Equivariant oriented cohomology of flag varieties

Documenta Mathematica, Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday, 2015, p. 113-144
(1^st online appearance of the preprint on the 24-9-2014)

This article contains the geometric interpretation of the algebraic formalism introduced in A coproduct structure on the formal affine Demazure algebra and Push-pull operators on the formal affine Demazure algebra and its dual. It thus completes the algebraic description of the following elements.

• h_T(G/P), a T-équivariant oriented cohomology ring, where T is a maximal torus contained in a parabolic subgroup P of a split semi-simple algebraic group G over a base field);
• The restriction to the fixed points of G/B under the action of T going from h_T(G/B) to a direct sum of copies of the cohomology of the base, indexed by the Weyl group;
• The pull-back map from h_T(G/P) to h_T(G/B) where B is a Borel subgroup containing P;
• The push-forward map from h_T(G/B) to h_T(G/P);
• The pairing on h_T(G/B) given by product followed by push-forward to the base.