Spherical Shalika models on ${\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2}$ and the theta correspondence for $({\mathrm{P}\mathrm{G}\mathrm{S}\mathrm{p}}_4,{\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2})$

We study Shalika models for generic unramified representations of ${\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2}$ over non-archimedean local fields of characteristic zero. We show that they are unique up to constant by means of the theta correspondence for $({\mathrm{P}\mathrm{G}\mathrm{S}\mathrm{p}}_4,{\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2})$. We then prove a Casselman-Shalika formula which relates the values of spherical Shalika functionals on ${\mathrm{P}\mathrm{G}\mathrm{U}}_{2,2}$ to the values of finite dimensional complex representations of the dual group of ${\mathrm{P}\mathrm{G}\mathrm{S}\mathrm{p}}_4$.