Graded Multiplicities in the Macdonald kernel and a (q,t) coinvariant algebra

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.

In this talk, I will describe a general method for decomposing the Macdonald kernel into irreducible characters, and some of the consequences of this decomposition (e.g., explicit formulas, some proved, some conjectured). The most interesting result relates the q,t-multiplicities in the Macdonald kernel with a bi-graded generalization of the co-invariant algebra of the associated Weyl group. This generalizes work of A. Broer (1995), Y. Bazlov (2001), and J. Stembridge (Ph.D. thesis, 1985), and both proves and generalizes a conjecture of M. Reeder (1997).