Determine distinction for standard modules and Langlands quotients of \(G_2\)

Determine, for every irreducible tempered representation \(\bm{\sigma}\) of the Heisenberg-parabolic Levi \(M_E\simeq (GL_2^3)^{\det}\) with central character trivial on the diagonal \(F^\times\), and every \(s>0\), the dimensions of \(\operatorname{Hom}_{H}(I_{P_E}(\bm{\sigma},s),\mathbf{1})\) and \(\operatorname{Hom}_{H}(J_{P_E}(\bm{\sigma},s),\mathbf{1})\), where \(H=G_2(F)\), \(I_{P_E}(\bm{\sigma},s)\) is the corresponding standard module of \(G_E=Spin_8(F)\), and \(J_{P_E}(\bm{\sigma},s)\) is its Langlands quotient.

Background

The paper introduces a second distinction problem arising from the Heisenberg parabolic of the split group $G_E=Spin_8$, whose Levi subgroup is identified with $M_E\simeq (GL_2^3)^{\det}$, and the embedding of $H=G_2(F)$ into $G_E$.

For a tempered representation $\bm{\sigma}$ of $M_E$, the standard module and Langlands quotient are induced with parameter $s>0$. The paper explains that Mackey theory yields five $H\backslash G_E/P_E$ double cosets and that, on a dense open set of parameters containing $s=1/2$, the non-closed-orbit contributions vanish. This gives a generic reduction to the first distinction problem, but the full determination requested here is stated as a separate problem.