In type A, the Macdonald polynomials and the integral from Macdonald polynomials are related by a plethystic transformation. We interpret this plethystic transformation geometrically as a relationship between nilpotent parabolic Springer fibres and nilpotent Lusztig varieties. This points the way to a generalization of modified Macdonald polynomials and integral form Macdonald polynomials to all Lie types. But these generalizations are not polynomials, they are elements of the Iwahori-Hecke algebra of the finite Weyl group. This work concerns the generalization of, and connection between, a 1997 paper of Halverson-Ram (which counts points of nilpotent Lusztig varieties over a finite field) and a 2017 paper of Mellit (which counts points of nilpotent parabolic affine Springer fibres over a finite field).
This video is part of the conference Representation Theory and Geometry that took place at the University of Georgia.
