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

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