Duke Mathematical Journal

Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves

I. Gordon and J. T. Stafford

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Abstract

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)

Article information

Source
Duke Math. J., Volume 132, Number 1 (2006), 73-135.

Dates
First available in Project Euclid: 28 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1141136437

Digital Object Identifier
doi:10.1215/S0012-7094-06-13213-1

Mathematical Reviews number (MathSciNet)
MR2219255

Zentralblatt MATH identifier
1096.14003

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes) 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 32S45: Modifications; resolution of singularities [See also 14E15]
Secondary: 16S80: Deformations of rings [See also 13D10, 14D15] 05E10: Combinatorial aspects of representation theory [See also 20C30]

Citation

Gordon, I.; Stafford, J. T. Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves. Duke Math. J. 132 (2006), no. 1, 73--135. doi:10.1215/S0012-7094-06-13213-1. https://projecteuclid.org/euclid.dmj/1141136437


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