Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 635–659 | Cite as

Modular representations of exceptional supergroups

  • Shun-Jen ChengEmail author
  • Bin Shu
  • Weiqiang Wang


We classify the simple modules of the exceptional algebraic supergroups over an algebraically closed field of prime characteristic.


Exceptional supergroups Simple modules Odd reflections 

Mathematics Subject Classification

Primary 20G05 17B25 



S.-J.C. is partially supported by a MoST and an Academia Sinica Investigator grant; B.S. is partially supported by the National Natural Science Foundation of China (Grant Nos. 11671138, 11771279) and Shanghai Key Laboratory of PMMP (No. 13dz2260400); W.W. is partially supported by an NSF Grant DMS-1702254. We thank East China Normal University and Institute of Mathematics at Academia Sinica for hospitality and support.


  1. 1.
    Brundan, J.: Modular representations of the supergroup \(Q(n)\), II. Pacific J. Math. 224, 65–90 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brundan, J., Kleshchev, A.: Modular representations of the supergroup \(Q(n)\), I. J. Algebra 260, 64–98 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brundan, J., Kujawa, J.: A new proof of the Mullineux conjecture. J. Algebraic Combin. 18, 13–39 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cheng, S.-J., Wang, W.: Dualities and Representations of Lie Superalgebras, Graduate Studies in Mathematics. Am. Math. Soc. 144 (2012)Google Scholar
  5. 5.
    Cheng, S.-J., Wang, W.: Character formulae in category \(\cal{O}\) for exceptional Lie superalgebras \(D(2|1;\zeta )\). Transform. Groups (to appear). arXiv:1704.00846v3
  6. 6.
    Fioresi, R., Gavarini, F.: Chevalley supergroups. Mem. Am. Math. Soc. 215, 1014 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Freund, P., Kaplansky, I.: Simple supersymmetries. J. Math. Phys. 17, 228–231 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gavarini, F.: Chevalley Supergroups of type \(D(2,1;a)\). Proc. Edinb. Math. Soc. 57, 465–491 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jantzen, J.C.: Reresentations of Algebraic Groups, 2nd edn. American Mathematical Society, Providence (2003)Google Scholar
  10. 10.
    Kac, V.: Lie superalgebras. Adv. Math. 16, 8–96 (1977)CrossRefzbMATHGoogle Scholar
  11. 11.
    Leites, D., Saveliev, M., Serganova, V.: Embedding of \( osp (N/2)\) and the associated non-linear supersymmetric equations. Group theoretical methods in physics, Vol. I (Yurmala: 255–297, p. 1986. Press, Utrecht, VNU Sci) (1985)Google Scholar
  12. 12.
    Martirosyan, L.: The representation theory of the exceptional Lie superalgebras \(F(4)\) and \(G(3)\). J. Algebra 419, 167–222 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Masuoka, A.: Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field. Transform. Groups 17, 1085–1121 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Masuoka, A., Shibata, T.: Algebraic supergroups and Harish-Chandra pairs over a commutative ring. Trans. Am. Math. Soc. 369, 3443–3481 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Masuoka, A., Zubkov, A.N.: Solvability and nilpotency for algebraic supergroups. J. Pure Appl. Algebra 221, 339–365 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Serganova, V.: Kac-Moody superalgebras and integrability. Developments and Trends in Infinite-Dimensional Lie Theory. Progr. Math. 288, 169–218 (2011). (Birkhäuser)zbMATHGoogle Scholar
  17. 17.
    Shu, B., Wang, W.: Modular representations of the ortho-symplectic supergroups. Proc. London Math. Soc. 96, 251–271 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zubkov, A.N.: On some properties of Noetherian superschemes (Russian), Algebra Logic, to appearGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsAcademia SinicaTaipeiTaiwan
  2. 2.Department of mathematicsEast China Normal UniversityShanghaiChina
  3. 3.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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