Twists of representations of complex reflection groups and rational Cherednik algebras

TL;DR

This paper explores how cocycle twists deform algebra structures and representations, specifically focusing on complex reflection groups and rational Cherednik algebras, revealing new relationships and characterizations.

Contribution

It provides general results on cocycle twists of algebra factorisations and applies them to reflection groups and Cherednik algebras, including character actions and module characterizations.

Findings

Twist actions on Coxeter group characters are described.

Standard modules of Cherednik algebras are characterized as braided Cherednik modules.

An analogue of Chevalley's theorem is established for noncommutative coinvariant algebras.

Abstract

Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to reflection groups and rational Cherednik algebras. In particular, we describe how a twist acts on characters of Coxeter groups of type $B_n$ and $D_n$ and relate them to characters of mystic reflection groups. This is used to characterise twists of standard modules of rational Cherednik algebras as standard modules for certain braided Cherednik algebras. We introduce the coinvariant algebra of a mystic reflection group and use a twist to show that an analogue of Chevalley's theorem holds for these noncommutative algebras. We also discuss several cases where the negative braided Cherednik algebras are, and are not, isomorphic to rational Cherednik algebras.