The Degenerate Eisenstein Series Attached to the Heisenberg Parabolic Subgroups of Quasi-Split Forms of $Spi{n}_8$

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 $Spi{n}_8$ 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 $\pi$.