Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by $r$-partitions $\lambda, \mu$ and a sign $+, -$. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov defined alternate functions $K_{\lambda,\mu}(t)$ by using an analogue of Lusztig's partition function, and showed that $K_{\lambda,\mu}(t)$ are polynomials in $t$ with non-negative integer coefficients. They conjecture that their $K_{\lambda,\mu}(t)$ coincide with $K^-_{\lambda,\mu}(t)$. In this paper, we show that their conjecture holds. We also discuss a multi-variable version of Kostka functions.