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Zbl 1213.20007
Rouquier, Raphaël
$q$-Schur algebras and complex reflection groups. (English)
[J] Mosc. Math. J. 8, No. 1, 119-158 (2008). ISSN 1609-3321

The paper under review develops a new aspect of the representation theory of Hecke algebras of complex reflection groups, namely, the study of quasi-hereditary covers (analogous to $q$-Schur algebras in the symmetric group case). The main idea of the paper is uniqueness of certain types of quasi-hereditary covers. Using this the author shows that in type $A$ and when the parameter is not in $\tfrac 12+\bbfZ$, the category $\mathcal O$ for a rational Cherednik algebra is equivalent to the module category of a $q$-Schur algebra, confirming the earlier conjecture of Ginzburg, Guay, Opdam and Rouquier. As a consequence, the author obtains character formulae for simple objects in this category $\mathcal O$ and a general translation principle. Along the way the author develops a general theory of split highest weight categories over a commutative ring (a categorical version of Cline-Parshall-Scott's integral quasi-hereditary algebras).
[Volodymyr Mazorchuk (Uppsala)]

MSC 2000:: *20C08 Hecke algebras and their representations
20G05 Representation theory of linear algebraic groups
17B20 Simple and semisimple Lie algebras
20F55 Coxeter groups
20G43
20C30 Representations of finite symmetric groups

Keywords: category $\cal O$; $q$-Schur algebras; complex reflection groups; rational Cherednik algebras; characters; translations; Hecke algebras; highest weight categories; quasi-hereditary covers

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