Supports of representations of the rational Cherednik algebra of type A

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 Sn 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.

On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$O

“…The results we obtain for finite Coxeter groups confirm, unify, and extend many previously known results in both classical and exceptional types. In type A, we recover Wilcox's description [W,Theorem 1.2] of the subquotients of the filtration by supports of the categories O c (S n , h). In type B, we show that the subquotients of our refined filtration are equivalent to categories of finite-dimensional modules over tensor products of Iwahori-Hecke algebras of type B, and we obtain similar descriptions in type D. These facts follows from results of Shan-Vasserot [SV], although their results do not generalize to the exceptional types.…”

Harish-Chandra bimodules over rational Cherednik algebras

“…Let us recall how [Wi,Theorem 1.8] is proved, as this will be important for our arguments. So let i and p be as in the previous paragraph.…”

“…So we can think of H reg−W ′ as H(W, X ), the rational Cherednik algebra associated to the action of W on the variety X , see e.g. [Wi,Section 2] for generalities on rational Cherednik algebras associated to the action of a complex reflection group on a smooth algebraic variety (not necessarily a vector space), in this paper we will only work with varieties that are disjoint unions of Zariski open sets inside a vector space. Note that X × h * = T * X is a symplectic algebraic variety, and the action of W on T * X is by symplectomorphisms.…”

“…and let X i be the union of the S n translates of X ′ i , so that [BE,Example 3.25], [Wi,Theorem 3.9], any module in category O c is supported on one of the X i . Denote by O i c the full subcategory of O c consisting of all modules whose support is contained in X i .…”

“…Bezrukavnikov-Etingof isomorphisms. We will need some isomorphisms ofétale lifts, that are essentially due to Bezrukavnikov and Etingof,[BE,Theorem 3.2], see also [L3, Lemma 2.1], [Wi,Proposition 2.6].…”