Spherical functions on symmetric spaces of Friedberg-Jacquet type

We give explicit models for spherical functions on $p$-adic symmetric spaces $X=H\mathrm{\backslash }G$ for pairs of $p$-adic groups $(G,H)$ of the form $(\mathrm{U}(2r),\mathrm{U}(r)\times \mathrm{U}(r)),$ $(\mathrm{O}(2r),\mathrm{O}(r)\times \mathrm{O}(r)),$ $(\mathrm{S}\mathrm{p}(4r),\mathrm{S}\mathrm{p}(2r)\times \mathrm{S}\mathrm{p}(2r))),$ $(\mathrm{U}(2r+1),\mathrm{U}(r+1)\times \mathrm{U}(r)),$ and $(\mathrm{O}(2r+1),\mathrm{O}(r+1)\times \mathrm{O}(r))\mathrm{.}$ As an application, we completely describe their Hecke module structure.