Schubert polynomials, Kazhdan-Lusztig basis and characters

A combinatorial formula for the characters of the homogeneous components of the coinvariant algebra is given. The formula is proved by considering the action of the simple reflections on the Schubert polynomials basis of this algebra. In the symmetric group case, the formula is equivalent to a combinatorial rule for decomposing the homogeneous components into irreducible representations. The proof of the equivalence involves permutation statistics and Kazhdan-Lusztig theory. The formula is very similar to an analogous one for Kazhdan-Lusztig representations of these groups.