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

Abstract

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 GL ⁢ ( 3 ) {\mathrm{GL}(3)} Eisenstein series and Whittaker functions which are non-trivial on SO ⁢ ( 3 , ℝ ) {\mathrm{SO}(3,\mathbb{R})} . The final goal for the series of papers is a complete and completely explicit spectral expansion for L 2 ⁢ ( SL ⁢ ( 3 , ℤ ) ∖ SL ⁢ ( 3 , ℝ ) ) {L^{2}(\mathrm{SL}(3,\mathbb{Z})\setminus\mathrm{SL}(3,\mathbb{R}))} 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.