Symplectic model for ladder and unitary representations

TL;DR

This paper classifies ladder representations of $GL_n(D)$ with symplectic models over a quaternion division algebra, confirms Prasad's conjecture for Steinberg representations, and explores hereditary properties of symplectic models.

Contribution

It provides a classification of symplectic models for ladder representations and confirms parts of Prasad's conjecture for discrete series representations.

Findings

Classified ladder representations with symplectic models.

Confirmed Prasad's conjecture for Steinberg representations.

Established hereditary properties of symplectic models in induced 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.