Transformation Groups

, Volume 16, Issue 1, pp 71–89 | Cite as

PBW filtration and bases for irreducible modules in type A n

  • Evgeny FeiginEmail author
  • Ghislain Fourier
  • Peter Littelmann


We study the PBW filtration on the highest weight representations V(λ) of \( \mathfrak{s}{\mathfrak{l}_{n + 1}} \). This filtration is induced by the standard degree filtration on \( {\text{U}}\left( {{\mathfrak{n}^{-} }} \right) \). We give a description of the associated graded \( S\left( {{\mathfrak{n}^{-} }} \right) \)-module gr V(λ) in terms of generators and relations. We also construct a basis of gr V(λ). As an application we derive a graded combinatorial character formula for V(λ), and we obtain a new class of bases of the modules V(λ) conjectured by Vinberg in 2005.


Positive Root Simple Root Total Order Vertex Operator Algebra High Weight Vector 
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  1. [ABS]
    F. Ardila, T. Bliem, D. Salazar, Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes, arXiv:1008.2365.Google Scholar
  2. [B]
    N. Bourbaki, Groupes et Algèbres de Lie, Chaps. IV, V, VI, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968. Russian transl.: Н. Бурбаки, Груnnы u алгебры Лu, гл. IV, V, VI, Мир, M., 1972.Google Scholar
  3. [Br]
    R.-K. Brylinski, Limits of weight spaces, Lusztig's q-analogs and fiberings of adjoint orbits, J. Amer. Math. Soc. 2 (1989), no. 3, 517–533.zbMATHMathSciNetGoogle Scholar
  4. [F1]
    E. Feigin, The PBW filtration, Represent. Theory 13 (2009), 165–181.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [F2]
    E. Feigin, The PBW filtration, Demazure modules and toroidal current algebras, SIGMA 4 (2008), 070, 21 pp.MathSciNetGoogle Scholar
  6. [FFJMT]
    B. Feigin, E. Feigin, M. Jimbo, T. Miwa, Y. Takeyama, A ϕ1,3-filtration on the Virasoro minimal series M(p, p) with 1 < p=p < 2, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 213–257.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [FFL]
    B. Feigin, E. Feigin, P. Littelmann, Zhu's algebras, C 2-algebras and abelian radicals, arXiv:0907.3962 (2009).Google Scholar
  8. [FL]
    E. Feigin, P. Littelmann, Zhu's algebras, C 2-algebras and abelian radicals, arXiv:0907.3962 (2009).Google Scholar
  9. [FH]
    W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1991.zbMATHGoogle Scholar
  10. [GG]
    M. R. Gaberdiel, T. Gannon, Zhu's algebra, the C 2 algebra, and twisted modules, arXiv:0811.3892.Google Scholar
  11. [GT]
    И. М. Гельфанд, М. Л. Цетлин, Конечномерные nредсmавленuя груnnы унuмодулярных маmрuц, ДАН СССР 71 (1950), 825–828. English transl. I. M. Gelfand, M. L. Tsetlin, Finite-dimensional representations of the group of unimodular matrices, I. M. Gelfand, Collected Papers, Vol. II, Springer-Verlag, Berlin, 1988, pp. 653–656.Google Scholar
  12. [H]
    J. E. Humphreys, Introduction to Lie algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York, 1970. Russian transl.: Дж. Хамфрис, Введенuе в mеорuю алгебр Лu u uх nредсмавленuй, МЦНМО, М., 2003.Google Scholar
  13. [K]
    B. Kostant, Lie groups representations on polynomial rings, Amer. J. Math, 85, 327–404 (1963).CrossRefzbMATHMathSciNetGoogle Scholar
  14. [V]
    E. Vinberg, On some canonical bases of representation spaces of simple Lie algebras, Conference Talk, Bielefeld, 2005.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Evgeny Feigin
    • 1
    • 2
    Email author
  • Ghislain Fourier
    • 3
  • Peter Littelmann
    • 3
  1. 1.Department of MathematicsUniversity Higher School of EconomicsMoscowRussia
  2. 2.Lebedev Physics InstituteMoscowRussia
  3. 3.Mathematisches InstitutUniversität zu KölnKölnGermany

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