Push-pull operators on the formal affine Demazure algebra and its dual

Manuscripta Mathematica, 160, 2019, p. 9-50
(1^st online appearance of the preprint on the 29-11-2013)

This paper is a sequel to the articles Invariants, torsion indices and oriented cohomology of flag varieties, A coproduct structure on the formal affine Demazure algebra and Formal Hecke algebras and algebraic oriented cohomology theories. As in these papers, our goal is to use algebraic and combinatorial methods to define and describe a model for the ring structure of an oriented cohomology theory applied to a split projective homogeneous variety, in order to get a clear understanding of the ring structure, to obtain algorithms for computations, and to make use of the connexion between different oriented cohomology theories. The descriptions are uniform and only depend on the formal group law of the oriented cohomology theory and the usual combinatorics of semi-simple groups (root systems, Weyl groups, etc.).

Here, we study algebraic objets corresponding to

• 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.