Poles, Residues and Siegel-Weil Identities of Degenerate Eisenstein Series on Split Exceptional Groups of Type $E_n$

Abstract

This manuscript has two goals: 1. To write an explicit description of the degenerate residual spectrum of the split, simple, simply-connected, exceptional groups of type $E_n$ (for $n=6,7,8$). 2. To set a practical guide for similar calculations and, in particular, to describe various methods of ``computational representation theory'' relevant to the study of residues of automorphic Eisenstein series. In Part I we supply background information and notations from the theory of automorphic representations as well as concrete information on the exceptional groups of type $E_n$ and their representation theory over non-Archimedean local fields. In Part II we make a systematic study of the residual spectrum of these groups where each chapter is devoted to a certain aspect of theory, it begins with methodical section and continues with sections devoted to the results for each of these groups. We describe completely the residual (square-integrable and non-square integrable) spectrum of the groups of type $E_6$ and $E_7$ and an almost complete description in the case of the group of type $E_8$. We also study and list Siegel-Weil like identities between residual representations of these groups and list the Arthur parameters for their square-integrable residual representations.