On the local behavior of weighted orbital integrals and the affine Springer fibers

TL;DR

This paper explores the local behavior of weighted orbital integrals and their connection to affine Springer fibers, proving homogeneity properties and showing that point counts form rational generating series with bounded denominators.

Contribution

It establishes the homogeneity of the Arthur-Shalika germ expansion and proves the rationality of the generating series for point counts on affine Springer fibers.

Findings

Proved homogeneity of Arthur-Shalika germ expansion.

Established parabolic descent for weighted orbital integrals.

Showed the generating series is a rational fraction with bounded denominator.

Abstract

The paper deals with two problems that seem to be of different nature: the local behavior of the weighted orbital integrals and the dependence of the affine Springer fibers on the root valuation datum of their defining elements. The intersection happens when we try to count the number of points on the fundamental domains of the affine Springer fibers: the number of points can be expressed in terms of certain weighted orbital integrals and vice versa. For the first problem, we prove the homogeneity of the Arthur-Shalika germ expansion of the weighted orbital integrals, and establish its parabolic descent. This is then applied to our conjecture about the dependence of the Poincar\'e polynomials of the fundamental domains on the root valuation datum. We show that the generating series of the number of points on the fundamental domains with respect to the root valuation datum of the defining elements is a rational fraction, and we bound its denominator with datum associated to the nilpotent orbits.