Symplectic model for ladder and unitary representations

Abstract

Let $D$ denote a quaternion division algebra over a non-archimedean local field $F$ with characteristic zero. Let $Sp_n(D)$ be the unique non-split inner form of the symplectic group $Sp_{2n}(F)$. An irreducible admissible representation $(\pi, V)$ of $GL_{n}(D)$ is said to have a symplectic model (or said to be $Sp_n(D)$-distinguished) if there exists a linear functional $\phi$ on $V$ such that $\phi(\pi(h)v) = \phi(v)$ for all $v \in V$ and $h \in Sp_n(D)$. This article classifies those ladder representations of $GL_n(D)$ that possess a symplectic model (i.e., those representations that are $Sp_n(D)$-distinguished). Recently, Prasad conjectured that non-supercuspidal discrete series representations of $GL_n(D)$ do not admit a symplectic model. We confirm this for the Steinberg representations, which serve as canonical examples of discrete series representations. Furthermore, we demonstrate the hereditary nature of the symplectic model for induced representations derived from finite-length representations. In addition, we prove a part of Prasad's conjecture, which provides a family of irreducible unitary representations, all equipped with a symplectic model.