Deligne-Lusztig Theory: an Overview*

  • Cédric BonnaféEmail author
Part of the Algebra and Applications book series (AA, volume 13)


This chapter gives a very succinct overview of Deligne-Lusztig theory. We will recall some of the principal results of the theory (including the parametrisation of characters and partition into blocks) with the goal of connecting this general theory with what we have seen for \({\mathrm{SL}}_{2}({{\mathbb{F}}_{\!q}})\). This chapter requires some knowledge of algebraic groups, for which we refer the reader to Borel 1991 or Digne and Michel 1991. For more details on the subjects covered in this chapter the reader is referred to the books Lusztig 1978, Carter 1985, Digne and Michel 1991 or Cabanes and Enguehard 2004. We also recommend the magnificent and foundational article on the subject, written by Deligne and Lusztig in 1976.


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  1. [Bor]
    A. Borel, Linear algebraic groups, deuxième édition, Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991, xii + 288pp. zbMATHGoogle Scholar
  2. [CaEn]
    M. Cabanes & M. Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs 1, Cambridge University Press, Cambridge, 2004, xviii + 436pp. zbMATHGoogle Scholar
  3. [Carter]
    R.W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Pure and Applied Mathematics, John Wiley & Sons (New York), 1985, xii + 544pp. zbMATHGoogle Scholar
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    P. Deligne & G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103–161. CrossRefMathSciNetGoogle Scholar
  5. [DiMi]
    F. Digne & J. Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts 21, Cambridge University Press, 1991, iv + 159pp. Google Scholar
  6. [Lu1]
    G. Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference (Madison 1977), CBMS Regional Conference Series in Mathematics 39, American Mathematical Society, Providence, 1978, v + 48pp. zbMATHGoogle Scholar

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© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Institut de Mathématiques et de Modélisation de MontpellierCNRS (UMR 5149), Université Montpellier 2Montpellier CedexFrance

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