Quaternionic symplectic model for discrete series representations

Let $D$ be the quatenion division algebra over a non-Archimedean local field $F$ of characteristic zero and odd residual characterisitc. We show that an irreducible discrete series representation of $\mathrm{GL}_n(D)$ is $\mathrm{Sp}_n(D)$-distinguished only if it is supercuspidal. Here, $\mathrm{Sp}_n(D)$ is the quaternionic symplectic group. Combined with the recent study on $\mathrm{Sp}_n(D)$-distinguished supercuspidal representations by S\'echerre and Stevens, this completes the classification of $\mathrm{Sp}_n(D)$-distinguished discrete series representations, as predicted by Dipendra Prasad.