Quaternionic symplectic model for discrete series representations

Abstract

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 Sécherre and Stevens, this completes the classification of ${\mathrm{Sp}}_n(D)$-distinguished discrete series representations, as predicted by Dipendra Prasad.

Has been superceded by the paper arXiv:2503.08955