On representations of ${\rm GL}_{2n}(F)$ with a symplectic period

TL;DR

This paper classifies certain irreducible admissible representations of ${ m GL}_{4}(F)$ and ${ m GL}_{6}(F)$ that have nontrivial symplectic invariance, and proposes conjectures for the general case.

Contribution

It provides a classification for specific low-rank cases and introduces conjectures for the broader classification problem.

Findings

Classified irreducible admissible representations with symplectic invariance for ${ m GL}_{4}(F)$ and ${ m GL}_{6}(F)$

Proposed conjectures for the general case of ${ m GL}_{2n}(F)$

Identified conditions for the existence of invariant linear forms

Abstract

The main aim of this paper is to classify the irreducible admissible representations of ${\rm GL}_{4}(F)$ and ${\rm GL}_{6}(F)$ for a nonarchimedean local field $F$, which bear a nontrivial linear form invariant under the groups ${\rm Sp}_{4}(F)$ and ${\rm Sp}_{6}(F)$ respectively. We propose a few conjectures for the general case.