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

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.