Skip to main content
SpringerLink
Log in

Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We define the formal affine Demazure algebra and a formal affine Hecke algebra associated to a Kac-Moody root system. We prove structure theorems for these algebras, and use them to extend several results and constructions (presentation in terms of generators and relations, coproduct and product structures, filtration by codimension of Bott-Samelson classes, root polynomials and multiplication formulas) that were previously known for finite root systems.

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. Billey, S.: Kostant polynomials and the cohomology ring for G/B. Duke Math, J. 96(1), 205–224 (1999)

    Article  MathSciNet  Google Scholar 

  2. Calmès, B., Petrov, V., Zainoulline, K.: Invariants, torsion indices and oriented cohomology of complete flags. Annales scientifiques de l’École normale supérieure (4) 46(3), 405–448 (2013)

    Article  MathSciNet  Google Scholar 

  3. Calmès, B., Zainoulline, K., Zhong, C.: A coproduct structure on the formal affine Demazure algebra. Math. Z. 282(3), 1191–1218 (2016)

    Article  MathSciNet  Google Scholar 

  4. Calmès, B., Zainoulline, K., Zhong, C.: Push-pull operators on the formal affine Demazure algebra and its dual. Manuscripta Math (to appear)

  5. Calmès, B., Zainoulline, K., Zhong, C.: Equivariant oriented cohomology of flag varieties, Documenta Mathematica Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday, pp. 113–144 (2015)

  6. Deshpande, D: Algebraic Cobordism of Classifying Spaces. arXiv:0907.4437v1 (2009)

  7. Ginzburg, V., Kapranov, M., Vasserot, E.: Residue construction of Hecke algebras. Adv. Math. 128(1), 1–19 (1997)

    Article  MathSciNet  Google Scholar 

  8. Heller, J., Malagón-López, J.: Equivariant algebraic cobordism. J. Reine Angew. Math. 684, 87–112 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Hoffnung, A., Malagón-López, J., Savage, A., Zainoulline, K.: Formal Hecke algebras and algebraic oriented cohomology theories. Sel. Math. New Ser. 20(4), 1213–1245 (2014)

    Article  MathSciNet  Google Scholar 

  10. Kac, V.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990). xxii+ 400 pp

    Book  Google Scholar 

  11. Kostant, B., Kumar, S.: The nil Hecke ring and cohomology of G/P for a Kac-Moody group G∗. Adv. Math. 62, 187–237 (1986)

    Article  MathSciNet  Google Scholar 

  12. Kostant, B., Kumar, S.: T-equivariant K-theory of generalized flag varieties. J. Differential Geom. 32, 549–603 (1990)

    Article  MathSciNet  Google Scholar 

  13. Kumar, S.: Kac-Moody Groups, Their Flag Varieties and Representation Theory Progress in Mathematics, vol. 204. Birkhäuser, Boston (2002)

    Book  Google Scholar 

  14. Hornbostel, J., Kiritchenko, V.: Schubert calculus for algebraic cobordism. J. Reine Angew. Math. 656, 59–85 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Kiritchenko, V., Krishna, A.: Equivariant cobordism of flag varieties and of symmetric varieties. Transform Groups 18(2), 391–413 (2013)

    Article  MathSciNet  Google Scholar 

  16. Krishna, A.: Equivariant cobordism of schemes. Doc. Math. 17, 95–134 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Lenart, C., Zainoulline, K.: A Schubert basis in equivariant elliptic cohomology, New York. J. Math. 23, 711–737 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Lenart, C., Zainoulline, K.: Towards generalized cohomology Schubert calculus via formal root polynomials. Math. Res. Lett. 24(3), 839–877 (2017)

    Article  MathSciNet  Google Scholar 

  19. Leclerc, M.-A.: The hyperbolic formal affine Demazure algebra. Algebras Represent. Theory 19(5), 1043–1057 (2016)

    Article  MathSciNet  Google Scholar 

  20. Milne, J.: Étale Cohomology, Princeton Mathematical Series 33. Princeton University Press, Princeton (1980). xiii+ 323 pp

    Google Scholar 

  21. Su, C., Zhao, G., Zhong, C.: On the K-theoretic stable bases of Springer Resolutions, Annales scientifiques de l’École normale supérieure (to appear)

  22. Zhao, G., Zhong, C.: Geometric representations of the formal affine Hecke algebra. Adv. Math. 317, 50–90 (2017)

    Article  MathSciNet  Google Scholar 

  23. Zhong, C.: On the formal affine Hecke algebra. J. Instit. Math. Jussieu 14(4), 837–855 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

B. C. was supported by the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005. K. Z. was partially supported by the NSERC Discovery Grant RGPIN-2015-04469. C. Z. was supported by PIMS and NSERC grants of S. Gille and V. Chernousov in 2014, and would like to thank S. Kumar, T. Zhang and G. Zhao for helpful discussions, and to the Max Planck Institute for Mathematics for hospitality during the visit in May, 2014.

Author information

Authors and Affiliations

  1. Faculté des Sciences Jean Perrin, Université d’Artois, Rue Jean Souvraz SP 18, 62307, Lens Cedex, France

    Baptiste Calmès

  2. University of Ottawa, 585 King Edward, Ottawa, ON, K1N 6N5, Canada

    Kirill Zainoulline

  3. State University of New York, Albany, 1400 Washington Ave, Albany, NY, 12222, USA

    Changlong Zhong

Authors
  1. Baptiste Calmès
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kirill Zainoulline
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Changlong Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changlong Zhong.

Additional information

Presented by: Michel Brion

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Calmès, B., Zainoulline, K. & Zhong, C. Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems. Algebr Represent Theor 23, 1031–1050 (2020). https://doi.org/10.1007/s10468-019-09880-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-019-09880-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation