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 GLₙ(D) is Spₙ(D)-distinguished only if it is supercuspidal. Here, Spₙ(D) is the quaternionic symplectic group. Combined with the recent study on Spₙ(D)-distinguished supercuspidal representations by Sécherre and Stevens, this completes the classification of Spₙ(D)-distinguished discrete series representations, as predicted by Dipendra Prasad.