Abstract:
We show that the category $\mathbb O$ for a rational Cherednik algebra of type $A$ is equivalent to modules over a $q$-Schur algebra (parameter $\notin\frac12+\mathbb Z$), 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 weight categories covering Hecke algebras. We also provide a semi-simplicity criterion for Hecke algebras of complex reflection groups and show the isomorphism type of Hecke algebras is invariant under field automorphisms acting on parameters.
Key words and phrases:Hecke algebra, reflection group, Schur algebra, Cherednik algebra, highest weight category.
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