# Truncated affine Springer fibers and Arthur's weighted orbital integrals

## TL;DR

This paper presents an algorithm to compute Arthur's weighted orbital integrals by counting rational points on truncated affine Springer fibers using two reduction methods, enabling explicit calculations for certain groups.

## Contribution

It introduces a novel approach combining Arthur-Kottwitz and Harder-Narasimhan reductions to relate rational point counts to weighted orbital integrals, with explicit examples for GL2 and GL3.

## Key findings

- Derived recurrence relations between rational points and orbital integrals.
- Successfully computed orbital integrals for GL2 and GL3.
- Established a new method for calculating orbital integrals via point counting.

## Abstract

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 us 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 group $\mathrm{GL}_{2}$ and $\mathrm{GL}_{3}$.