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Representation Theory

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Cherednik algebras and Hilbert schemes in characteristic $ p$

Authors: Roman Bezrukavnikov, Michael Finkelberg and Victor Ginzburg; with an Appendix by Pavel Etingof
Journal: Represent. Theory 10 (2006), 254-298
Published electronically: April 17, 2006
MathSciNet review: 2219114
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Abstract: We prove a localization theorem for the type $ \mathbf{A}_{n-1}$ rational Cherednik algebra $ \mathsf{H}_c=\mathsf{H}_{1,c}(\mathbf{A}_{n-1})$ over $ \overline{\mathbb{F}}_p$, an algebraic closure of the finite field. In the most interesting special case where $ c\in \mathbb{F}_p$, we construct an Azumaya algebra $ \mathscr{H}_c$ on $ \operatorname{Hilb}^n{\mathbb{A}}^2$, the Hilbert scheme of $ n$ points in the plane, such that $ \Gamma(\operatorname{Hilb}^n{\mathbb{A}}^2, \,\mathscr{H}_c)=\mathsf{H}_c$. Our localization theorem provides an equivalence between the bounded derived categories of $ \mathsf{H}_c$-modules and sheaves of coherent $ \mathscr{H}_c$-modules on $ \operatorname{Hilb}^n{\mathbb{A}}^2$, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.

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Additional Information

Roman Bezrukavnikov
Affiliation: Department of Mathematics, M.I.T., 77 Massachusetts Ave, Cambridge, Massachusetts 02139

Michael Finkelberg
Affiliation: Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002 Moscow, Russia

Victor Ginzburg
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Pavel Etingof
Affiliation: Department of Mathematics, M.I.T., 77 Massachusetts Ave, Cambridge, Massachusetts 02139

Received by editor(s): May 4, 2005
Received by editor(s) in revised form: February 19, 2006
Published electronically: April 17, 2006
Dedicated: To David Kazhdan with admiration
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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