Selecta Mathematica

, Volume 24, Issue 2, pp 1593–1631 | Cite as

Categorical geometric symmetric Howe duality

Article
  • 34 Downloads

Abstract

We provide a natural geometric setting for symmetric Howe duality. This is realized as a (loop) \(\mathfrak {sl}_n\) action on derived categories of coherent sheaves on certain varieties arising in the geometry of the Beilinson–Drinfeld Grassmannian. The main construction parallels our earlier work on categorical \(\mathfrak {sl}_n\) actions and skew Howe duality. In that case the varieties involved arose in the geometry of the affine Grassmannian. We discuss some relationships between the two actions.

Mathematics Subject Classification

14F05 22E46 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves. http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf
  2. 2.
    Bezrukavnikov, R., Riche, S.: Affine braid group actions on derived categories of Springer resolutions. Ann. Sci. de l’ENS 45, 535–599 (2012). arXiv:1101.3702
  3. 3.
    Braverman, A., Finkelberg, M.: Pursuing the double affine Grassmannian I: transversal slices via instantons on \(A_k\)-singularities. Duke Math. J. 152(2), 175–206 (2010). arXiv:0711.2083 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bezrukavnikov, R., Mirković, I.: Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution. Ann. Math. (2) 178(3), 835–919 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cautis, S.: Clasp technology to knot homology via the affine Grassmannian. Math. Ann. 363(3), 1053–1115 (2015). arxiv:1207.2074
  6. 6.
    Cautis, S.: Rigidity in higher representation theory. arXiv:1409.0827
  7. 7.
    Cautis, S., Kamnitzer, J.: Knot homology via derived categories of coherent sheaves I, \({\mathfrak{sl}}_2\) case. Duke Math. J. 142(3), 511–588 (2008). arXiv:math.AG/0701194 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cautis, S., Kamnitzer, J.: Knot homology via derived categories of coherent sheaves II, \({\mathfrak{sl}}_m\) case. Invent. Math. 174(1), 165–232 (2008). arXiv:0710.3216 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cautis, S., Kamnitzer, J.: Braiding via geometric categorical Lie algebra actions. Compos. Math. 148(2), 464–506 (2012). arXiv:1001.0619 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cautis, S., Kamnitzer, J.: Quantum K-theoretic geometric Satake. Compositio Math. arXiv:1509.00112 (to appear)
  11. 11.
    Cautis, S., Kamnitzer, J., Licata, A.: Categorical geometric skew Howe duality. Invent. Math. 180(1), 111–159 (2010). arXiv:0902.1795 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cautis, S., Kamnitzer, J., Licata, A.: Derived equivalences for cotangent bundles of Grassmannians via categorical \({\mathfrak{sl}}_2\) actions. J. Reine Angew. Math. 675, 53–99 (2013). arXiv:0902.1797 MathSciNetMATHGoogle Scholar
  13. 13.
    Cautis, S., Kamnitzer, J., Licata, A.: Coherent sheaves on quiver varieties and categorification. Math. Ann. 357(3), 805–854 (2013). arXiv:1104.0352 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cautis, S., Kamnitzer, J., Morrison, S.: Webs and quantum skew Howe duality. Math. Ann. 360(1), 351–390 (2014). arXiv:1210.6437 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cautis, S., Koppensteiner, C.: Exotic t-structures and actions of quantum affine algebras. arXiv:1611.02777
  16. 16.
    Cautis, S., Licata, A.: Vertex operators and 2-representations of quantum affine algebras. arXiv:1112.6189
  17. 17.
    Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and \({\mathfrak{sl}} _2\)-categorification. Ann. Math. 167, 245–298 (2008). arXiv:math.RT/0407205 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups III. Quantum Topol. 1(1), 1–92 (2010). arXiv:0807.3250 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mirković, I., Vybornov, M.: On quiver varieties and affine Grassmannians of type A. C. R. Math. Acad. Sci. Paris 336(3), 207–212 (2003). arXiv:math.AG/0206084 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mackaay, M., Webster, B.: Categorified skew Howe duality and comparison of knot homologies. arXiv:1502.06011
  21. 21.
    Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Am. Math. Soc. 14(1), 145–238 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Nandakumar, V., Zhao, G.: Categorification via blocks of modular representations of \({\mathfrak{sl}}_n\). arXiv:1612.06941
  23. 23.
    Queffelec, H., Rose, D.: The \({\mathfrak{sl}}_n\) foam 2-category: a combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality. Adv. Math. 302, 1251–1339 (2016). arXiv:1405.5920 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Riche, S.: Koszul duality and modular representations of semi-simple Lie algebras. Duke Math. J. 154(1), 31–134 (2010). arXiv:0803.2076 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Rose, D., Tubbenhauer, D.: Symmetric webs, Jones–Wenzl recursions and q-Howe duality. Int. Math. Res. Not. 17, 5249–5290 (2016). arXiv:1501.00915 MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rouquier, R.: 2-Kac–Moody Algebras. arXiv:0812.5023
  27. 27.
    Toledano Loredo, V.: A Kohno–Drinfeld theorem for quantum Weyl groups. Duke Math. J. 112, 421–451 (2002)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Tubbenhauer, D., Vaz, P., Wedrich, P.: Super q-Howe duality and web categories. arXiv:1504.05069

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations