Equivariant oriented cohomology of flag varieties

  • Baptiste Calmès

  • Kirill Zainoulline

  • Changlong Zhong

Abstract

Given an equivariant oriented cohomology theory h, a split reductive group GG, a maximal torus TT in GG, and a parabolic subgroup PP containing TT, we explain how the TT-equivariant oriented cohomology ring \ssfhT(G/P){\ssf h}_T(G/P) can be identified with the dual of a coalgebra defined using exclusively the root datum of (G,T)(G,T), a set of simple roots defining PP and the formal group law of \ssfh\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 TT-fixed points of G/PG/P which embeds the cohomology ring in question into a direct product of a finite number of copies of the TT-equivariant oriented cohomology of a point.