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.
Similar content being viewed by others
References
Bressler, P., Evens, S.: The Schubert calculus, braid relations, and generalized cohomology. Trans. Am. Math. Soc. 317(2), 799–811 (1990)
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]
Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Modern Birkhäuser Classics. Birkhäuser, Boston (2010) [Reprint of the 1997 edition]
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 Barbasch
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])
Calmés, B., Zainoulline, K., Zhong, C.: A Coproduct Structure on the Formal Affine Demazure Algebra. arXiv:arXiv:1209.1676 [math.RA]
Demazure, M.: Invariants symétriques entiers des groupes de Weyl et torsion. Invent. Math. 21, 287–301 (1973)
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]
Evens, S., Bressler, P.: On certain Hecke rings. Proc. Nat. Acad. Sci. USA 84(3), 624–625 (1987)
Fröhlich, A.: Formal Groups. Lecture Notes in Mathematics, No. 74. Springer, Berlin (1968)
Ginzburg, V.: Geometric Methods in the Representation Theory of Hecke Algebras and Quantum Groups (Notes by V. Baranovsky). arXiv:math/9802004v3 [math.AG]
Ginzburg, V., Kapranov, M., Vasserot, E.: Elliptic Algebras and Equivariant Elliptic Cohomology. arXiv:q-alg/9505012
Ginzburg, V., Kapranov, M., Vasserot, E.: Residue construction of Hecke algebras. Adv. Math. 128(1), 1–19 (1997)
Gille, S., Zainoulline, K.: Equivariant pretheories and invariants of torsors. Transform. Groups 17(2), 471–498 (2012)
Humphreys, J.E.: Reflection Groups and Coxeter Groups, Volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)
Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
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)
Kleshchev, A.: Linear and Projective Representations of Symmetric Groups, Volume 163 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2005)
Lang, S.: Elliptic Functions, Volume 112 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1987). [With an appendix by J. Tate]
Levine, M., Morel, F.: Algebraic Cobordism. Springer Monographs in Mathematics. Springer, Berlin (2007)
Lusztig, G.: Equivariant \(K\)-theory and representations of Hecke algebras. Proc. Am. Math. Soc. 94(2), 337–342 (1985)
Lusztig, G.: Affine Hecke algebras and their graded version. J. Am. Math. Soc. 2(3), 599–635 (1989)
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]
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)
Rouquier, R.: 2-Kac-Moody Algebras. arXiv:math/0812.5023v1 [math.RT]
Silverman, J.H.: The Arithmetic of Elliptic Curves, Volume 106 of Graduate Texts in Mathematics, 2nd edn. Springer, Dordrecht (2009)
Tate, J.T.: The arithmetic of elliptic curves. Invent. Math. 23, 179–206 (1974)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hoffnung, A., Malagón-López, J., Savage, A. et al. Formal Hecke algebras and algebraic oriented cohomology theories. Sel. Math. New Ser. 20, 1213–1245 (2014). https://doi.org/10.1007/s00029-013-0132-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-013-0132-8