Duke Mathematical Journal

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

I. Gordon and J. T. Stafford

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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

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

First available in Project Euclid: 28 February 2006

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Zentralblatt MATH identifier

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]


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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  • Y. Berest, P. Etingof, and V. Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279--337.
  • —, Finite-dimensional representations of rational Cherednik algebras, Internat. Math. Res. Notices 2003, no. 19, 1053--1088.
  • J.-E. BjöRk, Rings of Differential Operators, North-Holland Math. Library 21, North-Holland, Amsterdam, 1979.
  • —, ``Filtered noetherian rings'' in Noetherian Rings and Their Applications (Oberwolfach, West Germany, 1983), Math. Surveys Monogr. 24, Amer. Math. Soc., Providence, 1987, 59--97.
  • W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces, III: Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math. 80 (1985), 1--68.
  • T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), 535--554.
  • J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387--410.
  • N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston, 1997.
  • N. Conze, Algèbres d'opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France 102 (1974), 379--415.
  • C. DezéLéE, Représentations de dimension finie de l'algèbre de Cherednik rationnelle, Bull. Soc. Math. France 131 (2003), 465--482.
  • C. F. Dunkl, Singular polynomials for the symmetric groups, Internat. Math. Res. Notices 2004, no. 67, 3607--3635.
  • P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243--348.
  • O. Gabber, The integrability of the characteristic variety, Amer. J. Math. 103 (1981), 445--468.
  • W. L. Gan and V. Ginzburg, Almost commuting variety, $\mathcalD$-modules, and Cherednik algebras, to appear in Int. Math. Res. Not., preprint.
  • A. M. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244.
  • —, Some natural bigraded $S\sb n$-modules and $q,t$-Kostka coefficients, Electron. J. Combin. 3 (1996), R24.
  • V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier, On the category $\mathcalO$ for rational Cherednik algebras, Invent. Math. 154 (2003), 617--651.
  • I. Gordon, On the quotient by diagonal invariants, Invent. Math. 153 (2003), 503--518.
  • I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), 222--274.
  • I. Grojnowski, Instantons and affine algebras, I: The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275--291.
  • A. Grothendieck, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961).
  • N. Guay, Projective modules in the category $\OO$ for the Cherednik algebra, J. Pure Appl. Algebra 182 (2003), 209--221.
  • M. Haiman, $t,q$-Catalan numbers and the Hilbert scheme, Discrete Math. 198 (1998), 201--224.
  • —, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941--1006.
  • —, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371--407.
  • A. Joseph and J. T. Stafford, Modules of $\mathfrakk$-finite vectors over semisimple Lie algebras, Proc. London Math. Soc. (3) 49 (1984), 361--384.
  • G. R. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, rev. ed., Grad. Stud. Math. 22, Amer. Math. Soc., Providence, 2000.
  • B. Leclerc and J.-Y. Thibon, Canonical bases of $q$-deformed Fock spaces, Internat. Math. Res. Notices 1996, no. 9, 447--456.
  • T. Levasseur, ``Relèvements d'opérateurs différentiels sur les anneaux d'invariants'' in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math. 92, Birkhäuser, Boston, 1990.
  • J. C. Mcconnell and J. C. Robson, Noncommutative Noetherian Rings, rev. ed., Grad. Stud. Math. 30, Amer. Math. Soc., Providence, 2001.
  • I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, Oxford, 1995.
  • H. Matsumura, Commutative Algebra, 2nd ed., Math. Lecture Note Ser. 56, Benjamin/Cummings, Reading, Mass., 1980.
  • H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser. 18, Amer. Math. Soc., Providence, 1999.
  • R. Rouquier, $q$-Schur algebras and complex reflection groups, I, preprint.
  • M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267--297.
  • N. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc. 6 (1993), 779--816.
  • K. Watanabe, Certain invariant subrings are Gorenstein, I, Osaka J. Math. 11 (1974), 1--8.