Advertisement

Selecta Mathematica

, 25:16 | Cite as

Elliptic and K-theoretic stable envelopes and Newton polytopes

  • R. Rimányi
  • V. Tarasov
  • A. VarchenkoEmail author
Article

Abstract

In this paper we consider the cotangent bundles of partial flag varieties. We construct the \(K\)-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the \(K\)-theoretic stable envelopes and our elliptic stable envelopes. We show that the \(K\)-theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the \({\mathfrak {gl}}_2\) case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the \(K\)-theoretic stable envelopes.

Mathematics Subject Classification

55N34 14M15 17B37 

Notes

References

  1. 1.
    Aganagic, M., Okounkov, A.: Elliptic stable envelopes. Preprint (2016). arXiv:1604.00423
  2. 2.
    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry, Modern Birkhäuser Classics. Birkhäuser Inc., Boston, MA (2010)CrossRefGoogle Scholar
  3. 3.
    Felder, G.: Elliptic quantum groups. In: Iagolnitzer, D. (ed.) Proceedings of the ICMP, Paris 1994, pp. 211–218. Intern. Press, Cambridge, MA (1995)Google Scholar
  4. 4.
    Feher, L., Rimanyi, R.: Calculation of Thom polynomials and other cohomological obstructions for group actions. Gaffney, T., Ruas, M. (ed.) Real and Complex Singularities (Sao Carlos, 2002) Contemporary Mathematics,#354, pp. 69–93. American Mathematical Society, Providence, RI, (2004)Google Scholar
  5. 5.
    Felder, G., Rimanyi, R., Varchenko, A.: Elliptic dynamical quantum groups and equivariant elliptic cohomology. SIGMA 14(2018), 41 (2018)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Felder, G., Tarasov, V., Varchenko, A.: Solutions of the elliptic QKZB equations and Bethe ansatz I. In: Topics in Singularity Theory, V.I.Arnold’s 60th Anniversary Collection, Advances in the Mathematical Sciences, AMS Translations, Series 2, vol. 180, pp. 45–76 (1997)Google Scholar
  7. 7.
    Felder, G., Tarasov, V., Varchenko, A.: Monodromy of solutions of the elliptic quantum Knizhnik–Zamolodchikov–Bernard difference equations. Int. J. Math. 10, 943–975 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fulton, W., Pragacz, P.: Schubert Varieties and Degeneracy Loci, Appendix J by the authors in collaboration with I. Ciocan-Fontanine, Lecture Notes in Mathematics 1689, pp. 1–148 (1998)Google Scholar
  9. 9.
    Ganter, N.: The elliptic Weyl character formula. Compos. Math. 150(7), 1196–1234 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ginzburg, V., Kapranov, M., Vasserot, E.: Elliptic Algebras and Equivariant Elliptic Cohomology I. arXiv:q-alg/9505012
  11. 11.
    Gorbounov, V., Rimányi, R., Tarasov, V., Varchenko, A.: Cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra. J. Geom. Phys. 74, 56–86 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1), 25–83 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Grojnowski, I.: Delocalized equivariant elliptic cohomology, (preprint 1994). In: Elliptic Cohomology, London Mathematical Society, Lecture Note Series, vol. 342, pp. 114–121. Cambridge University Press, Cambridge (2007)Google Scholar
  14. 14.
    Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. Preprint, pp. 1–276 (2012). arXiv:1211.1287
  15. 15.
    Maulik, D., Okounkov, A.: in preparation (2015)Google Scholar
  16. 16.
    Mukhin, E., Tarasov, V., Varchenko, A.: Bethe eigenvectors of higher transfer matrices. J. Stat. Mech. (8), P08002, 1–44 (2006)Google Scholar
  17. 17.
    Mumford, D.: Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Oxford University Press, London (1970)Google Scholar
  18. 18.
    Okounkov, A., Lectures on K-theoretic computations in enumerative geometry. Geometry of moduli spaces and representation theory. IAS/Park City Math. Ser., vol. 24, pp. 251–380. American Mathematical Society, Providence, RI (2017)Google Scholar
  19. 19.
    Okounkov, A., Smirnov, A.: Quantum difference equation for Nakajima varieties. Preprint (2016). arXiv:1602.09007
  20. 20.
    Pushkar, P.P., Smirnov, A., Zeitlin, A.M.: Baxter Q-operator from quantum \(K\)-theory. Preprint (2016). arXiv:1612.08723
  21. 21.
    Rimányi, R., Tarasov, V., Varchenko, A.: Partial flag varieties, stable envelopes and weight functions. Quantum Topol. 6(2), 333–364 (2015).  https://doi.org/10.4171/QT/65 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rimanyi, R., Tarasov, V., Varchenko, A.: Trigonometric weight functions as \(K\)-theoretic stable envelope maps for the cotangent bundle of a flag variety. J. Geom. Phys. 94, 81–119 (2015).  https://doi.org/10.1016/j.geomphys.2015.04.002. arXiv:1411.0478 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rimanyi, R., Varchenko, A.: Dynamical Gelfand–Zetlin algebra and equivariant cohomology of Grassmannians. J. Knot Theory Ramif. 25(12) (2016).  https://doi.org/10.1142/S021821651642013X. arXiv:1510.03625
  24. 24.
    Rimanyi, R., Varchenko, A.: Equivariant Chern-SchwartzMacPherson classes in partial flag varieties: interpolation and formulae. In: Buczynski, J., Michalek, M., Postingel, E. (eds.) Schubert Varieties, Equivariant Cohomology and Characteristic Classes, IMPANGA2015, EMS, pp. 225–235 (2018)Google Scholar
  25. 25.
    Rosu, I.: Equivariant K-theory and equivariant cohomology, with an appendix by Allen Knutson and Ioanid Rosu. Math. Z. 243(3), 423–448 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schechtman, V., Varchenko, A.: Arrangements of hyperplanes and lie algebra homology. Invent. Math. 106, 139–194 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Varchenko, A., Tarasov, V.: Jackson integral representations for solutions of the Knizhnik–Zamolodchikov quantum equation. Leningr. Math. J. 6, 275–313 (1994)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Tarasov, V., Varchenko, A.: Geometry of \(q\)-hypergeometric functions as a bridge between Yangians and quantum affine algebras. Invent. Math. 128(3), 501–588 (1997)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tarasov, V., Varchenko, A.: Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups. Asterisque 246, 1–135 (1997)zbMATHGoogle Scholar
  30. 30.
    Tarasov, V., Varchenko, A.: Combinatorial formulae for nested Bethe vectors. SIGMA 9(048), 1–28 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Varchenko, A.: Quantized KZ equations, quantum YBE, and difference equations for q-hypergeometric functions. Commun. Math. Phys. 162, 499–528 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA
  2. 2.Faculty of Mathematics and MechanicsLomonosov Moscow State UniversityMoscow GSP-1Russia
  3. 3.Department of Mathematical SciencesIndiana University – Purdue University IndianapolisIndianapolisUSA
  4. 4.St. Petersburg Branch of Steklov Mathematical InstituteSt. PetersburgRussia

Personalised recommendations