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15 March 2006 Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves
I. Gordon, J. T. Stafford
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Duke Math. J. 132(1): 73-135 (15 March 2006). DOI: 10.1215/S0012-7094-06-13213-1


Let Hc be the rational Cherednik algebra of type An-1 with spherical subalgebra Uc=eHce. Then Uc is filtered by order of differential operators with associated graded ring grUc=C[hh*]W, where W is the nth symmetric group. Using the Z-algebra construction from [GS], it is also possible to associate to a filtered Hc- or Uc-module M a coherent sheaf Φ(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and we relate it to Hilb(n) and to the resolution of singularities τ:Hilb(n)hh*/W. For example, we prove the following.

• If c=1/n so that Lc(triv) is the unique one-dimensional simple Hc-module, then Φ(eLc(triv))OZn, where Zn=τ-1(0) is the punctual Hilbert scheme.

• If c=1/n+k for kN, then under a canonical filtration on the finite-dimensional module Lc(triv), greLc(triv) has a natural bigraded structure that coincides with that on H0(Zn,Lk), where LOHilb(n)(1); this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].

• Under mild restrictions on c, the characteristic cycle of Φ(eΔc(μ)) equals λKμλ[Zλ], where Kμλ are Kostka numbers and the Zλ are (known) irreducible components of τ-1(h/W)


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I. Gordon. J. T. Stafford. "Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves." Duke Math. J. 132 (1) 73 - 135, 15 March 2006.


Published: 15 March 2006
First available in Project Euclid: 28 February 2006

zbMATH: 1096.14003
MathSciNet: MR2219255
Digital Object Identifier: 10.1215/S0012-7094-06-13213-1

Primary: 14C05, 16D90, 32S45
Secondary: 05E10, 16S80

Rights: Copyright © 2006 Duke University Press


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Vol.132 • No. 1 • 15 March 2006
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