Poles of degenerate Eisenstein series and Siegel-Weil identities for exceptional split groups

TL;DR

This thesis investigates the poles of degenerate Eisenstein series for split exceptional groups, determines conditions for square integrability of their Laurent expansion, and develops an algorithm to find identities between leading terms of Eisenstein series.

Contribution

It introduces new results on Eisenstein series poles and identities specifically for split exceptional groups, with an algorithm implemented in SAGE for this purpose.

Findings

Identified poles of Eisenstein series in the region Re s > 0.

Determined when Laurent expansion leading terms are square integrable.

Developed an algorithm for finding identities between Eisenstein series leading terms.

Abstract

Let $G$ be a linear split algebraic group. The degenerate Eisenstein series associated to a maximal parabolic subgroup $E_{P}(f^{0},s,g)$ with the spherical section $f^{0}$ is studied in the first part of the thesis. In this part, we study the poles of $E_{P}(f^{0},s,g)$ in the region $\operatorname{Re} s >0$. We determine when the leading term in the Laurent expansion of $E_{P}(f^{0},s,g)$ around $s=s_0$ is square integrable. The second part is devoted to finding identities between the leading terms of various Eisenstein series at different points. We present an algorithm to find this data and implement it on \textit{SAGE}. While both parts can be applied to a general algebraic group, we restrict ourself to the case where $G$ is split exceptional group of type $F_4,E_6,E_7$, and obtain new results.