Geometric representations of the formal affine Hecke algebra

For any formal group law, there is a formal affine Hecke algebra defined by Hoffnung-Malagón-López-Savage-Zainoulline. Coming from this formal group law, there is also an oriented cohomology theory. We identify the formal affine Hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the Steinberg variety. As a consequence, this algebra acts on the corresponding cohomology of the Springer fibers. This generalizes the action of classical affine Hecke algebra on the K-theory of the Springer fibers constructed by Lusztig. We also give a residue interpretation of the formal affine Hecke algebra, which generalizes the residue construction of Ginzburg-Kapranov-Vasserot when the formal group law comes from a 1-dimensional algebraic group.