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

Kostka functions K _λ,μ ^± (t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_λ,μ(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig’s partition function. Finkelberg and Ionov (2016) defined alternate functions K_λ,μ(t) by using an analogue of Lusztig’s partition function, and showed that K_λ,μ(t) ∈ Z>0t] for generic μ by making use of a coherent realization. They conjectured that K_λ,μ(t) coincide with K _λ,μ ^- (t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K _λ,μ ^± (t₁,..., t _r ).