Formal Hecke algebras and algebraic oriented cohomology theories

TL;DR

This paper extends the construction of Hecke algebras to a broad class of algebraic oriented cohomology theories, introducing formal Demazure and Hecke algebras parameterized by formal group laws, unifying various algebraic structures.

Contribution

It generalizes the nil Hecke ring to arbitrary algebraic oriented cohomology theories, defining formal Demazure and Hecke algebras that unify classical cases.

Findings

Specializations recover classical nil and 0-Hecke rings.

Specializations yield degenerate and usual affine Hecke algebras.

All formal affine Demazure and Hecke algebras become isomorphic over certain rings.

Abstract

In the present paper 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 and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call 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 show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.