A comparison problem for abelian surfaces and descent for symplectic orbital integrals

To answer a question about the distribution of products of elliptic curves in isogeny classes of abelian surfaces defined over finite fields, we compute specific orbital integrals in the group ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$. More precisely, we compute integrals over the orbits of elements in the subgroup ${\mathrm{G}\mathrm{L}}_2{\times }_{\det }{\mathrm{G}\mathrm{L}}_2$. As a first step towards a complete solution of the problem, this article contains explicit computations for arbitrary orbital integrals of spherical functions over this subgroup, and also compute orbital integrals over ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ in a large number of cases.