JournalsjemsVol. 16, No. 5pp. 1017–1079

On category O\mathcal{O} for cyclotomic rational Cherednik algebras

  • Iain G. Gordon

    University of Edinburgh, UK
  • Ivan Losev

    Northeastern University, Boston, USA
On category $\mathcal{O}$ for cyclotomic rational Cherednik algebras cover
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Abstract

We study equivalences for category Op\mathcal{O}_p of the rational Cherednik algebras Hp{\bf H}_p of type G(n)=(μ)nSnG_{\ell}(n) = (\mu_{\ell})^n\rtimes \mathfrak{S}_n: a highest weight equivalence between Op\mathcal{O}_p and Oσ(p)\mathcal{O}_{\sigma(p)} for σS\sigma\in \mathfrak{S}_{\ell} and an action of S\mathfrak{S}_{\ell} on an explicit non-empty Zariski open set of parameters pp; a derived equivalence between Op\mathcal{O}_p and Op\mathcal{O}_{p'} whenever pp and pp' have integral difference; a highest weight equivalence between Op\mathcal{O}_p and a parabolic category O\mathcal{O} for the general linear group, under a non-rationality assumption on the parameter pp. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.

Cite this article

Iain G. Gordon, Ivan Losev, On category O\mathcal{O} for cyclotomic rational Cherednik algebras. J. Eur. Math. Soc. 16 (2014), no. 5, pp. 1017–1079

DOI 10.4171/JEMS/454