Quaternionic symplectic model for discrete series representations

TL;DR

This paper proves that irreducible discrete series representations of ig( ext{GL}_n(D)ig) are ig( ext{Sp}_n(D)ig)-distinguished only if they are supercuspidal, completing a classification predicted by Prasad in the context of quaternionic symplectic models.

Contribution

It establishes a necessary condition for ig( ext{GL}_n(D)ig) discrete series representations to be ig( ext{Sp}_n(D)ig)-distinguished, advancing the classification of such representations.

Findings

Discrete series representations are ig( ext{Sp}_n(D)ig)-distinguished only if supercuspidal.

Completes the classification of ig( ext{Sp}_n(D)ig)-distinguished discrete series.

Supports the prediction by Dipendra Prasad.

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