Residues of Certain Eisenstein Series of Classical Groups and the First Term Identities

Abstract

In this thesis, we are going to study certain degenerate Eisenstein series for the Symplectic groups and Orthogonal groups. We are going to prove the analytical properties, particularly the pole locations and exact order, as well as the first term identity. In order to understand the pole information, we first compute the normalizing factor for the Eisenstein series using the Gindikin-Karpalevich formula. Then we get a recursion formula with normalized sections to connect the Eisenstein series for Sp(2n) (or O(2n)) with Sp(2(n−1)) (or O(2(n−1)), correspondingly) by unfolding the Eisenstein series. From the recursion formula, we can get the range for pole locations for the Eisenstein series. The recursion formula also leads to a functional equation that demonstrates a symmetry between two Eisenstein series at different points. By using this functional equation as well as the recursion formula, we can get the exact pole order as well as the first term identity for spherical sections.