Differential operators and Cherednik algebras

• V. Ginzburg ^[1] ; I. Gordon ^[3] ; J.T. Stafford ^[2]
  1. [1] University of Chicago

     University of Chicago

     City of Chicago, Estados Unidos

     
  2. [2] University of Michigan–Ann Arbor

     University of Michigan–Ann Arbor

     City of Ann Arbor, Estados Unidos

     
  3. [3] School of Mathematics and Maxwell Institute for Mathematical Sciences, Edinburgh University, Reino Unido

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 14, Nº. 3-4, 2009, págs. 629-666
• Idioma: inglés
• DOI: 10.1007/s00029-009-0515-z
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• Resumen

  • 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].