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 $\mathbb{C}[X_1,\dots ,X_m]$. 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. $74$ (1983)] (or rather its homological variant [K. Ann. Sci. ENS $48$(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 $40$ (2004)]. This particularly settles a conjecture of Shoji [$loc.cit.$ §3.13] and answers a question in [Shoji, Sci. China Math. $61$ (2018)].

27pp, v2: corrected typos and minor errors, slightly changed some notations, and amplified examples