The first coefficient of Langlands Eisenstein series for $\hbox{SL}(n,\mathbb Z)$

TL;DR

This paper introduces an elementary approach to defining Langlands Eisenstein series for SL(n,Z) and explicitly computes their first Fourier coefficient, simplifying previous complex methods.

Contribution

It provides a straightforward, explicit formula for the first Fourier coefficient of Langlands Eisenstein series for SL(n,Z), using Borel Eisenstein series as a template.

Findings

Explicit formula for the first Fourier coefficient of Langlands Eisenstein series

Simplified proofs using Borel Eisenstein series as a template

Elementary derivation of Eisenstein series for SL(n,Z)

Abstract

Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands-Shahidi method, and in various other aspects of the theory of automorphic forms and representations. In this paper, we define Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$ in an elementary manner, and then determine the first Fourier coefficient of these series in a very explicit form. Our proofs and derivations are short and simple, and use the Borel Eisenstein series as a template to determine the first Fourier coefficient of other Langlands Eisenstein series.