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

Abstract

Kostka functions $K_{\lambda ,\mu }^{\pm }(t)$ 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.

47 pages, v3: Final version, some references are added. To appear in SCIENCE CHINA Mathematics

References in corpus (1)

• Quiver Hall-Littlewood functions and Kostka-Shoji polynomials