We generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel (e.g. Chow groups, Grothendieck's K_0, connective K-theory and algebraic cobordism). The resulting object, called a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We apply it to construct an algebraic model of the T-equivariant algebraic oriented cohomology of the variety of complete flags. The talk is based on two recent preprints arXiv:1208.4114, arXiv:1209.1676
and the paper [Invariants, torsion indices and cohomology of complete flags. Ann. Sci. Ecole Norm. Sup. (4) 46 (2013), no.3.].