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

TL;DR

This paper introduces push-pull operators on the formal affine Demazure algebra and its dual, establishing a non-degenerate pairing that advances the algebraization of equivariant oriented cohomology theories.

Contribution

It develops push-pull operators on the formal affine Demazure algebra and constructs a pairing that connects algebraic and geometric aspects of equivariant cohomology.

Findings

Defined push-pull operators on the algebra and its dual

Established a non-degenerate pairing analogous to geometric cohomology pairing

Progressed the algebraization program for equivariant oriented cohomology

Abstract

In the present paper we introduce and study the push pull operators on the formal affine Demazure algebra and its dual. As an application we provide a non-degenerate pairing on the dual of the formal affine Demazure algebra which serves as an algebraic counterpart of the natural pairing on the T-equivariant oriented cohomology of G/B induced by multiplication and push-forward to a point. This paper can be viewed as the next step towards the `algebraization program' for equivariant oriented cohomology theories started in arXiv:0905.1341 and continued in arXiv:1208.4114 and arXiv:1209.1676; the general idea being to match cohomology rings of algebraic varieties and elements of classical interest in them (such as classes of Schubert varieties) with algebraic and combinatorial objects that can be introduced in the spirit of [Demazure, Invariants sym\'etriques entiers des groupes de Weyl et torsion, Invent. Math. 21:287-301, 1973] and [Kostant, Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Advances in Math. 62:187-237, 1986].