Push-pull operators on the formal affine Demazure algebra and its dual

  • Baptiste Calmès
  • Kirill Zainoulline
  • Changlong Zhong


In the present paper we introduce and study the push pull operators on the formal affine Demazure algebra and its dual. As an application we provide a non-degenerate pairing on the dual of the formal affine Demazure algebra which serves as an algebraic counterpart of the natural pairing on the equivariant oriented cohomology of the complete flag variety induced by multiplication and push-forward to a point.

Mathematics Subject Classification

20C08 14F43 57T15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



One of the ingredients of this paper, the push-pull formulas in the context of Weyl group actions, arose in discussions between the first author and Victor Petrov, whose unapparent contribution we therefore gratefully acknowledge.


  1. 1.
    Arabia, A.: Cycles de Schubert et cohomologie équivariante de \(K/T\) (French). Invent. Math. 85(1), 39–52 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arabia, A.: Cohomologie \(T\)-équivariante de la variété de drapeaux d’un groupe de Kac–Moody. Bull. Soc. Math. France 117(2), 129–165 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atiyah, M., Bott, R.: The moment map and equivariant cohomology. Topology 23(1), 1–28 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourbaki, N.: Éléments de mathématique. Algèbre, Hermann, Paris (1958)zbMATHGoogle Scholar
  5. 5.
    Bourbaki, N.: Éléments de mathématique. Groupes et algèbres de Lie, Hermann, Paris (1968)zbMATHGoogle Scholar
  6. 6.
    Bressler, P., Evens, S.: The Schubert calculus, braid relations and generalized cohomology. Trans. Am. Math. Soc. 317(2), 799–811 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brion, M.: Equivariant Chow groups for torus actions. Transf. Groups 2(3), 225–267 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buchstaber, V., Bunkova, E.: Krichever formal groups. Funct. Anal. Appl. 45(2), 23–44 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buchstaber, V., Kholodov, A.: Formal groups, functional equations and generalized cohomology theories (Russian). Mat. Sb. 181 (1990), 1:75–94; translation in Math. USSR-Sb. 69 (1991), 1:77–97Google Scholar
  10. 10.
    Calmès, B., Petrov, V., Zainoulline, K.: Invariants, torsion indices and oriented cohomology of complete flags. Ann. Sci. École Norm. Sup. (4) 46(3), 405–448 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Calmès, B., Zainoulline, K., Zhong, C.: A coproduct structure on the formal affine Demazure algebra. Math. Z. 282(3), 1191–1218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Calmès, B., Zainoulline, K., Zhong, C.: Equivariant Oriented Cohomology of Flag Varieties, Documenta Math, pp. 113–144. Alexander S. Merkurjev’s Sixtieth Birthday, Extra Volume (2015)Google Scholar
  13. 13.
    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry, Modern Birkhauser Classics, Birkhauser Boston Inc., Boston, MA, 2010. Reprint of the 1997 editionGoogle Scholar
  14. 14.
    Demazure, M., Grothendieck, A.: Schémas en groupes III: Structure des schémas en groupes réductifs (SGA 3, Vol. 3), Lecture Notes in Math. 153, viii+529 pp. Springer, Berlin-New York (1970)Google Scholar
  15. 15.
    Demazure, M.: Invariants symétriques entiers des groupes de Weyl et torsion. Invent. Math. 21, 287–301 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Demazure, M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4) 7, 53–88 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Deodhar, V.: Some characterizations of bruhat ordering on a coxeter group and determination of the relative Möbius function. Invent. Math. 39, 187–198 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ganter, N., Ram, A.: Generalized Schubert Calculus. J. Ramanujan Math. Soc. 28A, 149–190 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ginzburg, V., Kapranov, M., Vasserot, E.: Residue construction of Hecke algebras. Adv. Math. 128(1), 1–19 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Harada, M., Henriques, A., Holm, T.: Computation of generalized equivariant cohomologies of Kac–Moody flag varieties. Adv. Math. 197, 198–221 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hazewinkel, M.: Formal Groups and Applications, Pure and Applied Mathematics, vol. 78, xxii+573pp. Acad. Press, New-York-London (1978)Google Scholar
  22. 22.
    Hirzebruch, F.: Topological Methods in Algebraic Geometry, Grundl. der Math. Wiss. vol. 131. Springer (1966)Google Scholar
  23. 23.
    Hoffnung, A., Malagón-López, J., Savage, A., Zainoulline, K.: Formal Hecke algebras and algebraic oriented cohomology theories. Selecta Math. 20(4), 1213–1245 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Humphreys, J.: Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Math. vol. 29. Cambridge University Press, Cambridge (1990)Google Scholar
  25. 25.
    Kiritchenko, V., Krishna, A.: Equivariant cobordism of flag varieties and of symmetric varieties. Transf. Groups 18, 391–413 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kostant, B., Kumar, S.: \(T\)-equivariant \(K\)-theory of generalized flag varieties. J. Differ. Geom. 32, 549–603 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kumar, S.: Kac–Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics, vol. 204. Birkhäuser, Boston, MA (2002)CrossRefzbMATHGoogle Scholar
  29. 29.
    Leclerc, M.-A.: The hyperbolic formal affine Demazure algebra. Algebras Represent. Theory 19(5), 1043–1057 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lenart, C., Zainoulline, K.: Towards generalized cohomology Schubert calculus via formal root polynomials. Math. Res. Lett. 24(3), 839–877 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Levine, M., Morel, F.: Algebraic Cobordism Springer Monographs in Mathematics. Springer, Berlin (2007)Google Scholar
  32. 32.
    Zhao, G., Zhong, C.: Geometric representations of the formal affine Hecke algebra. Adv. Math. 317, 50–90 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhong, C.: On the formal affine Hecke algebra. J. Inst. Math. Jussieu 14(4), 837–855 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de LensUniversité d’ArtoisArrasFrance
  2. 2.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  3. 3.Department of Mathematics and StatisticsState University of New York at AlbanyAlbanyUSA

Personalised recommendations