A stratification and smoothening method gives exact stable orbital integral formulas for gl2, gl3, and u2, and new lower bounds for all n.
An explicit formula for the orbital integrals on the spherical Hecke algebra of $\mathrm{GL}_3$
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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.
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Stable orbital integrals for classical Lie algebras and smooth integral models
A stratification and smoothening method gives exact stable orbital integral formulas for gl2, gl3, and u2, and new lower bounds for all n.