Formal Hecke algebras and algebraic oriented cohomology theories

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.Comment: 28 pages. v2: Some results strengthened and references added. v3: Minor corrections, section numbering changed to match published version. v4: Sign errors in Proposition 6.8(d) corrected. This version incorporates an erratum to the published versio

Smooth Schubert varieties and generalized Schubert polynomials in algebraic cobordism of Grassmannians

“…Although this algebra generalizes the ones of [11], [4] and others, note that it is different from the formal Demazure algebras studied in [7], [20]. See Remark 3.18 below for more details on this.…”

“…Remark 3.18. Note that the computations from [11] cannot be done in the formal Demazure algebra of [20]. E.g., the equality…”

A coproduct structure on the formal affine Demazure algebra

“…We recall the notions of twisted formal group algebra and formal affine Demazure algebra introduced in [HMSZ,§6]. The purpose of this section is to generalize [KK86,Prop.…”

“…Since the Weyl group W preserves the set Σ, its action on S extends to Q. Following [KK86, §4.1] and [HMSZ,Def. 6.1], we define the twisted formal group algebra to be the R-module Q W := Q ⊗ R R[W ] with the multiplication given by…”

“…In section 7, we describe the formal affine Demazure algebra D F in terms of generators and relations, hence, generalizing [HMSZ,Thm. 6.14].…”