The degenerate Eisenstein series attached to the Heisenberg parabolic subgroups of quasi-split forms of spin8

In [J. Inst. Math. Jussieu 14 (2015), 149-184] and [Int. Math. Res. Not. IMRN 7 (2017), 2014-2099] a family of Rankin-Selberg integrals was shown to represent the twisted standard L-function L(s, π, χ, st) of a cuspidal representation π of the exceptional group of type G₂. These integral representations bind the analytic behavior of this L-function with that of a family of degenerate Eisenstein series for quasi-split forms of Spin8 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 these degenerate Eisenstein series in the right half-plane Re(s) > 0. In Part 2 we use the results of Part 1 to prove the conjecture, made by J. Hundley and D. Ginzburg in [Israel J. Math. 207 (2015), 835-879], for stable poles and also to give a criterion for π to be a CAP representation with respect to the Borel subgroup of G₂ in terms of the analytic behavior of L(s, π, χ, st) at s = 3/2.