Differential operators and Cherednik algebras

V. Ginzburg, I. Gordon, J. T. Stafford

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Abstract

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A : one based on a noncommutative Proj construction [GS1]; the other involving quantum hamiltonian reduction of an algebra of differential operators [GG]. In this paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result [GS1, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem [Ha1]. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG].

Original languageEnglish
Pages (from-to)629-666
Number of pages38
JournalSelecta Mathematica (New Series)
Volume14
Issue number3-4
DOIs
Publication statusPublished - May 2009

Keywords

  • Cherednik algebra
  • Hilbert scheme
  • characteristic varieties
  • FINITE-DIMENSIONAL REPRESENTATIONS
  • HILBERT SCHEMES

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