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 L -function L ( s , π , χ , s t ) L(s,π ,χ ,st) of a cuspidal representation π π of the exceptional group of type G 2 G_2 . These integral representations bind the analytic behavior of this L L -function with that of a family of degenerate Eisenstein series for quasi-split forms of S p i n 8 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 these degenerate Eisenstein series in the right half-plane R e ( s ) > 0 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 π to be a CAP representation with respect to the Borel subgroup of G 2 G_2 in terms of the analytic behavior of L ( s , π , χ , s t ) L(s,π ,χ ,st)