Formal Hecke algebras and algebraic oriented cohomology theories
Alex Hoffnung,Alex Hoffnung,Jose Malagon-Lopez,Jose Malagon-Lopez,Alistair Savage,Kirill Zainoulline +5 more
TLDR
In this paper, the authors 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 cohomologies, and algebraic cobordism.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.read more
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Equivariant oriented cohomology of flag varieties
TL;DR: In this article, the authors show how to match equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms.
Journal ArticleDOI
A coproduct structure on the formal affine Demazure algebra
TL;DR: In this paper, the authors generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant and Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law.
Equivariant Oriented Cohomology of Flag Varieties Посвящается А.С. Меркурьеву, Ученому и Учителю
TL;DR: In this paper, the authors explain how the T-equivariant oriented cohomology ring hT(G/P) can be identified with the dual of a coal-gebra defined using exclusively the root datum of (G,T), a set of simple roots defining P and the formal group law.
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Towards generalized cohomology Schubert calculus via formal root polynomials
TL;DR: In this paper, the authors define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory) and study some of the properties of such polynomial structures.
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Generalized Schubert Calculus
Nora Ganter,Arun Ram +1 more
TL;DR: In this article, the T-equivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model, was studied and the differences between the Schubert classes and the Bott-Samelson classes were compared.
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TL;DR: It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.
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