Lowest-weight representations of Cherednik algebras in positive characteristic
نویسنده
چکیده
Lowest-weight representations of Cherednik algebras H~,c have been studied in both characteristic 0 and positive characteristic. However, the case of positive characteristic has been studied less, because of a lack of general tools. In positive characteristic the lowest-weight representation Lc(τ) of the Cherednik algebra is finite-dimensional. The representation theory of complex reflection groups becomes more complicated in positive characteristic, which makes the representation theory of the associated Cherednik algebras more interesting. The Cherednik algebra for the rank 1 group Z/l has been studied by Latour in [Latour]. Martina Balagović and Harrison Chen have studied the Cherednik algebras for GLn(Fq) and SLn(Fq) in [BC]. The nonmodular case for the symmetric group Sn (where the characteristic does not divide the order of the group) has been studied by Roman Bezrukavnikov, Michael Finkelberg, and Victor Ginzburg in the context of algebraic geometry in [BFG]. The nonmodular case has also been studied in [Gordon] for the symmetric group Sn (permutation matrices). Unlike previous work, this paper studies Cherednik algebras for complex reflection groups in characteristic 0 reduced modulo p (the characteristic), mostly in the modular case (where p divides the order of the group) which is the most difficult case. The groups studied here are also of higher rank than previous work. We study representations of Cherednik algebras of complex reflection groups G(m,m, n) and G(m, 1, n) where the parameter ~ for the Cherednik algebra is equal to 0, focusing on finding generators for the submodule Jc and then describing the quotient Mc/Jc = Lc that is a representation of the Cherednik algebra.
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تاریخ انتشار 2012