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Wall-crossings and a categorification of K-theory stable bases of the Springer resolution

Part of: Special varieties Representation theory of groups Lie algebras and Lie superalgebras Topological $K$-theory

Published online by Cambridge University Press:  06 October 2021

Changjian Su ,
Gufang Zhao  and
Changlong Zhong
Changjian Su
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Room 6290, Toronto, ONM5S 2E4, [email protected]
Gufang Zhao
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, Parkville, VIC3010, [email protected]
Changlong Zhong
Affiliation:
State University of New York at Albany, 1400 Washington Avenue, ES 110, Albany, NY12222, [email protected]
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Abstract

We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

Keywords

stable basisSpringer resolutionHecke algebraVerma modulewall-crossing

MSC classification

Primary: 19L47: Equivariant $K$-theory 20C08: Hecke algebras and their representations 14M15: Grassmannians, Schubert varieties, flag manifolds
Secondary: 17B50: Modular Lie (super)algebras
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 11 , November 2021 , pp. 2341 - 2376
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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References

Aganagic, M. and Okounkov, A., Elliptic stable envelope, J. Amer. Math. Soc. 34 (2021), 79–133.10.1090/jams/954CrossRefGoogle Scholar
Aluffi, P. and Mihalcea, L., Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, Compos. Math. 152 (2016), 2603–2625.10.1112/S0010437X16007685CrossRefGoogle Scholar
Aluffi, P., Mihalcea, L., Schürmann, J. and Su, C., Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells, Preprint (2017), arXiv:1709.08697.Google Scholar
Aluffi, P., Mihalcea, L., Schürmann, J. and Su, C., Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem, Preprint (2019), arXiv:1902.10101.Google Scholar
Arkhipov, S. and Bezrukavnikov, R., Perverse sheaves on affine flags and Langlands dual group, Israel J. Math. 170 (2009), 135–183.10.1007/s11856-009-0024-yCrossRefGoogle Scholar
Beilinson, A. and Bernstein, J., Localisation de ${\frak g}$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 15–18.Google Scholar
Ben-Zvi, D. and Frenkel, E., Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, vol. 88, second edition (American Mathematical Society, Providence, RI, 2004).Google Scholar
Bezrukavnikov, R., On two geometric realizations of an affine Hecke algebra, Publ. Math. Inst. Hautes Études Sci. 123 (2016), 1–67.10.1007/s10240-015-0077-xCrossRefGoogle Scholar
Bezrukavnikov, R. and Mirković, I., Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution, Ann. of Math. (2) 178 (2013), 835–919.10.4007/annals.2013.178.3.2CrossRefGoogle Scholar
Bezrukavnikov, R., Mirković, I. and Rumynin, D., Singular localization and intertwining functors for reductive lie algebras in prime characteristic, Nagoya Math. J. 184 (2006), 1–55.10.1017/S0027763000009302CrossRefGoogle Scholar
Bezrukavnikov, R., Mirković, I. and Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2) 167 (2008), 945–991.10.4007/annals.2008.167.945CrossRefGoogle Scholar
Bezrukavnikov, R. and Riche, S., Affine braid group actions on derived categories of Springer resolutions, Ann. Sci. Éc. Norm. Supér. (4) 45 (2013), 535–599.10.24033/asens.2173CrossRefGoogle Scholar
Brasselet, J., Schürmann, J. and Yokura, S., Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010), 1–55.10.1142/S1793525310000239CrossRefGoogle Scholar
Braverman, A., Maulik, D. and Okounkov, A., Quantum cohomology of the Springer resolution, Adv. Math. 227 (2011), 421–458.10.1016/j.aim.2011.01.021CrossRefGoogle Scholar
Bump, D. and Nakasuji, M., Casselman's basis of Iwahori vectors and the Bruhat order, Canad. J. Math. 63 (2011), 1238–1253.10.4153/CJM-2011-042-3CrossRefGoogle Scholar
Bump, D. and Nakasuji, M., Casselman's basis of Iwahori vectors and Kazhdan–Lusztig polynomials, Canad. J. Math. 71 (2019), 1351–1366.10.4153/CJM-2018-011-1CrossRefGoogle Scholar
Cartier, P., Representations of p-adic groups: a survey, in Automorphic forms, representations and L-functions, Proc. sympos. pure math., Corvallis, OR, 1977, Part 1, Proceedings of Symposia in Pure Mathematics, vol. 33 (American Mathematical Society, Providence, RI, 1979), 111–155.Google Scholar
Casselman, W., The unramified principal series of p-adic groups. I. The spherical function, Compos. Math. 40 (1980), 387–406.Google Scholar
Casselman, W. and Shalika, J., The unramified principal series of $p$-adic groups. II. The Whittaker function, Compos. Math. 41 (1980), 207–231.Google Scholar
Cherednik, I., Double affine Hecke algebras, vol. 319 (Cambridge University Press, 2005).10.1017/CBO9780511546501CrossRefGoogle Scholar
Chriss, N. and Ginzburg, V., Representation theory and complex geometry (Birkhäuser, Boston, MA, 1997).Google Scholar
Deodhar, V. V., Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), 187–198.10.1007/BF01390109CrossRefGoogle Scholar
Feher, L., Rimanyi, R. and Weber, A., Motivic Chern classes and K-theoretic stable envelopes, Proc. Lond. Math. Soc. (3) 122 (2021), 153–189.10.1112/plms.12374CrossRefGoogle Scholar
Gorsky, E. and Negut, A., Infinitesimal change of stable basis, Selecta Math. 23 (2017), 1909–1930.10.1007/s00029-017-0327-5CrossRefGoogle Scholar
Janzten, J. C., Modular representations of reductive Lie algebras, J. Pure Appl. Algebra 152 (2000), 133–185.Google Scholar
Kaledin, D., Derived equivalences by quantization, Geom. Funct. Anal. 17 (2008), 1968–2004.10.1007/s00039-007-0623-xCrossRefGoogle Scholar
Kaledin, D., Geometry and topology of symplectic resolutions, Algebraic Geometry, Seattle, WA, 2005, Part 2, Proceedings of Symposia in Pure Mathematics, vol. 80 (American Mathematical Society, Providence, RI, 2009), 595–628.Google Scholar
Kazhdan, D. and Lusztig, G., Proof of the Deligne–Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153–215.10.1007/BF01389157CrossRefGoogle Scholar
Lenart, C., Zainoulline, K. and Zhong, C., Parabolic Kazhdan–Lusztig basis, Schubert classes, and equivariant oriented cohomology, J. Inst. Math. Jussieu 19 (2020), 1889–1929.10.1017/S1474748018000592CrossRefGoogle Scholar
Lusztig, G., Hecke algebras and Jantzen's generic decomposition patterns, Adv. Math. 37 (1980), 121–164.10.1016/0001-8708(80)90031-6CrossRefGoogle Scholar
Lusztig, G., Cuspidal local systems and graded Hecke algebras, I, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 145–202.10.1007/BF02699129CrossRefGoogle Scholar
Lusztig, G., Bases in equivariant K-theory, Represent. Theory 2 (1998), 298–369.10.1090/S1088-4165-98-00054-5CrossRefGoogle Scholar
MacDonald, I. G., Affine Hecke algebras and orthogonal polynomials, in Séminaire Bourbaki, volume 1994/95, Exposé 797, Astérisque, vol. 237 (Société Mathématique de France, 1996), 189–207.Google Scholar
MacPherson, R., Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432.10.2307/1971080CrossRefGoogle Scholar
Maulik, D. and Okounkov, A., Quantum groups and quantum cohomology, Astérisque 408 (2019).Google Scholar
Nakajima, H., Quiver varieties and finite dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145–238.10.1090/S0894-0347-00-00353-2CrossRefGoogle Scholar
Okounkov, A., Geometry of moduli spaces and representation theory, in Lectures on K-theoretic computations in enumerative geometry, IAS/Park City Mathematics Series, vol. 24 (American Mathematical Society, Providence, RI, 2017), 251–380.Google Scholar
Okounkov, A., Enumerative geometry and geometric representation theory. Algebraic geometry: Salt Lake City 2015, Part 1, Proceedings of Symposia in Pure Mathematics, vol. 97 (American Mathematical Society, Providence, RI, 2018), 419–457.10.1090/pspum/097.1/01681CrossRefGoogle Scholar
Okounkov, A., On the crossroads of enumerative geometry and geometric representation theory, in Proceedings of the international congress of mathematicians—Rio de Janeiro 2018, Plenary Lectures, vol. 1 (World Scientific Publications, Hackensack, NJ, 2018), 839–867.Google Scholar
Okounkov, A., Inductive construction of stable envelopes and applications, I (2020), Preprint arXiv:2007.09094.Google Scholar
Okounkov, A., Inductive construction of stable envelopes and applications, II. Nonabelian actions. Integral solutions and monodromy of quantum difference equations, Preprint (2020), arXiv:2010.13217.Google Scholar
Okounkov, A. and Smirnov, A., Quantum difference equation for Nakajima varieties, Preprint (2016), arXiv:1602.09007.Google Scholar
Riche, S., Geometric braid group action on derived categories of coherent sheaves, Represent. Theory 12 (2008), 131–169.10.1090/S1088-4165-08-00325-7CrossRefGoogle Scholar
Rimanyi, R., Tarasov, V. and Varchenko, A., Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety, J. Geom. Phys. 94 (2015), 81–119.10.1016/j.geomphys.2015.04.002CrossRefGoogle Scholar
Rimanyi, R. and Varchenko, A., Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae, in Schubert varieties, equivariant cohomology and characteristic classes, IMPANGA 2015, eds J. Buczynski, M. Michalek and E. Postingel (European Mathematical Society, 2018), 225–235.10.4171/182-1/10CrossRefGoogle Scholar
Su, C., Equivariant quantum cohomology of cotangent bundle of $G/P$, Adv. Math. 289 (2016), 362–383.10.1016/j.aim.2015.11.026CrossRefGoogle Scholar
Su, C., Restriction formula for stable basis of the Springer resolution, Selecta Math. (N.S.) 23 (2017), 497–518.10.1007/s00029-016-0248-8CrossRefGoogle Scholar
Su, C., Zhao, G. and Zhong, C., On the $K$-theory stable bases of the Springer resolution, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), 663–711.Google Scholar