  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.
