#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 113-144

** Baptiste Calmès, Kirill Zainoulline, Changlong Zhong **
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

Keywords and Phrases:

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