Equivariant Oriented Cohomology of Flag Varieties

Given an equivariant oriented cohomology theory $\hh$, 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 $\hh_T(G/P)$ 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 $\hh$. In two papers [CZZ,CZZ2] 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.

2010 Mathematics Subject Classification: 14F43, 14M15, 19L41, 55N22, 57T15, 57R85

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