Generalized Shalika model on $\mathrm{SO}_{4n}(F),$ symplectic linear model on $\mathrm{Sp}_{4n}(F)$ and theta correspondence

Abstract

We show that if an irreducible admissible representation of $\mathrm{SO}_{4n}(F)$ has a generalized Shalika model, then its small theta lift to $\mathrm{Sp}_{4n}(F)$ has the symplectic linear model, thus answering a question posed by D. Jiang. Here $F$ is a non-archimedean field of characteristic zero.