Lusztig varieties and Macdonald polynomials

Abstract

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.