Enhanced variety of higher level and Kostka functions associated to complex reflection groups

Let $V$ be an $n$ dimensional vector space over an algebraic closure of a finite field $F_q$ and put $G=GL(V)$. For a positive integer $r$, we consider the variety $X_{uni}=G_{uni}\times V^{r-1}$, on which $G$ acts diagonally. $X_{uni}$ is the “unipotent part” of the enhanced variety of level $r$. $X_{uni}$ is partitioned into finitely many pieces $X_{\lambda }$ labelled by $r$-partitions $\lambda$ of $n$, and we consider the intersection cohomology $I{C}_{\lambda }$ associated to $X_{\lambda }$. In this paper, we show that the Frobenius trace functions (over $F_q$) associated to those $I{C}_{\lambda }$ satisfy certain orthogonality relations, which are very close to the equations characterizing the Kostka functions indexed by (a pair of) $r$-partitions. Using this we show, in some special cases, that the Kostka functions can be described in terms of those intersection cohomology, which is a (partial) generalization of the known results for the case $r=1,2$.