A template method for Fourier coefficients of Langlands Eisenstein series

TL;DR

This paper presents a template method for efficiently computing the first Fourier coefficient of Langlands Eisenstein series on general linear and Chevalley groups, simplifying complex calculations through elementary linear algebra.

Contribution

It introduces a novel template method that leverages the first coefficient of Borel Eisenstein series to compute coefficients of more general Eisenstein series.

Findings

Simplifies computation of Eisenstein series coefficients

Applicable to $ ext{GL}(n, ext{R})$ and Chevalley groups

Reduces complex calculations to elementary linear algebra

Abstract

This paper introduces the template method for computing the first coefficient of Langlands Eisenstein series on $\GL(n,\mathbb R)$ and more generally on Chevalley groups over the adele ring of $\mathbb Q.$ In brief, the first coefficient of Borel Eisenstein series can be used as a template to compute the first coefficient of more general Eisenstein series by elementary linear algebra calculations.