Kostka polynomials of $G(\ell,1,m)$

Abstract

For each integers $\ell > 1$ and $n \ge m \ge 1$, we prove an equivalence between the category of polynomial modules over a paraholic subalgebra $\mathfrak p$ of an affine Lie algebra of $\mathfrak{gl}(n\ell)$ and the module category of the smash product algebra $A$ of the complex reflection group $G(\ell,1,m)$ with $\mathbb C [X_1,\ldots,X_m]$. Then, we transfer the collection of $\mathfrak p$-modules considered in [Feigin-Makedonskyi-Khoroshkhin, arXiv:2311.12673] to $A$. Applying the Lusztig-Shoji algorithm [Shoji, Invent. Math. ${\bf 74}$ (1983)] (or rather its homological variant [K. Ann. Sci. ENS ${\bf 48}$(5) (2015)]), we conclude that the multiplicity counts of these modules yield the Kostka polynomials attached to the limit symbols in the sense of [Shoji, ASPM ${\bf 40}$ (2004)]. This particularly settles a conjecture of Shoji [${\it loc. cit.}$ §3.13] and answers a question in [Shoji, Sci. China Math. ${\bf 61}$ (2018)].