Symplectic model for Zelevinsky modules of GL(n,D)

TL;DR

This paper establishes the existence and uniqueness of symplectic models for Zelevinsky modules of GL(n,D) over quaternion division algebras, extending previous results beyond the case where D equals the base field.

Contribution

It proves the existence and uniqueness of symplectic models for a family of Zelevinsky modules of GL(n,D), generalizing prior work to quaternion division algebras.

Findings

Existence and uniqueness of symplectic models for Zelevinsky modules.

Identification of a necessary condition for symplectic model existence.

Extension of Offen and Sayag's results to quaternion division algebras.

Abstract

Let D be a quaternion division algebra over a non-archimedean local field F of characteristic zero. This article demonstrates the existence and uniqueness of the symplectic model for a family of Zelevinsky modules of GL(n, D) to a family of irreducible representations of GL(n, D). For this family of irreducible representations, we identify a necessary condition under which a symplectic model can exist. This work extends a result of Offen and Sayag beyond the case D = F.