Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semisimple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the "kappa" orbital integral that arises in the fundamental lemma is equal to the Lefschetz trace of the Frobenius acting on the étale cohomology of a related variety.
"Homology of affine Springer fibers in the unramified case." Duke Math. J. 121 (3) 509 - 561, 15 February 2004. https://doi.org/10.1215/S0012-7094-04-12135-9