Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

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
DOI: https://doi.org/10.1090/S1088-4165-06-00309-8
Published electronically: April 17, 2006
MathSciNet review: 2219114
Full-text PDF

Abstract | References | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AFLS] J. Alev, M.A. Farinati, T. Lambre, and A.L. Solotar, Homologie des invariants d'une algèbre de Weyl sous l'action d'un groupe fini. J. of Algebra 232 (2000), 564-577. MR 1792746 (2002c:16047)
  • [BB] A. Beilinson, J. Bernstein, A proof of Jantzen conjectures. I. M. Gelfand Seminar, 1-50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993. MR 1237825 (95a:22022)
  • [BGS] A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527. MR 1322847 (96k:17010)
  • [BEG] Yu. Berest, P. Etingof, V. Ginzburg, Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. 118 (2003), 279-337. MR 1980996 (2004f:16039)
  • [BEG2] Yu. Berest, P. Etingof, V. Ginzburg, Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 2003, no. 19, 1053-1088. MR 1961261 (2004h:16027)
  • [BK] R. Bezrukavnikov, D. Kaledin, MacKay equivalence for symplectic quotient resolutions of singularities, Proc. of the Steklov Inst. of Math. 246 (2004), 13-33. MR 2101282
  • [BMR] R. Bezrukavnikov, I. Mirkovic, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. Math. (2006). [arXiv:math.RT/0205144].
  • [BKR] T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc. 14 (2001), 535-554. MR 1824990 (2002f:14023)
  • [Br] J.-L. Brylinski, A differential complex for Poisson manifolds. J. Diff. Geom. 28 (1988), 93-114. MR 0950556 (89m:58006)
  • [CG] N. Chriss, V. Ginzburg, Representation theory and complex geometry. Birkhäuser Boston, 1997. MR 1433132 (98i:22021)
  • [Ch] I. Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald operators , IMRN (Duke Math. J.) 9 (1992), 171-180. MR 1185831 (94b:17040)
  • [DO] C. Dunkl, E. Opdam, Dunkl operators for complex reflection groups. Proc. London Math. Soc. 86 (2003), 70-108. MR 1971464 (2004d:20040)
  • [EG] P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147 (2002), 243-348. MR 1881922 (2003b:16021)
  • [FG] M. Finkelberg, V. Ginzburg, Character sheaves for Cherednik algebras. (in preparation).
  • [Ga] O. Gabber, Some theorems on Azumaya algebras. The Brauer group (Sem., Les Plans-sur-Bex, 1980), pp. 129-209, Lecture Notes in Math., 844, Springer, Berlin-New York, 1981. MR 0611868 (83d:13004)
  • [GIT] D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 34 Springer-Verlag, Berlin, 1994. MR 1304906 (95m:14012)
  • [GG] W. L. Gan, V. Ginzburg, Quantization of Slodowy slices. Int. Math. Res. Not. 2002, no. 5, 243-255. MR 1876934 (2002m:53129)
  • [GG2] W. L. Gan, V. Ginzburg, Almost-commuting variety, D-modules, and Cherednik Algebras, Int. Math. Res. Publ. (2006). arXiv:math.RT/0409262.
  • [GS] I. Gordon, T. Stafford, Rational Cherednik algebras and Hilbert schemes of points. Adv. in Math. (2006). [arXiv:math.RA/0407516].
  • [Gr] A. Grothendieck, SGA 1, Lecture Notes in Mathematics, 224 (1971). MR 0354651 (50:7129)
  • [H] M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371-407. MR 1918676 (2003f:14006)
  • [Har] R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics, 52. Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57:3116)
  • [Ha] T. Hayashi, Sugawara operators and Kac-Kazhdan conjecture. Invent. Math. 94 (1988), 13-52. MR 0958588 (90c:17035)
  • [J] N. Jacobson, Lie algebras. Interscience Tracts in Pure and Applied Mathematics, 10 Interscience Publishers (a division of John Wiley & Sons), New York-London 1962. MR 0143793 (26:1345)
  • [Ja] J.C. Jantzen, Representations of Lie algebras in prime characteristic. Notes by Iain Gordon. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Representation theories and algebraic geometry (Montreal, PQ, 1997), 185-235, Kluwer Acad. Publ., Dordrecht, 1998. MR 1649627 (99h:17026)
  • [K] N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin Publ. Math.IHES 39(1970). MR 0291177 (45:271)
  • [KKS] D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31(1978), 481-507. MR 478225 (57:7711)
  • [KT] S. Kumar, J. F. Thomsen, Frobenius splitting of Hilbert schemes of points on surfaces. Math. Ann. 319 (2001), 797-808. MR 1825408 (2002d:14004)
  • [La] F. Latour, Representations of rational Cherednik algebras of rank 1 in positive characteristic, J. Pure Appl. Algebra 195 (2005), 97-112. MR 2100312
  • [MVK] V. B. Mehta, W. van der Kallen, On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic $ p$. Invent. Math. 108 (1992), 11-13. MR 1156382 (93a:14017)
  • [Na1] H. Nakajima, Lectures on Hilbert schemes of points on surfaces. University Lecture Series, 18, American Mathematical Society, Providence, RI, 1999. MR 1711344 (2001b:14007)
  • [Na2] H. Nakajima, Quiver varieties and Kac-Moody algebras. Duke Math. J. 91 (1998), 515-560. MR 1604167 (99b:17033)
  • [Ob] A. Oblomkov, Double affine Hecke algebras and Calogero-Moser spaces. Represent. Theory 8 (2004), 243-266. MR 2077482 (2005e:20005)
  • [Pr] A. Premet, Special transverse slices and their enveloping algebras. Adv. Math. 170 (2002), 1-55. MR 1929302 (2003k:17014)
  • [PS] A. Premet, S. Skryabin, Representations of restricted Lie algebras and families of associative $ \mathscr{L}$-algebras. J. Reine Angew. Math. 507 (1999), 189-218. MR 1670211 (99m:17026)
  • [Q] D. Quillen, On the endormorphism ring of a simple module over an enveloping algebra. Proc. Amer. Math. Soc. 21 1969 171-172. MR 0238892 (39:252)
  • [Wi] G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian, Inv. Math. 133 (1998), 1-41. MR 1626461 (99f:58107)


Additional Information

Roman Bezrukavnikov
Affiliation: Department of Mathematics, M.I.T., 77 Massachusetts Ave, Cambridge, Massachusetts 02139
Email: bezrukav@math.mit.edu

Michael Finkelberg
Affiliation: Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002 Moscow, Russia
Email: fnklberg@mccme.ru

Victor Ginzburg
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: ginzburg@math.uchicago.edu

Pavel Etingof
Affiliation: Department of Mathematics, M.I.T., 77 Massachusetts Ave, Cambridge, Massachusetts 02139
Email: etingof@math.mit.edu

DOI: https://doi.org/10.1090/S1088-4165-06-00309-8
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.

American Mathematical Society