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

Abstract

Kostka functions K , ± (t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials Kλ,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 Kλ,μ(t) by using an analogue of Lusztig’s partition function, and showed that Kλ,μ(t) ∈ Z>0[t] for generic μ by making use of a coherent realization. They conjectured that Kλ,μ(t) coincide with K , - (t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K , ± (t1,..., t r ).