$\textrm {GL}(4,\textbf {R})$-Whittaker functions and ${}_ 4F_ 3(1)$ hypergeometric series
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- by Eric Stade PDF
- Trans. Amer. Math. Soc. 336 (1993), 253-264 Request permission
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 ${\text {GL}}(n,\mathbb {R})$.References
- Daniel Bump, Automorphic forms on $\textrm {GL}(3,\textbf {R})$, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 765698, DOI 10.1007/BFb0100147 A. Erdélyi et al., Higher transcendental functions, Vol. I, McGraw-Hill, 1953.
- Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495, DOI 10.1007/BFb0070263
- Michihiko Hashizume, Whittaker functions on semisimple Lie groups, Hiroshima Math. J. 12 (1982), no. 2, 259–293. MR 665496
- Hervé Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France 95 (1967), 243–309 (French). MR 271275, DOI 10.24033/bsmf.1654
- Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, DOI 10.1007/BF01390249
- Tomio Kubota, Elementary theory of Eisenstein series, Kodansha, Ltd., Tokyo; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. MR 0429749
- H. Neunhöffer, Über die analytische Fortsetzung von Poincaréreihen, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. (1973), 33–90 (German). MR 0352007
- Douglas Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973), 133–145. MR 337788, DOI 10.1017/S0027763000015932
- I. I. Pjateckij-Šapiro, Euler subgroups, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 597–620. MR 0406935
- A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
- J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
- Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
- Eric Stade, Poincaré series for $\textrm {GL}(3,\textbf {R})$-Whittaker functions, Duke Math. J. 58 (1989), no. 3, 695–729. MR 1016442, DOI 10.1215/S0012-7094-89-05833-X
- Eric Stade, On explicit integral formulas for $\textrm {GL}(n,\textbf {R})$-Whittaker functions, Duke Math. J. 60 (1990), no. 2, 313–362. With an appendix by Daniel Bump, Solomon Friedberg and Jeffrey Hoffstein. MR 1047756, DOI 10.1215/S0012-7094-90-06013-2 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.
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
Additional Information
- © Copyright 1993 American Mathematical Society
- 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