The Degenerate Eisenstein Series Attached to the Heisenberg Parabolic Subgroups of Quasi-Split Forms of $Spin_8$

TL;DR

This paper investigates the poles of a degenerate Eisenstein series on quasi-split forms of $Spin_8$ and relates these poles to the properties of certain automorphic representations, providing criteria for CAP representations and settling related conjectures.

Contribution

It analyzes the poles of the Eisenstein series and establishes a criterion for CAP representations based on $ ext{L}$-function poles, also settling a conjecture and linking $ ext{L}$-function behavior to Fourier coefficients.

Findings

Pole analysis of Eisenstein series in $ ext{Re}(s)>0$

Criterion for CAP representations via $ ext{L}$-function poles

Resolution of a conjecture by Hundley and Ginzburg

Abstract

In previews works, joint with N. Gurevitch, a family of Rankin-Selberg integrals were shown to represent the twisted standard $\mathcal{L}$-function $\mathcal{L}\left(s,\pi,\chi,\mathfrak{st}\right)$ of a cuspidal representation $ \pi$ of the exceptional group of type $G_2$. This integral representation binds the analytic behavior of this $\mathcal{L}$-functions with that of a degenerate Eisenstein series defined over the family of quasi-split forms of $Spin_8$ associated to an induction from a character on the Heisenberg parabolic subgroup. This paper is divided into two parts. In part 1 we study the poles of this degenerate Eisenstein series in the right half plane $\mathfrak{Re}(s)>0$. In part 2 we use the results of part 1 to give a criterion for $\pi$ to be a {\bf CAP} representation with respect to the Borel subgroup in terms of poles of $\mathcal{L}\left(s,\pi,\chi,\mathfrak{st}\right)$. We also settle a conjecture of J. Hundley and D. Ginzburg and prove a few results relating the analytic behavior of $\mathcal{L}\left(s,\pi,\chi,\mathfrak{st}\right)$ and the set of Fourier coefficients supported by $\pi$.