Raphaël Rouquier

On the category O for rational Cherednik algebras

Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

PICARD GROUPS FOR DERIVED MODULE CATEGORIES

We introduce in this paper a generalization of Picard groups to derived cat- egories of algebras. We study general properties of those groups and we construct braid group actions on these groups for particular classes of algebras.

Centers and simple modules for Iwahori-Hecke algebras

The work of Dipper and James on Iwahori-Hecke algebras associated with the finite Weyl groups of type A n has shown that these algebras behave in many ways like group algebras of finite groups. Moreover, there are “generic” features in the modular representation theory of these algebras which, at present, can only be verified in examples by explicit computations. This paper arose from an attempt to provide a conceptual explanation of these phenomena, in the general framework of the representation theory of (symmetric) algebras. We will study relations between the center of such algebras and properties of decomposition maps, and we will use this to obtain a general result about the “genericity” of the number of simple modules of Iwahori-Hecke algebras.

Quiver Hecke Algebras and 2-Lie Algebras

We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

Microlocalization of rational Cherednik algebras

We construct a microlocalization of the rational Cherednik algebras H of type Sn. This is achieved by a quantization of the Hilbert scheme Hilb C2 of n points in C2. We then prove the equivalence of the category of H -modules and that of modules over its microlocalization under certain conditions on the parameter.

On the category for rational Cherednik algebras

Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

Automorphismes, graduations et catégories triangulées

We give a moduli interpretation of the outer automorphism group Out of a finite dimensional algebra similar to that of the Picard group of a scheme. We deduce that Out^0 is invariant under derived and stable equivalences. This allows us to transfer gradings between algebras and gives rise to conjectural homological constructions of interesting gradings on block of finite groups with abelian defect. We give applications to the lifting of stable equivalences to derived equivalences. We give a counterpart of the invariance result for smooth projective varieties: the product Pic^0xAut^0 is invariant under derived equivalence.

Familles de caractères de groupes de réflexions complexes

Nous etudions certains types de blocs d’algebres de Hecke associees aux groupes de reflexions complexes qui generalisent les familles de caracteres definies par Lusztig pour les groupes de Weyl. Nous determinons ces blocs pour les groupes de reflexions spetsiaux et nous etablissons un theoreme de compatibilite entre familles et d-series de Harish-Chandra.

Complexes de chaı̂nes étales et courbes de Deligne–Lusztig

Abstract In a first part, we lift the usual constructions of functors between derived categories of etale sheaves over schemes with a sheaf of algebras to pure derived categories. For varieties with finite group actions, we recover, in a more functorial way, Rickards construction. We apply this to the case of Deligne–Lusztig varieties and show that Broues conjecture holds for curves. The additional ingredient we need is obtained from an easy property of the cohomology of etale covers of the affine line.

The Dynkin diagram cohomology of finite Coxeter groups

Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are carried out by filtering the Dynkin complex by the number of vertices of subgraphs of D. The corresponding graded complex turns out to be dual to the sum of the Coxeter complexes of all standard, irreducible parabolic subgroups of W.