Skip to main content
SpringerLink
Log in

Procesi Bundles and Symplectic Reflection Algebras

  • Conference paper
  • First Online:
Algebraic and Analytic Microlocal Analysis (AAMA 2013)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 269))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    After this survey was written, I have proved that \(e\mathbf{H}e\) is a universal graded deformation of \(\mathbb {C}[V_n]^{\Gamma _n}\) compatible with the Poisson bracket in a suitable sense, which can be used to prove the isomorphism theorem without appealing to Procesi bundles, see [42, Section 3] for details.

  2. 2.

    After this survey was written, I have established a shift equivalence for general symplectic reflection groups, [44]. The proof follows the scheme outlined in this section: Procesi bundles on symplectic resolutions are replaced with their generalizations, Procesi sheaves on \(\mathbb {Q}\)-factorial terminalizations.

  3. 3.

    The case of general complex reflection groups was done in [43] after this survey was written using different techniques.

References

  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. Bayen, F., Flato, M., Fronsdal, C., Lichnerowitz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111, 61–110 (1978)

    Google Scholar 

  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. 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. Bellamy, G.: On singular Calogero-Moser spaces. Bull. Lond. Math. Soc. 41(2), 315–326 (2009)

    Article  MathSciNet  Google Scholar 

  6. Bezrukavnikov, R., Etingof, P.: Parabolic induction and restriction functors for rational Cherednik algebras. Selecta Math. 14, 397–425 (2009)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  8. Bezrukavnikov, R., Finkelberg, M.: Wreath Macdonald polynomials and categorical McKay correspondence. Camb. J. Math. 2(2), 163–190 (2014)

    Google Scholar 

  9. Bezrukavnikov, R., Kaledin, D.: Fedosov quantization in positive characteristic. J. Am. Math. Soc. 21(2), 409–438 (2008)

    Article  MathSciNet  Google Scholar 

  10. Bezrukavnikov, R., Kaledin, D.: Fedosov quantization in the algebraic context. Moscow Math. J. 4, 559–592 (2004)

    Google Scholar 

  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. Bezrukavnikov, R., Losev, I.: Etingof conjecture for quantized quiver varieties. arXiv:1309.1716

  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. Boyarchenko, M.: Quantization of minimal resolutions of Kleinian singularities. Adv. Math. 211(1), 244–265 (2007)

    Article  MathSciNet  Google Scholar 

  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. Braden, T., Proudfoot, N., Webster, B.: Quantizations of conical symplectic resolutions I: local and global structure. Astérisque 384, 1–73 (2016)

    Google Scholar 

  17. Crawley-Boevey, W., Holland, M.: Noncommutative deformations of Kleinian singularities. Duke Math. J. 92, 605–635 (1998)

    Article  MathSciNet  Google Scholar 

  18. Crawley-Boevey, W.: Geometry of the moment map for representations of quivers. Comp. Math. 126, 257–293 (2001)

    Google Scholar 

  19. Crawley-Boevey, W.: Normality of Marsden-Weinstein reductions for representations of quivers. Math. Ann. 325(1), 55–79 (2003)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  21. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147(N2), 243–348 (2002)

    Article  MathSciNet  Google Scholar 

  22. Etingof, P.: Symplectic reflection algebras and affine Lie algebras. Mosc. Math. J. 12, 543–565 (2012)

    Google Scholar 

  23. Gan, W.L., Ginzburg, V.: Quantization of Slodowy slices. IMRN 5, 243–255 (2002)

    Google Scholar 

  24. Ginzburg, V., Guay, N., Opdam, E., Rouquier, R.: On the category \(\cal{O}\) for rational Cherednik algebras. Invent. Math. 154, 617–651 (2003)

    Article  MathSciNet  Google Scholar 

  25. Gordon, I., Losev, I.: On category \(\cal{O}\) for cyclotomic rational Cherednik algebras. J. Eur. Math. Soc. 16, 1017–1079 (2014)

    Article  MathSciNet  Google Scholar 

  26. Gordon, I., Stafford, T.: Rational Cherednik algebras and Hilbert schemes. Adv. Math. 198(1), 222–274 (2005)

    Article  MathSciNet  Google Scholar 

  27. Gordon, I.: A remark on rational Cherednik algebras and differential operators on the cyclic quiver. Glasg. Math. J. 48, 145–160 (2006)

    Article  MathSciNet  Google Scholar 

  28. Gordon, I.: Baby Verma modules for rational Cherednik algebras. Bull. Lond. Math. Soc. 35(3), 321–336 (2003)

    Article  MathSciNet  Google Scholar 

  29. Gordon, I.: Macdonald positivity via the Harish-Chandra D-module. Invent. Math. 187(3), 637–643 (2012)

    Article  MathSciNet  Google Scholar 

  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. Griffeth, S.: Orthogonal functions generalizing Jack polynomials. Trans. Am. Math. Soc. 362, 6131–6157 (2010)

    Article  MathSciNet  Google Scholar 

  32. Haboush, W.: Reductive groups are geometrically reductive. Ann. of Math. (2) 102(1), 67–83 (1975)

    Article  MathSciNet  Google Scholar 

  33. Haiman, M.: Combinatorics, Symmetric Functions, and Hilbert Schemes. Current developments in mathematics, 2002, pp. 39–111. International Press, Somerville (2002)

    Google Scholar 

  34. Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001)

    Google Scholar 

  35. Holland, M.: Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers. Ann. Sci. Ec. Norm. Super. IV Ser. 32, 813–834 (1999)

    Article  MathSciNet  Google Scholar 

  36. Kaledin, D., Verbitsky, M.: Period map for non-compact holomorphically symplectic manifolds. Geom. Funct. Anal. 12(6), 1265–1295 (2002)

    Article  MathSciNet  Google Scholar 

  37. Kaledin, D.: Symplectic singularities from the Poisson point of view. J. Reine Angew. Math. 600, 135–156 (2006)

    Google Scholar 

  38. Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316, 565–576 (2000)

    Article  MathSciNet  Google Scholar 

  39. Kashiwara, M., Rouquier, R.: Microlocalization of rational Cherednik algebras. Duke Math. J. 144, 525–573 (2008)

    Article  MathSciNet  Google Scholar 

  40. Losev, I.: Abelian localization for cyclotomic Cherednik algebras. Int. Math. Res. Not. 2015, 8860–8873 (2015)

    Article  MathSciNet  Google Scholar 

  41. Losev, I.: Completions of symplectic reflection algebras. Selecta Math. 18(N1), 179–251 (2012)

    Article  MathSciNet  Google Scholar 

  42. Losev, I.: Deformations of symplectic singularities and Orbit method for semisimple Lie algebras. arXiv:1605.00592

  43. Losev, I.: Derived equivalences for Rational Cherednik algebras. Duke Math J. 166(N1), 27–73 (2017)

    Article  MathSciNet  Google Scholar 

  44. Losev, I.: Derived equivalences for Symplectic reflection algebras. arXiv:1704.05144

  45. Losev, I.: Isomorphisms of quantizations via quantization of resolutions. Adv. Math. 231, 1216–1270 (2012)

    Article  MathSciNet  Google Scholar 

  46. Losev, I.: On Procesi bundles. Math. Ann. 359(N3), 729–744 (2014)

    Article  MathSciNet  Google Scholar 

  47. Losev. I.: Finite dimensional representations of W-algebras. Duke Math J. 159(1), 99–143 (2011)

    Google Scholar 

  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. McGerty, K., Nevins, T.: Derived equivalence for quantum symplectic resolutions. Selecta Math. 20, 675–717 (2014)

    Article  MathSciNet  Google Scholar 

  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. Nakajima, H.: Instantons on ALE spaces, quiver varieties and Kac-Moody algebras. Duke Math. J. 76, 365–416 (1994)

    Article  MathSciNet  Google Scholar 

  52. Nakajima, H.: Jack polynomials and Hilbert schemes of points on surfaces. arXiv:alg-geom/9610021

  53. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. University lecture series 18. AMS (1999)

    Google Scholar 

  54. Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. J. 91(3), 515–560 (1998)

    Article  MathSciNet  Google Scholar 

  55. Namikawa, Y.: Poisson deformations of affinne symplectic varieties, II. Kyoto J. Math. 50(4), 727–752 (2010)

    Article  MathSciNet  Google Scholar 

  56. Oblomkov, A.: Deformed Harish-Chandra homomorphism for the cyclic quiver. Math. Res. Lett. 14, 359–372 (2007)

    Article  MathSciNet  Google Scholar 

  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. Rouquier, R.: \(q\)- Schur algebras for complex reflection groups. Mosc. Math. J. 8, 119–158 (2008)

    Google Scholar 

  59. Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ. 14, 1–28 (1976)

    Article  MathSciNet  Google Scholar 

  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. Vologodsky, V.: Appendix to [BF]

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Ivan Losev

Authors
  1. Ivan Losev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Losev .

Editor information

Editors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, USA

    Michael Hitrik

  2. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Dmitry Tamarkin

  3. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Boris Tsygan

  4. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Steve Zelditch

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Losev, I. (2018). Procesi Bundles and Symplectic Reflection Algebras. In: Hitrik, M., Tamarkin, D., Tsygan, B., Zelditch, S. (eds) Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proceedings in Mathematics & Statistics, vol 269. Springer, Cham. https://doi.org/10.1007/978-3-030-01588-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation