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

TL;DR

This paper reviews and consolidates key results on GL(3) Eisenstein series and Whittaker functions, aiming to develop an explicit spectral expansion for the space of automorphic forms on SL(3).

Contribution

It provides new explicit formulas, proofs, and insights into GL(3) Eisenstein series and Whittaker functions, advancing the spectral theory of automorphic forms.

Findings

New formulas for Whittaker functions

Explicit proofs of functional equations

Progress towards spectral expansion of SL(3) automorphic forms

Abstract

The purpose of this paper is to collect and make explicit the results of Langlands, Bump, Miyazaki and Manabe, Ishii and Oda for the $GL(3)$ Eisenstein series and Whittaker functions which are non-trivial on $SO(3,\mathbb{R})$. The final goal for the series of papers is a complete and completely explicit spectral expansion for $L^2(SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R}))$ in the style of Duke, Friedlander and Iwaniec's paper on Artin L-functions. 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.