Local theta-lift identities for archimedean long root $A$-packets of $\boldsymbol{\rm G}_2$

Let $F$ be an archimedean local field, let $G$ be the group of type $\mathsf{G}_2$, and let $K/F$ be the quadratic algebra used to define $G'_J$. Let $\chi:K^\times\to S^1$ be a unitary conjugate-symplectic character, let $\tau$ be its automorphic induction to an irreducible smooth representation of $\operatorname{PGL}_2(F)$, and let $\pi^+$ be the unique irreducible quotient of $i_Q^G(\tau|\cdot|^{1/2})$. When $K$ is a field, write $\pi^-$ for the irreducible discrete series representation with Harish--Chandra parameter $((|N|+1)/2,(|N|-1)/2,-|N|)$, and let $\sigma^\pm$ be the associated representations of $G'_J(F)$. Let $\Omega$ be the minimal representation of $\widetilde{G}(F)$ and let $\theta(-)$ denote the exceptional theta lift. Local theta-lift identities. One has $\theta(\sigma^+)\cong\pi^+$ and

$$\dim\operatorname{Hom}_{(G\times G'_J)(F)}(\Omega,\sigma^+\otimes\pi^+)=1.$$

If $K$ is a field and $G'_J$ is anisotropic, then $\theta(\sigma^-)\cong\pi^-$ and

$$\dim\operatorname{Hom}_{(G\times G'_J)(F)}(\Omega,\sigma^-\otimes\pi^-)=1.$$

If $K$ is a field and $G'_J$ is quasi-split, then $\theta(\sigma^-)=0$. These are local structural identities describing the archimedean members of the packet; the supplied text gives no evidence that they are conjectural or resolved.