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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

MSC:
11F27 Theta series; Weil representation; theta correspondences
22E30 Analysis on real and complex Lie groups
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