We study projective objects in the category OHc(0) of the Cherednik algebra introduced recently by Berest, Etingof and Ginzburg. We prove that it has enough projectives and that it is a highest weight category in the sense of Cline, Parshall and Scott, and therefore satisfies an analog of the BGG-reciprocity formula for a semisimple Lie algebra.

In this paper we introduce a basic representation for the confluent Cherednik algebras HV, HIII, H7 III and H D8 III defined in arXiv:1307.6140. To prove faithfulness of this basic representation, we introduce the non-symmetric versions of the continuous dual qHahn, Al-Salam–Chihara, continuous big q-Hermite and continuous q-Hermite polynomials.

Let W be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that W is a wreath product of a symmetric group with a cyclic group of order l.

In the paper we calculate the images of the operators of multiplication by Laurent polynomials with respect to the Harish-Chandra transform and its non-symmetric generalization due to Opdam. It readily leads to a new simple proof of the Harish-Chandra inversion theorem in the zonal case (see [HC,He1]) and the corresponding theorem from [O1]. We assume that k > 0 and restrict ourselves to compac...