Joint work w. José Malagón Lopez, Alistair Savage, and Kirill Zainoulline

The affine Hecke algebra can be constructed geometrically on the equivariant K-theory of the Steinberg variety. Many related convolution constructions appear in geometric representation on various (co)homology theories. It seems natural to work towards a theory of higher representation theory in order to find a unified framework for such geometric constructions.

We implement formal methods to study the role of arbitrary oriented cohomology theories in analogues of geometric constructions in representation theory. We generalize the construction of the nil Hecke ring of Kostant and Kumar to the context of an arbitrary oriented cohomology theory of Levine and Morel, e.g. to Chow groups, connective K-theory, elliptic cohomology, or algebraic cobordism.

In particular, we define formal (affine) Demazure algebras and formal (affine) Hecke algebras. These depend on one-dimensional commutative formal group laws and specialize to known variants of the Hecke algebra at the additive and multiplicative formal group laws. In general, this family of formal Hecke algebras satisfies equations that we call ´oriented braid relations´, which agree with the usual braid relations at the additive and multiplicative formal group laws.