The Degenerate Residual Spectrum of Quasi-Split Forms of $Spin_8$ Associated to the Heisenberg Parabolic Subgroup

Abstract

In \cite{MR3284482} and \cite{MR3658191}, the twisted standard $\mathcal{L}$-function $\mathcal{L}(s,\pi,\chi,st)$ of a cuspidal representation $ \pi$ of the exceptional group of type $G_2$ was shown to be represented by a family of new-way Rankin-Selberg integrals. These integrals connect the analytic behaviour of $\mathcal{L}(s,\pi,\chi,st)$ with that of a family of degenerate Eisenstein series $\mathcal{E}_E(\chi, f_s, s, g)$ on quasi-split forms $H_E$ of $Spin_8$, induced from Heisenberg parabolic subgroups. The analytic behaviour of the series $\mathcal{E}_E(\chi, f_s, s, g)$ in the right half-plane $Re(s)>0$ was studied in \cite{SegalEisen}. In this paper we study the residual representations associated with $\mathcal{E}_E(\chi, f_s, s, g)$.