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$ {\rm GL}(4,{\bf R})$-Whittaker functions and $ {}\sb 4F\sb 3(1)$ hypergeometric series


Author: Eric Stade
Journal: Trans. Amer. Math. Soc. 336 (1993), 253-264
MSC: Primary 22E30; Secondary 11F55, 33C15, 33C20
DOI: https://doi.org/10.1090/S0002-9947-1993-1102226-1
MathSciNet review: 1102226
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Abstract: In this paper we consider spaces of $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions, which are special functions that arise in the study of $ {\text{GL}}(4,\mathbb{R})$ automorphic forms. Our main result is to determine explicitly the series expansion for a $ {\text{GL}}(4,\mathbb{R})$-Whittaker function that is "fundamental," in that it may be used to generate a basis for the space of all $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions of fixed eigenvalues.

The series that we find in the case of $ {\text{GL}}(4,\mathbb{R})$ is particularly interesting in that its coefficients are not merely ratios of Gamma functions, as they are in the lower-rank cases. Rather, these coefficients are themselves certain series-- namely, they are finite hypergeometric series of unit argument. We suspect that this is a fair indication of what will happen in the general case of $ {\text{GL}}(n,\mathbb{R})$.


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  • [1] D. Bump, Automorphic forms on $ GL(3,\mathbb{R})$, Lecture Notes in Math., vol. 1083, Springer, 1984. MR 765698 (86g:11028)
  • [2] A. Erdélyi et al., Higher transcendental functions, Vol. I, McGraw-Hill, 1953.
  • [3] R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer, 1972. MR 0342495 (49:7241)
  • [4] M. Hashizume, Whittaker functions on semisimple Lie groups, Hiroshima Math. J. 12 (1982), 259-293. MR 665496 (84d:22018)
  • [5] H. Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France 95 (1967), 243-309. MR 0271275 (42:6158)
  • [6] B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101-184. MR 507800 (80b:22020)
  • [7] T. Kubota, Elementary theory of Eisenstein series, Kodansha-Wiley, 1973. MR 0429749 (55:2759)
  • [8] H. Neunhöffer, Uber die analytische Fortsetzung von Poincaréreihen, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl. 2 (1973), 33-90. MR 0352007 (50:4495)
  • [9] D. Niebur, A class of nonanalytic automorphic functions, Nagoya Math J. 52 (1973), 133-145. MR 0337788 (49:2557)
  • [10] I. I. Piatetski-Shapiro, Euler subgroups, Lie Groups and Their Representations, Wiley, 1975. MR 0406935 (53:10720)
  • [11] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. MR 0088511 (19:531g)
  • [12] J. Shalika, The multiplicity one theorem for $ GL(n)$, Ann. of Math. (2) 100 (1974), 171-193. MR 0348047 (50:545)
  • [13] L. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, 1966. MR 0201688 (34:1570)
  • [14] E. Stade, Poincaré series for $ GL(3,\mathbb{R})$-Whittaker functions, Duke Math. J. 58 (1989), 131-165. MR 1016442 (90i:22022)
  • [15] -, On explicit integral formulas for $ {\text{GL}}(n,\mathbb{R})$-Whittaker functions, Duke Math. J. 60 (1990), 313-362. MR 1047756 (92a:11060)
  • [16] I. Vinogradov and L. Takhtadzhyan, Theory of Eisenstein series for the group $ SL(3,\mathbb{R})$ and its application to a binary problem, J. Soviet Math. 18 (1982), no. 3, 293-324.
  • [17] E. Whittaker and G. Watson, A course of modern analysis, Cambridge Univ. Press, 1902. MR 1424469 (97k:01072)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1102226-1
Keywords: Whittaker functions, automorphic forms, hypergeometric series
Article copyright: © Copyright 1993 American Mathematical Society

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