Twisted Verma Modules

  • H. H. Andersen
  • N. Lauritzen
Part of the Progress in Mathematics book series (PM, volume 210)


Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approaches lead to the same modules. Moreover, we demonstrate that they possess natural Jantzen type filtrations with corresponding sum formulae.


Exact Sequence Weyl Group Simple Root Short Exact Sequence Verma Module 
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  1. 1.
    Andersen H. H., On the structure of cohomology groups of line bundles onGIB J. Algebra,71 (1981), 245–258MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arkhipov S., A new construction of the semi-infinite BGG resolution, q-alg/9605043.Google Scholar
  3. 3.
    Arkhipov S., Algebraic construction of contragredient quasi-Verma modules in positive characteristicMax Planck Institute, Bonnpreprint (April 2001).Google Scholar
  4. 4.
    Bernstein J., Gelfand I. M., and Gelfand S., Category of g-modulesFunctional Anal. Appl. 10(1976), 87–92.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bernstein J. and Gelfand S., Tensor products of finite and infinite dimensional representations of semisimple Lie algebrasCompositio Math. 41 (1980), 245–285.MathSciNetMATHGoogle Scholar
  6. 6.
    Feigin B. and Frenkel E., Affine Kac—Moody algebras and semi-infinite flag manifoldsComm. Math. Phys. 128 (1990), 161–189.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Irving R., Shuffled Verma modules and principal series modules over complex semisimple Lie algebrasJ. London Math. Soc. 48 (1993), 263–277.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Irving R., The socle filtration of a Verma moduleAnn. Sci. École Norm. Sup. 21 (4) (1988), 47–65.MathSciNetMATHGoogle Scholar
  9. 9.
    Jantzen J. C.Moduln mit einem höchsten GewichtLecture Notes in Mathematics750 (1979), Springer.MATHGoogle Scholar
  10. 10.
    Jantzen J. C.Einhüllende Algebren halbeinfacher Lie-algebrenGrundlehren, (1983), Springer.MATHCrossRefGoogle Scholar
  11. 11.
    Kashiwara M., Kazhdan—Lusztig conjecture for symmetrizable Kac—Moody Lie algebrasThe Grothendieck Festschrift Vol. II (1990), 407–433.MathSciNetGoogle Scholar
  12. 12.
    Kempf G., The Grothendieck—Cousin complex of an induced representationAdv. in Math. 29 (1978), 310–396.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Soergel W., Character formulas for tilting modules over Kac—Moody algebrasRepresent. Theory (electronic) 2 (1998), 432–444.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Stroppel, C.Der Kombinatorikfunktor V. Graduirte Kategorie OHauptserien und Primitive Ideale, October 2001, Universität Freiburg.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • H. H. Andersen
    • 1
  • N. Lauritzen
    • 1
  1. 1.University of AarhusAarhusDenmark

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