  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
$IC_{\lambda}$ associated to $X_{\lambda}$. In this paper, we show that the
Frobenius trace functions (over $F_q$) associated to those $IC_{\lambda}$
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 $r = 1, 2$.
