  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].
