Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T06:39:11.523Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivariant K-theory of quaternionic flag manifolds

Published online by Cambridge University Press:  27 November 2009

Augustin-Liviu Mare  and
Matthieu Willems
Augustin-Liviu Mare
Affiliation:
Department of Mathematics and Statistics, University of Regina, College West 307.14, Regina, Saskatchewan, S4S 0A2, Canada, mareal@math.uregina.ca
Matthieu Willems
Affiliation:
Department of Mathematics, McGill University, 805 Sherbrooke Street West, Montréal, Québec, H3A 2K6, Canada, matthieu.willems@polytechnique.org
Get access

Abstract

We consider the manifold Fln(ℍ) = Sp(n)/Sp(1)n of all complete flags in ℍn, where ℍ is the skew-field of quaternions. We study its equivariant complex K-theory rings with respect to the action of two groups: Sp(1)n and a certain canonical subgroup T = (S1)n (a maximal torus). For the first group action we obtain a Goresky-Kottwitz-MacPherson type description. For the second one, we describe the ring KT(Fln(ℍ)) as a subring of KT(Sp(n)/T). This ring is well known, since Sp(n)/T is a complex flag variety.

Keywords

equivariant topological K-theoryquaternionic flag manifoldcomplex flag manifoldsGoresky-Kottwitz-MacPherson type presentation
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 3 , December 2009 , pp. 537 - 557
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Borel, A.Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts, Ann. of Math. (2), 57 (1953), 115–207CrossRefGoogle Scholar
2.Bourbaki, N., Groupes et algèbres de Lie, chap. 4–6, Hermann, Paris, 1968Google Scholar
3.Draxl, P. K., Skew Fields, Cambridge University Press, 1983CrossRefGoogle Scholar
4.Fulton, W. and Harris, J., Representation Theory, Springer-Verlag, 2004CrossRefGoogle Scholar
5.Griffeth, S. and Ram, A., Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 (2004), 1263–1283CrossRefGoogle Scholar
6.Hiller, H., Geometry of Coxeter Groups, Pitman Advanced Publishing Program, Boston, 1982Google Scholar
7.Husemoller, D., Fibre Bundles (Third Edition), Springer-Verlag, 1994CrossRefGoogle Scholar
8.Harada, M., Henriques, A. and Holm, T., Computation of generalized equivariant cohomologies of Kac-Moody flag varieties, Adv. Math. 197 (2005), 198–221Google Scholar
9.Kocherlakota, R. R., Integral homology of real flag manifolds and loop spaces of symmetric spaces, Adv. Math. 110 (1995), 1–46Google Scholar
10.Kostant, B. and Kumar, S., T-equivariant K-theory of generalized flag varieties, J. Differential Geom., 32 (1990), 549–603CrossRefGoogle Scholar
11.Lenart, C. and Postnikov, A., Affine Weyl groups in K-theory and representation theory, Int. Math. Res. Not. 2007, no. 12Google Scholar
12.Littelmann, P. and Seshadri., C. S., A Pieri-Chevalley type formula for K(G/B) and standard monomial theory, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math. 210, pp. 155–176, Birkhäuser Boston, 2003Google Scholar
13.Mare, A.-L., Equivariant cohomology of quaternionic flag manifolds, J. Algebra 319 (2008), 2830–2844CrossRefGoogle Scholar
14.Mare, A.-L. and Willems, M., Topology of the octonionic flag manifold, preprint 2008Google Scholar
15.McLeod, J., The Künneth formula in equivariant K-theory, Algebraic Topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978), Lecture Notes in Math. 741, 316–333, Springer-Verlag, 1979Google Scholar
16.Pittie, H., Homogeneous vector bundles on homogeneous spaces, Topology 11 (1972), 199–203CrossRefGoogle Scholar
17.Pittie, H. and Ram, A., A Pieri-Chevalley formula in the K-theory of a G/B-bundle, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 102–107 (electronic)CrossRefGoogle Scholar
18.Rosu, I., Equivariant K-theory and equivariant cohomology (With an appendix by A. Knutson and I. Rosu), Math. Z. 243 (2003), 423–448CrossRefGoogle Scholar
19.Segal, G., Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151CrossRefGoogle Scholar
20.Willems, M., K-théorie équivariante des tours de Bott. Application à la structure multiplicative de la K-théorie équivariante des variétés de drapeaux, Duke Math. J. 132 (2006), 271–309CrossRefGoogle Scholar
21.Willems, M., A Chevalley formula in equivariant K-theory, J. Algebra 308 (2007), 764–779CrossRefGoogle Scholar