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

We characterize the standard modules of ${\text{GL}}_{2m}(\text{C})$ that are distinguished by ${\text{GL}}_m(\text{HH})$. Let ${\delta }_1,\dots ,{\delta }_{2m}$ be characters of ${\mathbb{C}}^{\times }$. Assume that $S={\delta }_1\times \cdots \times {\delta }_{2m}$ is a standard module of ${\text{GL}}_{2m}(\mathbb{C})$. For each $i$, define ${\delta }_i^{\ast }(z)={\delta }_i(\bar{z}{)}^{-1}$ for $z\in {\mathbb{C}}^{\times }$. In particular, we conclude that a standard module for ${\text{GL}}_{2m}(\mathbb{C})$ is distinguished by ${\text{GL}}_m(\mathbb{H})$ if and only if there exists an involution $p\in S_{2m}$ without fixed points such that ${\delta }_{p(i)}={\delta }_i^{\ast }$ for every $i$. We first verify the hypotheses of the multiplicity estimate theorem of Suzuki and Tamori in \cite{ST}. The orbit calculation of Matringe, Offen, and Yang in \cite{MOYglobal} then gives the necessary condition and a dimension bound. Then local intertwining periods prove sufficiency.