Israel Journal of Mathematics

, Volume 212, Issue 2, pp 635–676 | Cite as

Generalised Jantzen filtration of exceptional Lie superalgebras

  • Yucai SuEmail author
  • R. B. ZhangEmail author


Let g be an exceptional Lie superalgebra, and let p be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic Verma module in the parabolic category Op, it is shown that the Jantzen filtration is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan–Lusztig polynomials. An explicit description of the submodule lattices of the parabolic Verma modules is given, and formulae for characters and dimensions of the finite-dimensional simple modules are obtained.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. H. Andersen, Filtrations of cohomology modules for Chevalley groups, Annales Scientifiques de l’École Normale Supérieure 16 (1983), 495–528MathSciNetzbMATHGoogle Scholar
  2. [2]
    H. H. Andersen, Jantzen’s filtrations of Weyl modules, Mathematische Zeitschrift 194 (1987), 127–142.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. A. Beilinson and J. Bernstein, Localisation de g-modules, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 292 (1981), 15–18.MathSciNetzbMATHGoogle Scholar
  4. [4]
    A. A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, in I. M. Gel'fand Seminar, Advances in Soviet Mathematics, Vol. 16, American Mathematical Society, Providence, RI, 1993, pp. 1–50.Google Scholar
  5. [5]
    B. D. Boe and D. H. Collingwood, Multiplicity free categories of highest weight representations. I, II, Communications in Algebra 18 (1990), 947–1032, 1033–1070.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Brundan, Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n), Journal of the American Mathematical Society 16 (2003), 185–231.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Brundan and C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup, Journal of the European Mathematical Society 14 (2012), 373–419.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J.-L. Brylinski and M. Kashiwara, Kazhdan–Lusztig conjecture and holonomic systems, Inventiones Mathematicae 64 (1981), 387–410.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S. J. Cheng and N. Lam, Irreducible characters of general linear superalgebra and super duality, Commoncations in Mathematical Physics 298 (2010), 645–672.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. J. Cheng, N. Lam and W. Wang, Super duality and irreducible characters of orthosymplectic Lie superalgebras, Inventiones Mathematicae 183 (2011), 189–224.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. J. Cheng, W. Wang and R. B. Zhang, Super duality and Kazhdan–Lusztig polynomials, Transactions of the American Mathematical Society 360 (2008), 5883–5924.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S. J. Cheng and R. B. Zhang, Analogue of Kostant’s u-cohomology formula for the general linear superalgebra, International Mathematics Research Notices (2004), 31–53.Google Scholar
  13. [13]
    D. H. Collingwood, R. S. Irving and B. Shelton, Filtrations on generalized Verma modules for Hermitian symmetric pairs, Journal für die Reine und Angewandte Mathematik 383 (1988), 54–86.MathSciNetzbMATHGoogle Scholar
  14. [14]
    P. Fiebig, Centers and translation functors for the category O over Kac–Moody algebras, Mathematische Zeitschrift 243 (2003), 689–717.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    O. Gabber and A. Joseph, Towards the Kazhdan–Lusztig conjecture, Annales Scientifiques de l’École Normale Supérieure 14 (1981), 261–302.MathSciNetzbMATHGoogle Scholar
  16. [16]
    M. Gorelik, Strongly typical representations of the basic classical Lie superalgebras, Journal of the American Mathematical Society 15 (2002), 167–184.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Gorelik and V. G. Kac, Characters of (relatively) integrable modules over affine Lie superalgebras, Japanese Journal of Mathematics 10 (2015), 135–235.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    C. Gruson and V. Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proceedings of the London Mathematical Society 101 (2010), 852–892.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 2008.Google Scholar
  20. [20]
    R. S. Irving, A filtered category O S and applications (and List of Errata), Memoirs of the American Mathematical Societty 83 (1990).Google Scholar
  21. [21]
    J. C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Mathematische Annalen 226 (1977), no. 1, 53–65.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, Vol. 750, Springer, Berlin, 1979.Google Scholar
  23. [23]
    V. G. Kac, Lie superalgebras, Advances in Mathematics 26 (1977), 8–96.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    V. G. Kac, Characters of typical representations of classical Lie superalgebras, Communications in Algebra 5 (1977), 889–897.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    V. G. Kac, Representations of classical Lie superalgebras, in Differrential Geometrical Methods in Mathematical Physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Lecture Notes in Mathematics, Vol. 676 Spirnger, Berlin, 1978, pp. 597–626.CrossRefGoogle Scholar
  26. [26]
    V. G. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, in Lie Theory and Geometry, Progress in Mathematics, Vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 415–456.CrossRefGoogle Scholar
  27. [27]
    D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones Mathematicae 53 (1979), 165–184.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    L. Martirosyan, The representation theory of the exceptional Lie superalgebras F(4) and G(3), Journal of Algebra 419 (2014), 167–222.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    M. Scheunert, The theory of Lie Superalgebras. An Introduction, Lecture Notes in Mathematics, Vol. 716, Springer, Berlin, 1979.Google Scholar
  30. [30]
    V. Serganova, Kazhdan–Lusztig polynomials and character formula for the Lie superalgebra gl(m|n), Selecta Mathematica 2 (1996), 607–654.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    W. Soergel, Andersen filtration and hard Lefschetz, Geometric and Functional Analysis 17 (2008), 2066–2089.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    C. Stroppel, Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Mathematica 145 (2009), 954–992.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Y. Su, J. W. B. Hughes and R. C. King, Primitive vectors in the Kac-module of the Lie superalgebra sl(m|n), Journal of Mathematical Physics 41 (2000), 5044–5087.MathSciNetGoogle Scholar
  34. [34]
    Y. Su and R. B. Zhang, Cohomology of Lie superalgebras slm|n and osp2|2n, Proceedings of the London Mathematical Society 94 (2007), 91–136.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Y. Su and R. B. Zhang, Character and dimension formulae for general linear superalgebra, Advances in Mathematics 211 (2007), 1–33.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Y. Su and R. B. Zhang, Generalised Jantzen filtration of Lie superalgebras. I, Journal of the European Mathematical Society 14 (2012), 1103–1133.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    Y. Su and R. B. Zhang, Generalised Verma modules for the orthosymplectic Lie superalgebra osp k|2, Journal of Algebra 357 (2012), 94–115.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    J. Van der Jeugt, Irreducible representations of the exceptional Lie superalgebras D(2, 1; α), Journal of Mathematical Physics 26 (1985), 913–924.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    J. Van der Jeugt and R. B. Zhang, Characters and composition factor multiplicities for the Lie superalgebra gl(m/n), Letters in Mathematical Physics 47 (1999), 49–61.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    Y. M. Zou, Categories of finite-dimensional weight modules over type I classical Lie superalgebras, Journal of Algebra 180 (1996), 459–482.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiChina
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

Personalised recommendations