Representations of rational Cherednik algebras

This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C^2. We insist on the analogy with the representation theory of complex semi-simple Lie algebras

Dimensions of triangulated categories

We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a scheme and we show in particularthat the bounded derived category of coherent sheaves over avariety has a finite dimension. For a self-injective algebra, a lowerbound for Auslander's representation dimension is given by the dimensionof the stable category. We use this to compute the representationdimension of exterior algebras. This provides the first known examplesof representation dimension >3. We deduce that theLoewy length of the group algebra over F_2 of a finite group is strictly bounded below by2-rank of the group (a conjecture of Benson).Comment: Several new section

q-Schur algebras and complex reflection groups, I

We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a q-Schur algebra (parameter not a half integer), providing thus character formulas for simple modules. We give some generalization to B_n(d). We prove an ``abstract'' translation principle. These results follow from the unicity of certain highest categories covering Hecke algebras. We also provide a semi-simplicity criterion for Hecke algebras of complex reflection groups.Comment: Corrected version, 5.2.3 and 5.2.4 are ne

Derived equivalences for symmetric groups and sl2- categorification

We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Brou´e’s abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also construct categorifications for category O of gln(C) and for rational representations of general linear groups over ¯Fp, where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard