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Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes (2312.17200v4)

Published 28 Dec 2023 in math.AG, math.CO, and math.RT

Abstract: We prove a Chevalley formula to multiply the motivic Chern classes of Schubert cells in a generalized flag manifold $G/P$ by the class of any line bundle $\mathcal{L}_\lambda$. Our formula is given in terms of the $\lambda$-chains of Lenart and Postnikov. Its proof relies on a change of basis formula in the affine Hecke algebra due to Ram, and on the Hecke algebra action on torus-equivariant K-theory of the complete flag manifold $G/B$ via left Demazure--Lusztig operators. We revisit some wall-crossing formulae for the stable envelopes in $T*(G/B)$. We use our Chevalley formula, and the equivalence between motivic Chern classes of Schubert cells and K-theoretic stable envelopes in $T*(G/B)$, to give formulae for the change of polarization, and for the change of slope for stable envelopes. We prove several additional applications, including Serre, star, and Dynkin, dualities of the Chevalley coefficients, new formulae for the Whittaker functions, and for the Hall--Littlewood polynomials. We also discuss positivity properties of Chevalley coefficients, and properties of the coefficients arising from multiplication by minuscule weights.

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References (51)
  1. Positivity and Kleiman transversality in equivariant KđŸKitalic_K-theory of homogeneous spaces. J. Eur. Math. Soc. (JEMS), 13(1):57–84, 2011.
  2. Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells. to appear in Duke Mathematical Journal, arXiv preprint arXiv:1709.08697, 2017.
  3. Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem. to appear in Ann. Sci. Éc. Norm. SupĂ©r., arXiv preprint arXiv:1902.10101, 2019.
  4. From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes. arXiv preprint arXiv:2212.12509, to appear in Comtemp. Math., 2022.
  5. Elliptic stable envelopes. Journal of the American Mathematical Society, 34(1):79–133, 2021.
  6. Colored vertex models and Iwahori Whittaker functions. arXiv preprint arXiv:1906.04140, 2019.
  7. A Chevalley formula for the equivariant quantum KđŸKitalic_K-theory of cominuscule varieties. Algebr. Geom., 5(5):568–595, 2018.
  8. Curve neighborhoods of Schubert varieties. J. Differential Geom., 99(2):255–283, 2015.
  9. Michel Brion. Positivity in the Grothendieck group of complex flag varieties. J. Algebra, 258(1):137–159, 2002. Special issue in celebration of Claudio Procesi’s 60th birthday.
  10. Michel Brion. Lectures on the geometry of flag varieties. In Topics in cohomological studies of algebraic varieties, Trends Math., pages 33–85. BirkhĂ€user, Basel, 2005.
  11. Sur les classes de Chern d’un ensemble analytique complexe. In The Euler–PoincarĂ© characteristic (French), volume 83 of AstĂ©risque, pages 93–147. Soc. Math. France, Paris, 1981.
  12. Hirzebruch classes and motivic Chern classes for singular spaces. J. Topol. Anal., 2(1):1–55, 2010.
  13. Anders Skovsted Buch. A Littlewood–Richardson rule for the KđŸKitalic_K-theory of Grassmannians. Acta Math., 189(1):37–78, 2002.
  14. Representation theory and complex geometry. Springer Science & Business Media, 2009.
  15. The unramified principal series of p𝑝pitalic_p-adic groups. ii. the Whittaker function. Compositio Mathematica, 41(2):207–231, 1980.
  16. Fourier transform and the Iwahori–Matsumoto involution. Duke Math. J., 86(3):435–464, 1997.
  17. Pieri and Murnaghan–Nakayama type rules for Chern classes of Schubert cells. arXiv preprint arXiv:2211.06802, 2022.
  18. Motivic Chern classes and K-theoretic stable envelopes. Proceedings of the London Mathematical Society, 122(1):153–189, 2021.
  19. Affine Hecke algebras and the Schubert calculus. European J. Combin., 25(8):1263–1283, 2004.
  20. James E Humphreys. Reflection groups and Coxeter groups. Number 29. Cambridge university press, 1990.
  21. Proof of the Deligne–Langlands conjecture for Hecke algebras. Inventiones mathematicae, 87(1):153–215, 1987.
