Advertisement

Procesi Bundles and Symplectic Reflection Algebras

  • Ivan LosevEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)

Abstract

In this survey we describe an interplay between Procesi bundles on symplectic resolutions of quotient singularities and Symplectic reflection algebras. Procesi bundles were constructed by Haiman and, in a greater generality, by Bezrukavnikov and Kaledin. Symplectic reflection algebras are deformations of skew-group algebras defined in complete generality by Etingof and Ginzburg. We construct and classify Procesi bundles, prove an isomorphism between spherical Symplectic reflection algebras, give a proof of wreath Macdonald positivity and of localization theorems for cyclotomic Rational Cherednik algebras.

MSC 2010

14E16 53D20 53D55 (Primary)05E05 16G20 16G99 16S36 17B63 20F55 (Secondary) 

Notes

Acknowledgements

This survey is a greatly expanded version of lectures I gave at Northwestern in May 2012. I would like to thank Roman Bezrukavnikov and Iain Gordon for numerous stimulating discussions. My work was supported by the NSF under Grant DMS-1161584.

References

  1. 1.
    Bardsley, P., Richardson, R.W.: Étale slices for algebraic transformation groups in characteristic \(p\). Proc. Lond. Math. Soc. 51(3), 295–317 (1985)Google Scholar
  2. 2.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowitz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111, 61–110 (1978)Google Scholar
  3. 3.
    Beilinson, A., Bernstein, J.: Localisation de \(\mathfrak{g}\)-modules. C. R. Acad. Sci. Paris Ser. I Math. 292(1), 15–18 (1981)Google Scholar
  4. 4.
    Bellamy, G., Schedler, T.: A new linear quotient of \(\mathbb{C}^4\) admitting a symplectic resolution. Math. Z. 273(3–4), 753–769 (2013)Google Scholar
  5. 5.
    Bellamy, G.: On singular Calogero-Moser spaces. Bull. Lond. Math. Soc. 41(2), 315–326 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bezrukavnikov, R., Etingof, P.: Parabolic induction and restriction functors for rational Cherednik algebras. Selecta Math. 14, 397–425 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bezrukavnikov, R., Finkelberg, M., Ginzburg, V.: Cherednik algebras and Hilbert schemes in characteristic \(p\). With an appendix by Pavel Etingof. Represent. Theory 10, 254–298 (2006)CrossRefGoogle Scholar
  8. 8.
    Bezrukavnikov, R., Finkelberg, M.: Wreath Macdonald polynomials and categorical McKay correspondence. Camb. J. Math. 2(2), 163–190 (2014)Google Scholar
  9. 9.
    Bezrukavnikov, R., Kaledin, D.: Fedosov quantization in positive characteristic. J. Am. Math. Soc. 21(2), 409–438 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bezrukavnikov, R., Kaledin, D.: Fedosov quantization in the algebraic context. Moscow Math. J. 4, 559–592 (2004)Google Scholar
  11. 11.
    Bezrukavnikov, R.V., Kaledin, D.B.: McKay equivalence for symplectic quotient singularities. Proc. Steklov Inst. Math. 246, 13–33 (2004). With erratum in [BF]Google Scholar
  12. 12.
    Bezrukavnikov, R., Losev, I.: Etingof conjecture for quantized quiver varieties. arXiv:1309.1716
  13. 13.
    Bondal, A., Orlov, D.: Derived categories of coherent sheaves. In: Proceedings of the International Congress of Mathematicians, Vol. II, pp. 47–56. Higher Ed. Press, Beijing (2002)Google Scholar
  14. 14.
    Boyarchenko, M.: Quantization of minimal resolutions of Kleinian singularities. Adv. Math. 211(1), 244–265 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Braden, T., Licata, A., Proudfoot, N., Webster, B.: Quantizations of conical symplectic resolutions II: category O and symplectic duality. Astérisque 384, 75–179 (2016)Google Scholar
  16. 16.
    Braden, T., Proudfoot, N., Webster, B.: Quantizations of conical symplectic resolutions I: local and global structure. Astérisque 384, 1–73 (2016)Google Scholar
  17. 17.
    Crawley-Boevey, W., Holland, M.: Noncommutative deformations of Kleinian singularities. Duke Math. J. 92, 605–635 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Crawley-Boevey, W.: Geometry of the moment map for representations of quivers. Comp. Math. 126, 257–293 (2001)Google Scholar
  19. 19.
    Crawley-Boevey, W.: Normality of Marsden-Weinstein reductions for representations of quivers. Math. Ann. 325(1), 55–79 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Etingof, P., Gan, W.L., Ginzburg, V., Oblomkov, A.: Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products. Publ. Math. IHES 105, 91–155 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147(N2), 243–348 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Etingof, P.: Symplectic reflection algebras and affine Lie algebras. Mosc. Math. J. 12, 543–565 (2012)Google Scholar
  23. 23.
    Gan, W.L., Ginzburg, V.: Quantization of Slodowy slices. IMRN 5, 243–255 (2002)Google Scholar
  24. 24.
    Ginzburg, V., Guay, N., Opdam, E., Rouquier, R.: On the category \(\cal{O}\) for rational Cherednik algebras. Invent. Math. 154, 617–651 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Gordon, I., Losev, I.: On category \(\cal{O}\) for cyclotomic rational Cherednik algebras. J. Eur. Math. Soc. 16, 1017–1079 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Gordon, I., Stafford, T.: Rational Cherednik algebras and Hilbert schemes. Adv. Math. 198(1), 222–274 (2005)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gordon, I.: A remark on rational Cherednik algebras and differential operators on the cyclic quiver. Glasg. Math. J. 48, 145–160 (2006)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Gordon, I.: Baby Verma modules for rational Cherednik algebras. Bull. Lond. Math. Soc. 35(3), 321–336 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Gordon, I.: Macdonald positivity via the Harish-Chandra D-module. Invent. Math. 187(3), 637–643 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Gordon, I.: Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras. Int. Math. Res. Pap. IMRP, Art. ID rpn006 (3), 69 (2008)Google Scholar
  31. 31.
    Griffeth, S.: Orthogonal functions generalizing Jack polynomials. Trans. Am. Math. Soc. 362, 6131–6157 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Haboush, W.: Reductive groups are geometrically reductive. Ann. of Math. (2) 102(1), 67–83 (1975)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Haiman, M.: Combinatorics, Symmetric Functions, and Hilbert Schemes. Current developments in mathematics, 2002, pp. 39–111. International Press, Somerville (2002)Google Scholar
  34. 34.
    Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001)Google Scholar
  35. 35.
    Holland, M.: Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers. Ann. Sci. Ec. Norm. Super. IV Ser. 32, 813–834 (1999)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kaledin, D., Verbitsky, M.: Period map for non-compact holomorphically symplectic manifolds. Geom. Funct. Anal. 12(6), 1265–1295 (2002)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Kaledin, D.: Symplectic singularities from the Poisson point of view. J. Reine Angew. Math. 600, 135–156 (2006)Google Scholar
  38. 38.
    Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316, 565–576 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kashiwara, M., Rouquier, R.: Microlocalization of rational Cherednik algebras. Duke Math. J. 144, 525–573 (2008)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Losev, I.: Abelian localization for cyclotomic Cherednik algebras. Int. Math. Res. Not. 2015, 8860–8873 (2015)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Losev, I.: Completions of symplectic reflection algebras. Selecta Math. 18(N1), 179–251 (2012)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Losev, I.: Deformations of symplectic singularities and Orbit method for semisimple Lie algebras. arXiv:1605.00592
  43. 43.
    Losev, I.: Derived equivalences for Rational Cherednik algebras. Duke Math J. 166(N1), 27–73 (2017)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Losev, I.: Derived equivalences for Symplectic reflection algebras. arXiv:1704.05144
  45. 45.
    Losev, I.: Isomorphisms of quantizations via quantization of resolutions. Adv. Math. 231, 1216–1270 (2012)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Losev, I.: On Procesi bundles. Math. Ann. 359(N3), 729–744 (2014)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Losev. I.: Finite dimensional representations of W-algebras. Duke Math J. 159(1), 99–143 (2011)Google Scholar
  48. 48.
    Maffei, A.: A remark on quiver varieties and Weyl groups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1, 649–689 (2002)Google Scholar
  49. 49.
    McGerty, K., Nevins, T.: Derived equivalence for quantum symplectic resolutions. Selecta Math. 20, 675–717 (2014)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 3rd edn, vol. 34. Springer, Berlin (1994)Google Scholar
  51. 51.
    Nakajima, H.: Instantons on ALE spaces, quiver varieties and Kac-Moody algebras. Duke Math. J. 76, 365–416 (1994)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Nakajima, H.: Jack polynomials and Hilbert schemes of points on surfaces. arXiv:alg-geom/9610021
  53. 53.
    Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. University lecture series 18. AMS (1999)Google Scholar
  54. 54.
    Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. J. 91(3), 515–560 (1998)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Namikawa, Y.: Poisson deformations of affinne symplectic varieties, II. Kyoto J. Math. 50(4), 727–752 (2010)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Oblomkov, A.: Deformed Harish-Chandra homomorphism for the cyclic quiver. Math. Res. Lett. 14, 359–372 (2007)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Popov, V.L., Vinberg, E.B.: Invariant theory. Itogi nauki i techniki. Sovr. Probl. Matem. Fund. Napr. 55, 137–309 (1989). Moscow, VINITI (in Russian). English translation in: Algebraic geometry 4, Encyclopaedia Math. Sci. 55. Springer, Berlin (1994)Google Scholar
  58. 58.
    Rouquier, R.: \(q\)- Schur algebras for complex reflection groups. Mosc. Math. J. 8, 119–158 (2008)Google Scholar
  59. 59.
    Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ. 14, 1–28 (1976)MathSciNetCrossRefGoogle Scholar
  60. 60.
    van den Bergh, M.: Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2002)Google Scholar
  61. 61.
    Vologodsky, V.: Appendix to [BF]Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations