Kostka functions associated to complex reflection groups

Kostka functions Kλ,μ±(t) associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair λ, μ of r-partitions of n (and by the sign +, −). It is expected that there exists a close relationship between those Kostka functions and the intersection cohomology associated to the enhanced variety X of level r. In this paper, we study combinatorial properties of Kλ,μ±(t) based on the geometry of X. In particular, we show that in the case where μ=(−,…,−,μ(r)) (and for arbitrary λ), Kλ,μ−(t) has a Lascoux-Schützenberger type combinatorial description.