Kostka functions associated to complex reflection groups

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $\lambda, \mu$ 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 $\mu$ is of the form $\mu = (-,\dots, -, \xi)$ and $\lambda$ is arbitrary, $K^-_{\lambda, \mu}(t)$ has a Lascoux-Sch\"utzenberger type combinatorial description.