Uniqueness conjecture for Ginzburg–Rallis models

Let $\mathbb K$ be a local field of characteristic zero, let ${\mathbb D}$ be a quaternion algebra over $\mathbb K$, and set $G_{\mathbb D}={\mathrm{GL}}_3({\mathbb D})$. Let $S_{\mathbb D}$ be the subgroup defined in the paper, let $\chi_{S_{\mathbb D}}$ be its specified character, and let $V_{\mathbb D}$ be an irreducible smooth representation when $\mathbb K$ is nonarchimedean, or an irreducible representation in the class $\mathcal{FH}$ when $\mathbb K$ is archimedean. Write $\mathbb C_{\chi_{S_{\mathbb D}}}$ for the one-dimensional representation of $S_{\mathbb D}$ given by this character.

Uniqueness conjecture for Ginzburg–Rallis models. The Ginzburg–Rallis model on $V_{\mathbb D}$ is unique up to scalar:

$$\dim \operatorname{Hom}_{S_{\mathbb D}}(V_{\mathbb D},\mathbb C_{\chi_{S_{\mathbb D}}})\leq 1.$$

This is the local multiplicity-one property needed in the study of the Ginzburg–Rallis conjecture. The paper proves the assertion in the Archimedean case, so the conjecture is solved in the setting stated there.