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Poincaré series for GL(3,\({\mathbb{R}})\)-Whittaker functions. (English) Zbl 0699.10041
Poincaré series may be used to construct \(SL_ 2({\mathbb{Z}})\)-invariant eigenfunctions of the Laplacian on the upper half plane, see e.g. D. Niebur [Nagoya Math. J. 52, 133-145 (1973; Zbl 0288.10010)]. Except in the special case of Eisenstein series these functions have exponential growth at the cusp.
In this paper the generalization to automorphic forms on \(GL_ 3({\mathbb{R}})/O_ 3({\mathbb{R}})\) is considered. The straightforward generalization is shown to lead to series that do not converge absolutely for any choice of the spectral parameters. Nevertheless, some perturbed form of the Poincaré series converges, although it does not give an eigenfunction of the invariant differential operators. The scalar product with a fixed cusp form exists. It is given by the product of a Fourier coefficient of the cusp form and some integral expression containing Whittaker functions for \(GL_ 3({\mathbb{R}})\). An explicit analysis shows that this expression makes sense for the unperturbed value of the parameter, and that for this value it can be described in terms of hypergeometric series. This gives a meromorphic continuation in the spectral parameters, and even a functional equation.
What can be done for one cusp form, can be done for finitely many, without losing uniformity in the estimates. In this way the author obtains an interpretation of Poincaré series on \(GL_ 3({\mathbb{R}})/O_ 3({\mathbb{R}})\) as linear forms on any finite dimensional space spanned by cusp forms. Thus interpreted, the Poincaré series has a meromorphic continuation and a functional equation.
Reviewer: R.W.Bruggeman

11F27 Theta series; Weil representation; theta correspondences
22E30 Analysis on real and complex Lie groups
Full Text: DOI
[1] D. Bump, Automorphic forms on \(\mathrm GL(3,\mathbf R)\) , Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. · Zbl 0543.22005
[2] D. Bump and S. Friedberg, On Mellin transforms of unramified Whittaker functions on \(GL(3,\mathbfC)\) , to appear in J. Math. Anal. Appl. · Zbl 0675.33008 · doi:10.1016/0022-247X(89)90239-4
[3] S. Friedberg, A global approach to the Rankin-Selberg convolution for \(\mathrm GL(3,\mathbf Z)\) , Trans. Amer. Math. Soc. 300 (1987), no. 1, 159-174. JSTOR: · Zbl 0621.10019 · doi:10.2307/2000593 · links.jstor.org
[4] D. Goldfeld, Kloosterman zeta functions for \(\mathrm GL(n,\mathbf Z)\) , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 417-424. · Zbl 0667.10027
[5] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1980. · Zbl 0521.33001
[6] M. Hashizume, Whittaker functions on semisimple Lie groups , Hiroshima Math. J. 12 (1982), no. 2, 259-293. · Zbl 0524.43005
[7] H. Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley , Bull. Soc. Math. France 95 (1967), 243-309. · Zbl 0155.05901 · numdam:BSMF_1967__95__243_0 · eudml:87096
[8] B. Kostant, On Whittaker vectors and representation theory , Invent. Math. 48 (1978), no. 2, 101-184. · Zbl 0405.22013 · doi:10.1007/BF01390249 · eudml:142586
[9] T. Kubota, Elementary theory of Eisenstein series , Kodansha Ltd., Tokyo, 1973. · Zbl 0268.10012
[10] R. Miatello and N. Wallach, Automorphic forms constructed from Whittaker vectors , · Zbl 0692.10029 · doi:10.1016/0022-1236(89)90059-1
[11] H. Neunhöffer, Über die analytische Fortsetzung von Poincaréreihen , S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. (1973), 33-90.
[12] D. Niebur, A class of nonanalytic automorphic functions , Nagoya Math. J. 52 (1973), 133-145. · Zbl 0288.10010
[13] I. I. Pjateckij-Shapiro, Euler subgroups , Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 597-620. · Zbl 0329.20028
[14] 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. · Zbl 0072.08201
[15] J. A. Shalika, The multiplicity one theorem for \(\mathrm GL\sbn\) , Ann. of Math. (2) 100 (1974), 171-193. JSTOR: · Zbl 0316.12010 · doi:10.2307/1971071 · links.jstor.org
[16] L. J. Slater, Generalized hypergeometric functions , Cambridge University Press, Cambridge, 1966. · Zbl 0135.28101
[17] E. Stade, Whittaker functions and Poincaré series for \(GL(3,\mathbfR)\) , Ph.D. thesis, Columbia University, 1988. · Zbl 0682.62058
[18] I. Vinogradov and L. Takhtadzhyan, Theory of Eisenstein series for the group \(SL(3,\mathbfR)\) and its application to a binary problem , J. Soviet Math. 18 (1982), 293-324. · Zbl 0476.10024 · doi:10.1007/BF01084842
[19] E. Whittaker and G. Watson, A Course of Modern Analysis , Cambridge University Press, Cambridge, 1902. · JFM 33.0390.01
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