The Degenerate Residual Spectrum of Quasi-Split Forms of $Spi{n}_8$ 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 $Spi{n}_8$, 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 ${\mathcal{E}}_E(\chi ,f_s,s,g)$.