-90. | Zbl 0272.10015

[9] D. Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J., 52 (1973), 133-145. | MR 49 #2557 | Zbl 0288.10010

[10] I.I. Piatetski-Shapiro, Euler subgroups, Lie Groups and their Representations, John Wiley and Sons, 1975. | MR 53 #10720 | Zbl 0329.20028

[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. | MR 19,531g | Zbl 0072.08201

[12] J. Shalika, The multiplicity one theorem for GL(n), Annals of Math., 100 (1974), 171-193. | MR 50 #545 | Zbl 0316.12010

[13] L. Slater, Generalized Hypergeometric Functionsl Cambridge University Press, 1966. | Zbl 0135.28101

[14] E. Stade, Poincaré series for GL(3,R)-Whittaker functions, Duke Mathematical Journal, 58-3 (1989), pages 131-165. | MR 90i:22022 | Zbl 0699.10041

[15] E. Stade, On explicit integral formulas for GL(n, R)-Whittaker functions, Duke Mathematical Journal, 60-2 (1990), 313-362. | Zbl 0731.11027

[16] E. Stade, GL(4, R)-Whittaker functions and 4F3(1) hypergeometric series, Trans. Amer. Math. Soc., 336-1 (1993), 253-264. | MR 93e:22018 | Zbl 0786.11027

[17] N. Wallach, Asymptotic expansions of generalized matrix coefficients of representations of real reductive groups, Springer Lecture Notes in Mathematics, n°1024 (1983). | MR 85g:22029 | Zbl 0553.22005

[18] E. Whittaker and G. Watson, A Course of Modern Analysis, Cambridge University Press, 1902. | JFM 45.0433.02