The University of Michigan Combinatorics Seminar
For each root system, there is a Macdonald kernel--it is the symmetric
bilinear form relative to which the Macdonald polynomials are orthogonal.
It may also be viewed as a virtual character with coefficients that are
formal power series in q and t. Various specializations of it are
connected to classical work of Kostant and Chevalley. For example,
when q=0, the graded multiplicities of the irreducible characters are
(up to normalization) polynomials in t with nonnegative coefficients.
Aside from the type A case, the combinatorics of these polynomials still
remain rather mysterious.