  22. Shrawan Kumar. Kac-Moody groups, their flag varieties and representation theory, volume 204 of Progress in Mathematics. BirkhÀuser Boston, Inc., Boston, MA, 2002.
  23. Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes. arXiv preprint arXiv:2301.12746, 2023.
  24. Cristian Lenart. Hall–Littlewood polynomials, alcove walks, and fillings of Young diagrams. Discrete mathematics, 311(4):258–275, 2011.
  25. Jian-Shu Li. Some results on the unramified principal series of p-adic groups. Mathematische Annalen, 292(1):747–761, 1992.
  26. Affine Weyl groups in KđŸKitalic_K-theory and representation theory. Int. Math. Res. Not. IMRN, (12):Art. ID rnm038, 65, 2007.
  27. A combinatorial model for crystals of Kac–Moody algebras. Transactions of the American Mathematical Society, 360(8):4349–4381, 2008.
  28. Robert D. MacPherson. Chern classes for singular algebraic varieties. Ann. of Math. (2), 100:423–432, 1974.
  29. Ian G Macdonald. Affine Hecke algebras and orthogonal polynomials. AstĂ©risque, SociĂ©tĂ© mathĂ©matique de France, 237:189–208, 1996.
  30. Ian Grant Macdonald. Symmetric functions and Hall polynomials. Oxford university press, 1998.
  31. Hook formulae from Segre–MacPherson classes. arXiv preprint arXiv:2203.16461, 2022.
  32. Left Demazure–Lusztig operators on equivariant (quantum) cohomology and K-theory. Int. Math. Res. Not. IMRN, (16):12096–12147, 2022.
  33. Quantum groups and quantum cohomology. Astérisque, 408, 2019.
  34. Whittaker functions from motivic Chern classes. Transformation Groups, pages 1–23, 2022.
  35. H. Andreas Nielsen. Diagonalizably linearized coherent sheaves. Bull. Soc. Math. France, 102:85–97, 1974.
  36. Skew hook formula for d𝑑ditalic_d-complete posets via equivariant K-theory. Algebraic Combinatorics, 2(4):541–571, 2019.
  37. Toru Ohmoto. Equivariant Chern classes of singular algebraic varieties with group actions. Math. Proc. Cambridge Philos. Soc., 140(1):115–134, 2006.
  38. Andrei Okounkov. Lectures on K-theoretic computations in enumerative geometry. In Geometry of moduli spaces and representation theory, volume 24 of IAS/Park City Math. Ser., pages 251–380. Amer. Math. Soc., Providence, RI, 2017.
  39. Quantum difference equation for Nakajima varieties. Inventiones mathematicae, 229(3):1203–1299, 2022.
  40. A Pieri–Chevalley formula in the KđŸKitalic_K-theory of a G/Bđșđ”G/Bitalic_G / italic_B-bundle. Electron. Res. Announc. Amer. Math. Soc., 5:102–107, 1999.
  41. Robert A. Proctor. Dynkin diagram classification of Î»đœ†\lambdaitalic_λ-minuscule Bruhat lattices and of d𝑑ditalic_d-complete posets. J. Algebraic Combin., 9(1):61–94, 1999.
  42. Arun Ram. Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux. Pure and Applied Mathematics Quarterly, 2(4):963–1013, 2006.
  43. Trigonometric weight functions as KđŸKitalic_K-theoretic stable envelope maps for the cotangent bundle of a flag variety. J. Geom. Phys., 94:81–119, 2015.
  44. Elliptic and K-theoretic stable envelopes and Newton polytopes. Selecta Mathematica, 25(1):16, 2019.
  45. Marie-HĂ©lĂšne Schwartz. Classes caractĂ©ristiques dĂ©finies par une stratification d’une variĂ©tĂ© analytique complexe. I. C. R. Math. Acad. Sci. Paris, 260:3262–3264, 1965.
  46. Marie-HĂ©lĂšne Schwartz. Classes caractĂ©ristiques dĂ©finies par une stratification d’une variĂ©tĂ© analytique complexe. II. C. R. Math. Acad. Sci. Paris, 260:3535–3537, 1965.
  47. Christoph Schwer. Galleries, Hall–Littlewood polynomials, and structure constants of the spherical Hecke algebra. International Mathematics Research Notices, 2006(9):75395–75395, 2006.
  48. John R Stembridge. Minuscule elements of Weyl groups. Journal of Algebra, 235(2):722–743, 2001.
  49. Changjian Su. Equivariant quantum cohomology of cotangent bundle of G/Pđș𝑃G/Pitalic_G / italic_P. Adv. Math., 289:362–383, 2016.
  50. On the K-theory stable bases of the Springer resolution. Ann. Sci. Éc. Norm. SupĂ©r. (4), 53(3):663–711, 2020.
  51. Wall-crossings and a categorification of K-theory stable bases of the Springer resolution. Compositio Mathematica, 157(11):2341–2376, 2021.
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