2024/04/09 by Syu Kato, Kato, Syu
Mathematics · #05E05 #20F55 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2404.06268
openalex publication_date 2024/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For each integers ℓ > 1 and n ≥ m ≥ 1, we prove an equivalence between the category of polynomial modules over a paraholic subalgebra \mathfrak p of an affine Lie algebra of \mathfrakgl(nℓ) and the module category of the smash product algebra A of the complex reflection group G(ℓ,1,m) with \mathbb C [X1,…,Xm]. 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)].