Integrals of polynomials associated with tableaux and the Garsia-Haiman conjecture

In a 1983 paper [M1], I. G. Macdonald introduced his well-known “constant term conjectures.” These conjectures concern a certain polynomial ∆ = ∆(G, k) that is indexed by a semisimple Lie algebra G and a positive integer k. The polynomial ∆ lives in Z[Φ, q], the group ring of the root lattice Φ of G over Z[q]. A basis for this ring, over Z[q], is the set of formal exponentials, ev, for v ∈ Φ that satisfy the relations ev · ew = ev+w. The conjecture asserts that the constant term of∆, meaning the part that is independent of the formal exponentials, has a nice factorization as a polynomial in q. Later, Macdonald [M2] generalized this work in the following way. He showed that there is a unique collection of polynomials Pν indexed by dominant weights ν, satisfying the following properties: