Distinguished standard modules for $\mathrm{GL}_{2m}(\mathbb{C})/\mathrm{GL}_m(\mathbb{H})$

We characterize the standard modules of $\GL_{2m}(\C)$ that are distinguished by $\GL_m(\HH)$. Let $δ_1,\dots, δ_{2m}$ be characters of $\mathbb{C}^\times$. Assume that $S = δ_1 \times \cdots \times δ_{2m}$ is a standard module of $\GL_{2m}(\mathbb{C})$. For each $i$, define ${δ_i^*} (z) = δ_i(\over