Parabolic degeneration of rational Cherednik algebras

TL;DR

This paper introduces parabolic degenerations of rational Cherednik algebras to establish conditions for finite-dimensionality of modules and maps between standard modules, providing new insights and proofs in the representation theory of these algebras.

Contribution

It develops parabolic degenerations of rational Cherednik algebras and applies them to classify finite-dimensional modules and analyze module maps, refining existing combinatorial frameworks.

Findings

Necessary conditions for finite-dimensionality of modules.

Enhanced combinatorial understanding of module maps.

New proof of classification for symmetric group Cherednik algebras.

Abstract

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between two standard modules. The latter condition reproduces and enhances, in the case of the symmetric group, the combinatorics of cores and dominance order, and in general shows that the c-ordering on category O may be replaced by a much coarser ordering. The former gives a new proof of the classification of finite dimensional irreducible modules for the Cherednik algebra of the symmetric group.