Supports of representations of the rational Cherednik algebra of type A

TL;DR

This paper extends the theory of rational Cherednik algebra representations to complex varieties, characterizing support sets and establishing equivalences with Hecke algebra representations.

Contribution

It generalizes support set classifications for Cherednik algebra representations beyond vector spaces and links support categories to Hecke algebra modules.

Findings

Support sets are classified for Cherednik algebra representations on complex varieties.

Categories of representations with fixed support are equivalent to Hecke algebra modules.

The paper determines support sets for irreducible modules when the group is S_n.

Abstract

We first consider the rational Cherednik algebra corresponding to the action of a finite group on a complex variety, as defined by Etingof. We define a category of representations of this algebra which is analogous to "category O" for the rational Cherednik algebra of a vector space. We generalise to this setting Bezrukavnikov and Etingof's results about the possible support sets of such representations. Then we focus on the case of $S_n$ acting on $\C^n$, determining which irreducible modules in this category have which support sets. We also show that the category of representations with a given support, modulo those with smaller support, is equivalent to the category of finite dimensional representations of a certain Hecke algebra.