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Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Higher weight on GL(3). I: The Eisenstein series

Jack Buttcane
• Corresponding author
• Mathematics Department, University at Buffalo – The State University of New York, 244 Mathematics Building, Buffalo, NY 14260, USA
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Published Online: 2017-09-20 | DOI: https://doi.org/10.1515/forum-2017-0060

Abstract

The purpose of this paper is to collect and make explicit the results of Langlands [16], Bump [3], Miyazaki [18] and Manabe, Ishii and Oda [17] for the $\mathrm{GL}\left(3\right)$ Eisenstein series and Whittaker functions which are non-trivial on $\mathrm{SO}\left(3,ℝ\right)$. The final goal for the series of papers is a complete and completely explicit spectral expansion for ${L}^{2}\left(\mathrm{SL}\left(3,ℤ\right)\setminus \mathrm{SL}\left(3,ℝ\right)\right)$ in the style of Duke, Friedlander and Iwaniec’s paper [8]. We derive a number of new results on the Whittaker functions and Eisenstein series, and give new, concrete proofs of the functional equations and spectral expansion in place of the general constructions of Langlands.

MSC 2010: 11F72; 11F30

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Revised: 2017-08-05

Published Online: 2017-09-20

Published in Print: 2018-05-01

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 681–722, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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