Japanese Journal of Mathematics

, Volume 7, Issue 1, pp 41–134 | Cite as

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

  • Maria Gorelik
  • Victor G. Kac
  • Pierluigi Möseneder Frajria
  • Paolo Papi
Original Articles


We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups, and, as an application of these formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.

Keywords and phrases

Lie superalgebra denominator identity arc diagram Howe duality 

Mathematics Subject Classification (2010)

17B20 22E46 05E10 


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  1. 1.
    J. Adams, The theta correspondence over \({\mathbb {R}}\) , In: Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory, Lecture Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 1–37.Google Scholar
  2. 2.
    Enright T.J.: Analogues of Kostant’s \({\mathfrak {u}}\) -cohomology formulas for unitary highest weight modules. J. Reine Angew. Math. 392, 27–36 (1988)MathSciNetMATHGoogle Scholar
  3. 3.
    Enright T.J., Willenbring J.F.: Hilbert series, Howe duality and branching for classical groups. Ann. of Math. (2) 159, 337–375 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    M. Gorelik, Weyl denominator identity for finite-dimensional Lie superalgebras, arXiv:0905.1181.Google Scholar
  5. 5.
    R. Howe, θ-series and invariant theory, In: Automorphic Forms, Representations and L-Functions. Part 1, Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, RI, 1979, pp. 275–285.Google Scholar
  6. 6.
    Howe R.: Remarks on classical invariant theory. Trans. Amer. Math. Soc. 313, 539–570 (1989)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Howe R.: Trascending classical invariant theory. J. Amer. Math. Soc. 2, 535–552 (1989)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Iohara K., Koga Y.: Central extension of Lie superalgebras. Comment. Math. Helv. 76, 110–154 (2001)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kac V.G.: Lie superalgebras. Advances in Math. 26, 8–96 (1977)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    V.G. Kac, Representations of classical Lie superalgebras, In: Differential Geometrical Methods in Mathematical Physics. II, Proc. Conf., Univ. Bonn, Bonn, 1977, Lecture Notes in Math., 676, Springer-Verlag, 1978, pp. 597–626.Google Scholar
  11. 11.
    Kac V.G.: Infinite-Dimensional Lie Algebras. Third ed.. Cambridge Univ. Press, Cambridge (1990)CrossRefGoogle Scholar
  12. 12.
    Kac V.G., Möseneder Frajria P., Papi P.: On the kernel of the affine Dirac operator. Mosc. Math. J. 8, 759–788 (2008)MathSciNetMATHGoogle Scholar
  13. 13.
    V.G. Kac, P. Möseneder Frajria and P. Papi, Denominator identities for Lie superalgebras (extended abstract), Proceedings of the FPSAC 2010, 839–850.Google Scholar
  14. 14.
    V.G. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, In: Lie Theory and Geometry, Progr. Math., 123, Birkhäuser Boston, Boston, MA, 1994, pp. 415–456.Google Scholar
  15. 15.
    Kashiwara M., Vergne M.: On the Segal–Shale–Weil representations and harmonic polynomials. Invent. Math. 44, 1–47 (1978)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kostant B.: Lie algebra cohomology and the generalized Borel–Weil theorem. Ann. of Math. (2) 74, 329–387 (1961)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Li J.-S., Paul A., Tan E.-C., Zhu C.-B.: The explicit duality correspondence of (Sp(p, q), O*(2n)). J. Funct. Anal. 200, 71–100 (2003)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Parker M.: Classification of real simple Lie superalgebras of classical type. J. Math. Phys. 21, 689–697 (1980)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Penkov I., Serganova V.V.: Representations of classical Lie superalgebras of type I. Indag. Math. (N.S.) 3, 419–466 (1992)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Serganova V.V.: Automorphisms of simple Lie superalgebras. Math. USSR-Izv. 24, 539–551 (1985)MATHCrossRefGoogle Scholar
  21. 21.
    Serganova V.V.: Classification of simple real Lie superagebras and symmetric superspaces. Funktsional. Anal. i Prilozhen. 17(3), 46–54 (1983)MathSciNetGoogle Scholar
  22. 22.
    van de Leur J.W.: A classification of contragredient Lie superalgebras of finite growth. Comm. Algebra 17, 1815–1841 (1989)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2012

Authors and Affiliations

  • Maria Gorelik
    • 1
  • Victor G. Kac
    • 2
  • Pierluigi Möseneder Frajria
    • 3
  • Paolo Papi
    • 4
  1. 1.Department of Mathematics, Faculty of Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Department of MathematicsMITCambridgeUSA
  3. 3.Politecnico di Milano, Polo regionale di ComoComoItaly
  4. 4.Dipartimento di MatematicaSapienza Università di RomaRomaItaly

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