are a two-parameter family of orthogonal polynomials indexed by a positive weight λ of a root system, introduced by Ian G. Macdonald (1987).
They are known to have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
The so-called q,t-Kostka polynomials are the coefficients of a resulting transition matrix.
In an attempt to prove Macdonald's conjecture, Garsia & Haiman (1993) introduced the bi-graded module
This breakthrough led to the discovery of many hidden connections and new aspects of symmetric group representation theory, as well as combinatorial objects (e.g., insertion tableaux, Haglund's inversion numbers, and the role of parking functions in representation theory).