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

Hariom Sharma

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.

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

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.
Paper Structure (4 sections, 8 theorems, 32 equations)

This paper contains 4 sections, 8 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Some useful lemmas
  4. Proof of Theorem \ref{['1']}

Key Result

Lemma 1.1

For $\pi \in \mathrm{Irr}(G_n)$, we have $\mathrm{dim} \mathrm{Hom}_{H_n}(\pi, \mathbb{C}) \leq 1$.

Theorems & Definitions (13)

  • Lemma 1.1
  • Theorem 2.1
  • Remark 2.1
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 3 more