Selecta Mathematica

, Volume 20, Issue 4, pp 1213–1245 | Cite as

Formal Hecke algebras and algebraic oriented cohomology theories

  • Alex Hoffnung
  • José Malagón-López
  • Alistair SavageEmail author
  • Kirill Zainoulline


In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to an arbitrary algebraic oriented cohomology theory of Levine–Morel and Panin–Smirnov (e.g., to Chow groups, Grothendieck’s \(K_0\), connective \(K\)-theory, elliptic cohomology, and algebraic cobordism). The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings, respectively. We also introduce a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra, respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.


Hecke algebra Oriented cohomology Formal group law Demazure operator 

Mathematics Subject Classification (1991)

20C08 14F43 



The authors would like to thank Sam Evens, Iain Gordon, Anthony Licata, and Erhard Neher for useful discussions. They would also like to thank Changlong Zhong for sharing with them some of his computations. The work of the second two authors was supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada. The first two authors were supported by the Discovery Grants of the last two. The first author was also partially supported by funds from the Centre de Recherches Mathématiques, and the last author was also supported by an Early Researcher Award from the Government of Ontario.


  1. 1.
    Bressler, P., Evens, S.: The Schubert calculus, braid relations, and generalized cohomology. Trans. Am. Math. Soc. 317(2), 799–811 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bourbaki, N.: Éléments de Mathématique. Masson, Paris (1981). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]Google Scholar
  3. 3.
    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Modern Birkhäuser Classics. Birkhäuser, Boston (2010) [Reprint of the 1997 edition]Google Scholar
  4. 4.
    Cherednik, I., Markov, Y., Howe, R., Lusztig, G.: Iwahori–Hecke Algebras and Their Representation Theory, volume 1804 of Lecture Notes in Mathematics. Springer, Berlin (2002). Lectures from the C.I.M.E. Summer School held in Martina-Franca, June 28–July 6, 1999, Edited by M. Welleda Baldoni and Dan BarbaschGoogle Scholar
  5. 5.
    Calmès, B., Petrov, V., Zainoulline, K.: Invariants, torsion indices and oriented cohomology of complete flags. Ann. Sci. Éc. Norm. Supér. (4), 46(3) (2013) (preprint available at arXiv:0905.1341v2 [math.AG])Google Scholar
  6. 6.
    Calmés, B., Zainoulline, K., Zhong, C.: A Coproduct Structure on the Formal Affine Demazure Algebra. arXiv:arXiv:1209.1676 [math.RA]Google Scholar
  7. 7.
    Demazure, M.: Invariants symétriques entiers des groupes de Weyl et torsion. Invent. Math. 21, 287–301 (1973)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Demazure, M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4), 7, 53–88 (1974). [Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I]Google Scholar
  9. 9.
    Evens, S., Bressler, P.: On certain Hecke rings. Proc. Nat. Acad. Sci. USA 84(3), 624–625 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Fröhlich, A.: Formal Groups. Lecture Notes in Mathematics, No. 74. Springer, Berlin (1968)Google Scholar
  11. 11.
    Ginzburg, V.: Geometric Methods in the Representation Theory of Hecke Algebras and Quantum Groups (Notes by V. Baranovsky). arXiv:math/9802004v3 [math.AG]Google Scholar
  12. 12.
    Ginzburg, V., Kapranov, M., Vasserot, E.: Elliptic Algebras and Equivariant Elliptic Cohomology. arXiv:q-alg/9505012Google Scholar
  13. 13.
    Ginzburg, V., Kapranov, M., Vasserot, E.: Residue construction of Hecke algebras. Adv. Math. 128(1), 1–19 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Gille, S., Zainoulline, K.: Equivariant pretheories and invariants of torsors. Transform. Groups 17(2), 471–498 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups, Volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  16. 16.
    Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kostant, B., Kumar, S.: The nil Hecke ring and cohomology of \(G/P\) for a Kac-Moody group \(G\). Adv. Math. 62(3), 187–237 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Kleshchev, A.: Linear and Projective Representations of Symmetric Groups, Volume 163 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  19. 19.
    Lang, S.: Elliptic Functions, Volume 112 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1987). [With an appendix by J. Tate]Google Scholar
  20. 20.
    Levine, M., Morel, F.: Algebraic Cobordism. Springer Monographs in Mathematics. Springer, Berlin (2007)Google Scholar
  21. 21.
    Lusztig, G.: Equivariant \(K\)-theory and representations of Hecke algebras. Proc. Am. Math. Soc. 94(2), 337–342 (1985)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lusztig, G.: Affine Hecke algebras and their graded version. J. Am. Math. Soc. 2(3), 599–635 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Panin, I.: Oriented cohomology theories of algebraic varieties. \(K\)-Theory 30(3), 265–314 (2003). [Special issue in honor of Hyman Bass on his seventieth birthday. Part III]Google Scholar
  24. 24.
    Pittie, H., Ram, A.: A Pieri–Chevalley formula in the \(K\)-theory of a \(G/B\)-bundle. Electron. Res. Announc. Am. Math. Soc. 5, 102–107 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Rouquier, R.: 2-Kac-Moody Algebras. arXiv:math/0812.5023v1 [math.RT]Google Scholar
  26. 26.
    Silverman, J.H.: The Arithmetic of Elliptic Curves, Volume 106 of Graduate Texts in Mathematics, 2nd edn. Springer, Dordrecht (2009)CrossRefGoogle Scholar
  27. 27.
    Tate, J.T.: The arithmetic of elliptic curves. Invent. Math. 23, 179–206 (1974)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Alex Hoffnung
    • 1
    • 3
  • José Malagón-López
    • 2
    • 3
  • Alistair Savage
    • 3
    Email author
  • Kirill Zainoulline
    • 3
  1. 1.Temple University, Dept. of MathematicsPhiladelphia PAUSA
  2. 2.University of Toronto Mississauga, Department of Mathematics and Computational SciencesMississaugaCanada
  3. 3.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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