Skip to main content
SpringerLink
Log in

Character varieties with Zariski closures of \(\mathrm{GL}_n\)-conjugacy classes at punctures

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

In the paper [20], we gave a conjectural formula for the mixed Hodge polynomials of character varieties with generic semisimple conjugacy classes at punctures and we prove a formula for the \(E\)-polynomial. We also proved that these character varieties are irreducible [21]. In this paper, we extend the results of [20, 21] to character varieties with Zariski closures of arbitrary generic conjugacy classes at punctures working with intersection cohomology. We also study Weyl group action on the intersection cohomology of the partial resolutions of these character varieties and give a conjectural formula for the two-variable polynomials that encode the trace of the elements of the Weyl group on the subquotients of the weight filtration. Finally, we compute the generating function of the stack count of character varieties with Zariski closure of unipotent regular conjugacy class at punctures.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fu, B.: Symplectic resolutions for nilpotent orbits. Invent. Math. 151, 167–186 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bardsley, P., Richardson, R.W.: Etale slices for algebraic transformation groups in characteristic \(p\). Proc. Lond. Math. Soc. 51, 295–317 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boden, H., Yokogawa, K.: Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves. I. Int. J. Math. 7(5), 573–598 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bohro, W. and MacPherson, R.: Partial resolutions of nilpotent varieties. In: Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Astérisque, vol. 101, pp. 23–74. (1983)

  5. Chaudouard, P.-H., Laumon, G.: Sur le comptage des fibrés de Hitchin. arXiv:1307.7273

  6. Crawley-Boevey, W.: On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero. Duke Math. J. 118(2), 339–352 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Crawley-Boevey, W.: Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity. Publ. Math. Inst. Hautes études Sci. 100, 171–207 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Crawley-Boevey, W., Van den Bergh, M.: Absolutely indecomposable representations and Kac-Moody Lie algebras. With an appendix by Hiraku Nakajima. Invent. Math. 155(3), 537–559 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. de Cataldo, M., Hausel, T., Migliorini, L.: Topology of Hitchin systems and Hodge theory of character varieties: the case \(A_1\). Ann. Math. 175, 1329–1407 (2012)

    Article  MATH  Google Scholar 

  10. Deligne, P.: Théorie de Hodge II. Inst. Hautes Etudes Sci. Publ. Math. 40, 5–47 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  11. Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. 103(2), 103–161 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  12. Etingof, P., Oblomkov, A., Rains, E.: Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces. Adv. Math. 212, 749–796 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Foth, P.: Geometry of moduli spaces of flat bundles on punctured surfaces. Int. J. Math. 9, 63–73 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. García-Prada, O. Gothen, P.B., Munõz, V.: Betti numbers of the moduli space of rank 3 parabolic Higgs bundles. Mem. Am. Math. Soc. 187(879), viii+80 pp (2007).

  15. García-Prada, O. Heinloth, J., Schmitt, A.: On the motives of moduli of chains and Higgs bundles. arXiv:1104.5558 (to appear in J. Eur. Math. Soc.)

  16. Garsia, A.M., Haiman, M.: A remarkable q, t-Catalan sequence and q-Lagrange inversion. J. Algebraic Comb. 5(3), 191–244 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gothen, P.B.: The Betti numbers of the moduli space of rank 3 Higgs bundles. Int. J. Math. 5, 861–875 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Göttsche, L., Soergel, W.: Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann. 296, 235–245 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hausel, T.: Kac conjecture from Nakajima’s quiver varieties. Invent. Math. 181, 21–37 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. 160(2), 323–400 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties II. Adv. Math. 234, 85–128 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Positivity for Kac polynomials and DT-invariants of quivers. Ann. Math. 177, 1147–1168 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hausel, T., Thaddeus, M.: Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles. Proc. Lond. Math. Soc. 88, 632–658 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties. Invent. Math. 174(3), 555–624 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hitchin, N.: The self-duality equations on a Riemann surfac. Proc. Lond. Math. Soc. 55(3), 59–126 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kac, V.: Root systems, representations of quivers and invariant theory. Lecture Notes in Mathematics, vol. 996, pp. 74–108. Springer, Berlin (1982)

  27. Kac, V.: Infinite Dimensional Algebra, 3rd edn. Cambridge University Press, Cambridge (2003)

  28. Kostov, V.P.: On the Deligne-Simpson problem. C. R. Acad. Sci. Paris Sér. I Math. 329, 657–662 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  29. Letellier, E.: Fourier transforms of invariant functions on finite reductive Lie Algebras. Lecture Notes in Mathematics, vol. 1859. Springer, Berlin (2005)

  30. Letellier, E.: Quiver varieties and the character ring of general linear groups over finite fields. J. Eur. Math. Soc. 15, 1375–1455 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Letellier, E.: Tensor products of unipotent characters of general linear groups over finite fields. Transform. Groups 18, 233–262 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  32. Logares, M., Munõz, V., Newstead, P.: Hodge polynomials of SL(2, C)-character varieties of surfaces of low genus. arXiv:1106.6011

  33. Lusztig, G.: On the finiteness of the number of unipotent classes. Invent. Math. 34, 201–213 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. Math. 42, 169–178 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lusztig, G.: Fourier transforms on a semisimple Lie algebra over \({\mathbb{F}}_q\), Algebraic groups Utrecht 1986, pp. 177–188. Springer, Berlin (1987)

  36. Lusztig, G., Srinivasan, B.: The characters of the finite unitary groups. J. Algebra 49, 167–171 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  37. Macdonald, I.G: Symmetric functions and hall polynomials, Oxford Mathematical Monographs, 2nd edn. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1995)

  38. Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, xiv+292 Springer, Berlin (1994)

  39. Peters, C., Steenbrink, J.: Mixed Hodge structures. vol. 52, xiv+470 pp Springer, Berlin (2008)

  40. Saito, M.: Mixed hodge modules. Publ. Res. Inst. Math. Sci. 26, 221–333 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  41. Simpson, C.T.: Nonabelian Hodge theory. In: Proceedings of the International Congress of Mathematicians, vols. I, II (Kyoto, 1990), pp. 747–756, Mathematical Society of Japan, Tokyo (1991)

Download references

Acknowledgments

This work is supported by the Grant ANR-09-JCJC-0102-01.

Author information

Authors and Affiliations

  1. Université de Caen, Caen, France

    Emmanuel Letellier

Authors
  1. Emmanuel Letellier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emmanuel Letellier.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Letellier, E. Character varieties with Zariski closures of \(\mathrm{GL}_n\)-conjugacy classes at punctures . Sel. Math. New Ser. 21, 293–344 (2015). https://doi.org/10.1007/s00029-014-0163-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-014-0163-9

Keywords

Mathematics Subject Classification

Search

Navigation