Deligne-Lusztig Theory

Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

13-17 August 2018
University of Oregon
Eugene, OR

Expected background and suggested reading

For this WARTHOG workshop to be effective, we decided that students should be familiar with the basics of algebraic groups and reductive groups before the workshop began. Similarly, we hope students are familiar with the basics of Coxeter groups. Do not worry if you do not have this prerequisite yet, it should not be difficult to read up before the workshop begins. Below we give suggested readings. For these two topics, the workshop will recall the needed facts and give examples — but we cannot expect anyone who's never seen an algebraic group to be able to follow most of the lectures.

The expected background for attendees of this year's WARTHOG is:

  • Finite groups and their representation theory over the complex numbers.
  • Algebraic groups and reductive groups.
  • Coxeter groups.

    The following topics are advantageous to know, but not necessary:

  • Basic algebraic geometry (e.g. affine and projective varieties, morphisms).
  • Familiarity with cohomology (long exact sequence of cohomology, cohomology of simple spaces like projective spaces, the circle etc).
  • Homological algebra: modules over a f.d. algebras, homotopy and derived categories.
  • Notions in the representation theory of finite groups over a field with positive characteristic (e.g. blocs and defect groups)
    In particular, these last two topics will be useful primarily for the final day of lectures, when the workshop branches out into modular representation theory.

    For the purpose of acquiring a prior knowledge of algebraic groups and Coxeter groups, we recommend skimming through one of the following chapters/surveys:

  • The first 3 chapters of Jay Taylor's monograph. (Note: spoilers are contained in the following chapters.)
  • The first part of
    [MT] G. Malle, D. Testerman, Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011.
  • The first 4 chapters of these lecture notes by Michel and Dudas.
  • Chapters I.1 to I.11 of [C] (see below).

    We will be setting up a Googlegroup for this workshop, and we hope this will be a useful forum to ask questions or find online study groups.

    Notes on further pre-reading

    Even though the topic of this workshop is inherently geometric (Deligne-Lusztig varieties), we will not need much algebraic geometry. We may point out some beautiful theorems using the language of algebraic geometry (e.g. smoothness, affineness) but we will not use them, so it is not worth trying to give yourself a crash course. In particular, we will not use the language of sheaves.

    It will probably help having seen beforehand the definition of the Iwahori-Hecke algebra associated with a Coxeter group. However, there is no need to learn about Kazhdan-Lusztig theory. One might try sections 7.1-7.4 of [H] (see below).

    Of course, it is always advantageous to have seen in advance the more specific aspects of the workshop (e.g. one might read about flag varieties, Deligne-Lusztig varieties, the Lefschetz trace formula, etcetera). However, we will not be assuming any background on these topics. We recommend trying to read ahead only after acquiring the expected background outlined above.

    Additional references

    The content of the lectures will be based on the following papers (but you will not need to read them):

  • [B] M. Broué, Reflection groups, braid groups, Hecke algebras, finite reductive groups. Current developments in mathematics, 2000, 1--107, Int. Press, Somerville, MA, 2001.
  • [BM] M. Broué, J. Michel, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées. (French) [Some regular elements of Weyl groups and the associated Deligne-Lusztig varieties] Finite reductive groups (Luminy, 1994), 73-139, Progr. Math., 141, Birkhäuser Boston, Boston, MA, 1997.
  • [D] O. Dudas, Coxeter orbits and Brauer trees. Adv. Math. 229 (2012), no. 6, 3398--3435
  • [DL] P. Deligne, G. Lusztig, Representations of reductive groups over finite fields. Ann. of Math. (2) 103 (1976), no. 1, 103--161.
  • [L1] G. Lusztig, Coxeter orbits and eigenspaces of Frobenius. Invent. Math. 38 (1976/77), no. 2, 101--159.

    These books contain a lot of material on Deligne--Lusztig theory, but we will not really use them:

  • [C] R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985.
  • [DM] F. Digne, J. Michel, Representations of finite groups of Lie type. London Mathematical Society Student Texts, 21. Cambridge University Press, Cambridge, 1991.
  • [L2] G. Lusztig, Characters of reductive groups over a finite field. Annals of Mathematics Studies, 107. Princeton University Press, Princeton, NJ, 1984.

    The classic reference on Coxeter groups is:

  • [H] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.