Quantum wreath products and Schur--Weyl duality II

TL;DR

This paper develops a framework for modules over quantum wreath products, unifying various algebraic modules and applying them to solve a significant problem in rational Cherednik algebra theory.

Contribution

It introduces wreath modules constructed via parabolic induction, unifying multiple algebraic modules and solving a key problem in Category O for rational Cherednik algebras.

Findings

Unified construction of modules over quantum wreath products

Recovery of modules over Ariki-Koike, Hu algebra, and affine Hecke algebra

Application to solve the Ginzburg-Guay-Opdam-Rouquier problem

Abstract

In the first part of this series, the authors introduced the quantum wreath product, providing a unified framework that encompasses numerous results previously addressed only through case-by-case analysis. This paper shifts focus to the fundamental construction of modules over these products, termed wreath modules. Our approach utilizes parabolic induction on tensor products combined with a sophisticated labeling scheme based on multipartitions. While the underlying constructions are technically involved, they offer a transparent realization of several prominent module families. Specifically, these wreath modules recover and unify: Simple modules over the Ariki-Koike algebra; Specht and simple modules over the Hu algebra; (anti)spherical modules and Kashiwara-Miwa-Stern modules over the affine Hecke algebra and its pro-p Iwahori variants. Finally, we demonstrate that these wreath modules for the Hu algebra serve as a critical component in solving the Ginzburg-Guay-Opdam-Rouquier problem. This solution enables a concrete realization of Category O for the rational Cherednik algebra in Type D.