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

Abstract

We provide the explicit formula for orbital integrals associated with elliptic regular semisimple elements in $\mathrm{GL}_n(F) \cap \mathrm{M}_n(\mathfrak{o})$ and associated with arbitrary elements of the spherical Hecke algebra of $\mathrm{GL}_n(F)$ 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.