Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

Mihalcea, Leonardo C.; Naruse, Hiroshi; Su, Changjian

doi:10.1093/imrn/rnab049

Cited by 13 publications

(15 citation statements)

References 44 publications

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“…In this section we prove a counterpart of theorem 6.9 concerning the left Weyl group action. The proof is similar to the proof [RW20, proposition 7.3], see also [MNS22].…”

Section: Left Inductionsupporting

confidence: 60%

“…Starting from formulas for fundamental classes in cohomology [BGG73] or in K-theory [LS82,KK90] the recursion based on word length became a standard feature of cohomological study of homogenous varieties. Among further important contributions we mention [Bri97,Knu03] for fundamental classes, [AM16,MNS22] for c SM classes, [AMSS19, MNS22,MS22] for motivic Chern classes, [AMSS22] for Hirzebruch-Todd classes, [SZZ20,SZZ21] for stable envelopes, [RW20,KRW20] for elliptic classes and [MNS22] for classes in the quantum cohomology. Most of the mentioned results are nicely reviewed in [MNS22].…”

Section: Introductionmentioning

confidence: 99%

“…Among further important contributions we mention [Bri97,Knu03] for fundamental classes, [AM16,MNS22] for c SM classes, [AMSS19, MNS22,MS22] for motivic Chern classes, [AMSS22] for Hirzebruch-Todd classes, [SZZ20,SZZ21] for stable envelopes, [RW20,KRW20] for elliptic classes and [MNS22] for classes in the quantum cohomology. Most of the mentioned results are nicely reviewed in [MNS22]. In [RW22a] the study of such recurrences allowed to observe an instance of mirror symmetry.…”

Section: Introductionmentioning

confidence: 99%

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Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Jakub¹,

Weber²

2023

Preprint

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Let G be a linear semisimple algebraic group and B its Borel subgroup. Let T ⊂ B be the maximal torus. We study the inductive construction of Bott-Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells in G/B. To this end we introduce two families of operators acting on the equivariant K-theory K T (G/B)[y], the right and left Demazure-Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the type A this algebra acts on the Laurent polynomials. This action is a natural lift of the action on K T (G/B)[y] with respect to the Kirwan map. We show that the left and right twisted Demazure-Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.Both authors supported by NCN grant 2016/23/G/ST1/04282 (Beethoven 2). 3.3. Reflections 4. Bott-Samelson resolution 4.1. Inductive construction 4.2. Pull-backs of line bundles 5. Twisted motivic Chern class 5.1. Motivic Chern class 5.2. Twisted class 5.3. Twisted classes of Schubert cells and stable envelopes 6. Twisted Demazure-Lusztig operators 6.1. The definitions and main properties 6.2. Right Demazure-Lusztig operators 6.3. Bott-Samelson recursion 7. Twisted Hecke algebra 8. Left induction 8.1. Recursive formula for G/B 8.2. Recursion for G/P 8.3. Wall crossing for change of the coweight chamber 9. Wall-crossing formula for a change of the slope 10. Upgrade from GL n /B to End(C n ). 10.1. The Kirwan map 10.2. Matrix Schubert varieties 10.3. The lift of boundary divisors 11. Universal algebra 11.1. The small algebra 11.2. Operators with parameters in t * n ⊂ gl * n 11.3. Comparison with K T (GL n /B n ) 12. Comparison with the matrix Schubert varieties 12.1. Corollaries from the work on square-zero matrices 12.2. Twisted motivic Chern classes of matrix Schubert varieties 13. Resolution of matrix Schubert varieties 13.1. The left resolution 13.2. The right resolution 14. Proof of theorem 12.6 14.1. Left induction 14.2. Right induction 15. Twisted double Hecke algebra of the general type 15.1. Type C 2 15.2. Type G 2 References NotationsAll considered varieties are complex and quasi-projective. We consider an algebraic torus T ≃ (C * ) rk T .2.1. Line bundles and divisors. Let X be a quasiprojective T-variety and L → X an equivariant line bundle i.e. a line bundle together with a linearization. For any fixed point x ∈ X T , the fiber L |x is a representation of the torus T.

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“…In this section we prove a counterpart of theorem 6.9 concerning the left Weyl group action. The proof is similar to the proof [RW20, proposition 7.3], see also [MNS22].…”

Section: Left Inductionsupporting

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Jakub¹,

Weber²

2023

Preprint

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“…This is called the left Hecke action. For finite flag varieties case it is studied in [MNS22] using geometric arguments (see also [B97,K03,T09,LZZ20]). For connective K-theory (which specializes to cohomology and K-theory), we compute the recursive formulas for certain basis in h T (Gr G ) (Theorem 2.3).…”

Section: Introductionmentioning

confidence: 99%

Equivariant oriented homology of the affine Grassmannian

Zhong¹

2023

Preprint

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We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the GKM condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the the formal affine Demazure algebra.

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“…These generalized Chern classes of Schubert cells are closely related to the corresponding stable bases of the cotangent bundle T * G/B, defined by Maulik and Okounkov [2019;2017] in their study of quantum cohomology/K -theory of Nakajima quiver varieties. These classes are permuted by various Demazure-Lusztig operators [Aluffi and Mihalcea 2016;Aluffi et al 2017;Su 2017;Mihalcea et al 2022], and are related to unramified principal series representations of the Langlands dual group over a nonarchimedean local field Aluffi et al 2019].…”

Section: Introductionmentioning

confidence: 99%

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2023

Alg. Number Th.

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Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

Cited by 13 publications

References 44 publications

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Equivariant oriented homology of the affine Grassmannian

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