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

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.