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From double affine Hecke algebras to quantized affine Schur algebras. (English) Zbl 1073.22011

Let \(F\) be a local non-Archimedean field of residual characteristic \(p\) and \(q\) the order of the residual field. Let \(k\) be an algebraically closed field of characteristic \(l\) (\(l = 0\) or \(> 0\) and \(\neq p\)). Let \(G\) be a reductive group and \({\mathcal{\underline{H}}}\) be the affine Hecke algebra of \(G\) over \(k\). Then \({\mathcal{\underline{H}}} = {\mathbf{\underline{H}}}/(\zeta - q)\) where \({\mathbf{\underline{H}}}\) is the corresponding generic affine Hecke algebra over k\([\zeta^{\pm 1}]\). Let \({\mathbf H}\) be a double affine Hecke algebra and \({\mathbf{\underline{Sc}}}\) be the generic quantized affine Schur algebra of \(G = GL(n)\). It is proved that under some general conditions, the blocks of the category \(\mathcal O\) in \({\mathbf H}/(\zeta-\zeta_o, \tau-\tau_o)\)-mod are equivalent to the blocks in \({\mathbf{\underline{Sc}}}/(\zeta - e^{h_o})\)-mod where \(\tau_o = e^{u_o}\) and \(\zeta_o = e^{u_o h_o}\).

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
20C08 Hecke algebras and their representations
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