The degenerate residual spectrum of quasi-split forms of SPIN₈ associated to the Heisenberg parabolic subgroup

In [J. Inst. Math. Jussieu 14 (2015), pp. 149-184] and [Int. Math. Res. Not. imrn 7 (2017), pp. 2014-2099], the twisted standard L-function L(s, π, χ, st) of a cuspidal representation π of the exceptional group of type G₂ was shown to be represented by a family of new-way Rankin-Selberg integrals. These integrals connect the analytic behaviour of L(s, π, χ, st) with that of a family of degenerate Eisenstein series ε_E(χ, f_s, s, g) on quasi-split forms H_E of Spin8, induced from Heisenberg parabolic subgroups. The analytic behaviour of the series ε_E(χ, f_s, s, g) in the right half-plane Re(s) > 0 was studied in [Tran. Amer. Math. Soc. 370 (2018), pp. 5983-6039]. In this paper we study the residual representations associated with ε_E(χ, f_s, s, g).