Character, Multiplicity, and Decomposition Problems in the Representation Theory of Complex Lie Algebras

  • Peter FiebigEmail author
Part of the UNIPA Springer Series book series (USS)


This essay is meant as an introduction to a very successful approach towards understanding the structure of certain highest weight categories appearing in algebraic Lie theory. For simplicity, the focus lies on the case of the category \(\mathcal{O}\) of representations of a simple complex Lie algebra. We show how the approach yields a proof of the classical Kazhdan–Lusztig conjectures that avoids the theory of D-modules on flag varieties.



The author was partially supported by the DFG grant SP1388.


  1. 1.
    H.H. Andersen, J.C. Jantzen, W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p. Astérisque 220, 321 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    A. Beilinson, J. Bernstein, Localisation de g-modules. C. R. Acad. Sci. Paris Sér. I 292, 15–18 (1981)MathSciNetzbMATHGoogle Scholar
  3. 3.
    T. Braden, R. MacPherson, From moment graphs to intersection cohomology. Math. Ann. 321(3), 533–551 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J.-L. Brylinski, M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math. 64, 387–410 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    B. Elias, G. Williamson, The Hodge theory of Soergel bimodules. Ann. Math. (2) 180(3), 1089–1136 (2014)Google Scholar
  6. 6.
    P. Fiebig, Centers and translation functors for category \(\mathcal{O}\) over Kac-Moody algebras. Math. Z. 243(4), 689–717 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    P. Fiebig, The combinatorics of category \(\mathcal{O}\) over symmetrizable Kac-Moody algebras. Transform. Groups 11(1), 29–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    P. Fiebig, Sheaves on moment graphs and a localization of Verma flags. Adv. Math. 217, 683–712 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P. Fiebig, The combinatorics of Coxeter categories. Trans. Am. Math. Soc. 360, 4211–4233 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    P. Fiebig, Sheaves on affine Schubert varieties, modular representations and Lusztig’s conjecture. J. Am. Math. Soc. 24, 133–181 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    P. Fiebig, Moment graphs in representation theory and geometry (2013, preprint). arXiv:1308.2873Google Scholar
  12. 12.
    J.E. Humphreys, Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29 (Cambridge University Press, Cambridge, 1990)Google Scholar
  13. 13.
    J.E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category \(\mathcal{O}\). Graduate Studies in Mathematics, vol. 94 (American Mathematical Society, Providence, RI, 2008)Google Scholar
  14. 14.
    J.C. Jantzen, Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics, vol. 750 (Springer, New York, 1979)Google Scholar
  15. 15.
    D. Juteau, C. Mautner, G. Williamson, Parity sheaves. J. Am. Math. Soc. 27(4), 1169–1212 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    D. Kazhdan, G. Lusztig, Schubert varieties and Poincaré duality, in Geometry of the Laplace operator (Proceedings of Symposia in Pure Mathematics) (University of Hawaii, Honolulu, 1979), pp. 185–203Google Scholar
  18. 18.
    W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic. J. Pure Appl. Algebra 152(1–3), 311–335 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    W. Soergel, Kategorie \(\mathcal{O}\), perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Am. Math. Soc. 3(2), 421–445 (1990)zbMATHGoogle Scholar
  20. 20.
    W. Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen. J. Inst. Math. Jussieu 6(3), 501–525 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    G. Williamson; with a joint appendix with Alex Kontorovich and Peter J. McNamara. Schubert calculus and torsion explosion. J. Am. Math. Soc. 30, 1023–1046 (2017)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department MathematikFAU Erlangen–NürnbergErlangenGermany

Personalised recommendations