The Hyperbolic Formal Affine Demazure Algebra

Leclerc, Marc-Antoine

doi:10.1007/s10468-016-9610-y

Cited by 5 publications

(11 citation statements)

References 21 publications

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“…By the main result of [12], the T -equivariant cohomology ring h T (G/B) of the variety of Borel subgroups G/B can be identified with the dual D F of the so-called formal affine Demazure algebra D F . Moreover, in the generic hyperbolic case F = F t , the algebra D F contains the classical Iwahori-Hecke algebra H (see [11, §9] and [28]). We briefly recall the definition of the algebra D F and the details of the above identification.…”

Section: Equivariant Oriented Cohomologymentioning

confidence: 99%

“…By the results of [22] and [10], once we fix a reduced decomposition I w for each w ∈ W the elements {Y I w } w∈W (respectively {X I w } w∈W ) form an S-basis of D F . Moreover, we know the complete set of relations in D F (see [22,Proposition 5.8] for a general F and [28,Example 4.12] in the hyperbolic case). For instance, the quadratic relations…”

Section: Formal Affine Demazure Algebrasmentioning

confidence: 99%

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Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Lenart

Zainoulline²,

Zhong

2019

J. Inst. Math. Jussieu

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We study the equivariant oriented cohomology ring hT (G/P ) of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in hT (G/P ) can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results by Brion, Knutson, Peterson, Tymoczko and others.We then focus on the equivariant oriented cohomology theory corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar's construction of the parabolic Kazhdan-Lusztig basis. Based on it, we define the parabolic Kazhdan-Lusztig (KL) Schubert classes independently of a reduced word. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We then prove several special cases. 3.6. An example 13 3.7. A generalized Deodhar's resolution 15 4. Deodhar's parabolic Hecke modules revisited 17 4.1. The case of a multiplicative formal group law 17 4.2. Modules over the Hecke algebra

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Section: Equivariant Oriented Cohomologymentioning

confidence: 99%

Section: Formal Affine Demazure Algebrasmentioning

confidence: 99%

Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Lenart

Zainoulline²,

Zhong

2019

J. Inst. Math. Jussieu

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“…We introduce notation and list basic properties of the root datum associated to a generalized Cartan matrix following [Ka90,Ku02]. We recall definition of the Demazure lattice due to [Le16].…”

Section: Preliminariesmentioning

confidence: 99%

“…Observe that all these lattices are invariant under the action by W . We refer to [Le16,§3] Let F be a one-dimensional commutative formal group law over a commutative ring R. We sometimes write x + F y = F (x, y), and denote by…”

Section: Preliminariesmentioning

confidence: 99%

“…Having the structure theorem in hands we extend most of the results and properties concerning D F following the same proofs as in the finite case. For instance, we provide the presentation of D F in terms of generators and relations (Corollary 4.4), hence, generalizing the result of [Le16]. We introduce the coproduct structure on D F and, hence, the product on its dual D * F , hence, extending [CZZ1].…”

Section: Introductionmentioning

confidence: 99%

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Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems

Calmès

Zainoulline

Zhong

2019

Algebr Represent Theor

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We define the formal affine Demazure algebra and formal affine Hecke algebra associated to a Kac-Moody root systems. We prove the structure theorems of these algebras, hence, extending several results and constructions (presentation in terms of generators and relations, coproduct and product structures, filtration by codimension of Bott-Samelson classes, root polynomials and multiplication formulas) that were previously known for finite root systems.

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Push-pull operators on the formal affine Demazure algebra and its dual

2018

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Contents 1. Introduction 1 2. Formal Demazure and push-pull operators 4 3. Two bases of the formal twisted group algebra 7 4. The Weyl and the Hecke actions 9 5. Push-pull operators and elements 11 6. The push-pull operators on the dual 13 7. Relations between bases coefficients 15 8. Another basis of the W Ξ -invariant subring 17 9. The formal Demazure algebra and the Hecke algebra 18 10. The algebraic restriction to the fixed locus on G/B 20 11. The algebraic restriction to the fixed locus on G/P 24 12. The push-pull operators on D ⋆ F 27 13. An involution 28 14. The non-degenerate pairing on the W Ξ -invariant subring 29 15. Push-forwards and pairings on D ⋆ F,Ξ 32 References 34

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The Hyperbolic Formal Affine Demazure Algebra

Cited by 5 publications

References 21 publications

Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Parabolic Kazhdan–lusztig Basis, Schubert Classes, and Equivariant Oriented Cohomology

Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems

Push-pull operators on the formal affine Demazure algebra and its dual

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