Equivariant oriented cohomology of flag varieties

Abstract

Given an equivariant oriented cohomology theory h, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-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 $(G,T)$, a set of simple roots defining $P$ 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 $T$-fixed points of $G/P$ which embeds the cohomology ring in question into a direct product of a finite number of copies of the $T$-equivariant oriented cohomology of a point.