  This paper uses Lusztig varieties to give central elements of the
Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the
finite Chevalley group $GL_n(\mathbb{F}_q)$. We explain how these central
elements are related to Macdonald polynomials and how this provides a framework
for generalizing integral form and modified Macdonald polynomials to Lie types
other than $GL_n$. The key steps are to recognize (a) that counting points in
Lusztig varieties is equivalent to computing traces on the Hecke algebras, (b)
that traces on the Hecke algebra determine elements of the center of the Hecke
algebra, (c) that the Geck-Rouquier basis elements of the center of the Hecke
algebra produce an `expansion matrix', (d) that the parabolic subalgebras of
the Hecke algebra produce a `contraction matrix' and (e) that the combination
`expansion-contraction' is the plethystic transformation that relates integral
form Macdonald polynomials and modified Macdonald polynomials.
