Constructions of global integrals in the exceptional groups

TL;DR

This paper explores a new framework for constructing global integrals associated with reductive groups, especially exceptional groups, revealing structural insights and introducing two novel Eulerian integrals.

Contribution

It introduces a two-dimensional array of global integrals linked to reductive groups, detailing their structure, Fourier coefficients, and unfolding constraints, with new examples including Eulerian integrals.

Findings

Identification of a two-dimensional array of global integrals.

Detailed analysis of Fourier coefficients and automorphic representations.

Introduction of two new Eulerian integrals.

Abstract

Motivated by known examples of global integrals which represent automorphic L-functions, this paper initiates the study of a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other. Fourier coefficients attached to unipotent classes, Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process are presented in detail with examples selected from the exceptional groups. Two new Eulerian integrals are included among these examples.