Kostka functions associated to complex reflection groups

Kostka functions $K^{\pm}_{λ, μ}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $λ, μ$ of $r$-partitions and a sign $+, -$. It is expected that there exists a close connection 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 Kostka functions by making use of the geometry of $X$. In particular, we show that if $μ$ is of the form $μ= (-,\dots, -, ξ)$ and $λ$ is arbitrary, $K^-_{λ, μ}(t)$ has a Lascoux-Schützenberger type combinatorial description.