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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 Savage
  • Kirill Zainoulline
Article

Abstract

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.

Keywords

Hecke algebra Oriented cohomology Formal group law Demazure operator 

Mathematics Subject Classification (1991)

20C08 14F43 

Notes

Acknowledgments

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.

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Alex Hoffnung
    • 1
    • 3
  • José Malagón-López
    • 2
    • 3
  • Alistair Savage
    • 3
  • 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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