Equivariant oriented cohomology of flag varieties

  • Baptiste Calmès

  • Kirill Zainoulline

  • Changlong Zhong


Given an equivariant oriented cohomology theory h, a split reductive group , a maximal torus in , and a parabolic subgroup containing , we explain how the -equivariant oriented cohomology ring \( {\ssf h}_T(G/P) \) can be identified with the dual of a coalgebra defined using exclusively the root datum of , a set of simple roots defining and the formal group law of \( \ssf h \). In two papers [Math. Z. 282, No. 3--4, 1191--1218 (2016; Zbl 1362.14024); "Push-pull operators on the formal affine Demazure algebra and its dual", Preprint, arXiv:1312.0019] we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the -fixed points of which embeds the cohomology ring in question into a direct product of a finite number of copies of the -equivariant oriented cohomology of a point.