TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

We explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

Components of Affine Springer Fibers

Let G be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $\gamma _0\in (\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $\gamma =t\gamma _0$. Using methods from p-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $\gamma$ modulo $Z_{\mathbf{G}((t))}(\gamma )$ is equal to the order of the Weyl group.