An explicit formula for the orbital integrals on the spherical Hecke algebra of ${\mathrm{G}\mathrm{L}}_3$

We provide the explicit formula for orbital integrals associated with elliptic regular semisimple elements in ${\mathrm{G}\mathrm{L}}_n(F)\cap {\mathrm{M}}_n(\mathfrak{o})$ and associated with arbitrary elements of the spherical Hecke algebra of ${\mathrm{G}\mathrm{L}}_n(F)$ when $n=2,3$, using results of [CKL]. Here $F$ is a non-Archimedean local field of any characteristic with $\mathfrak{o}$ its ring of integers.