Transformation Groups

, Volume 4, Issue 2, pp 157–218

Towards spetses I

  • M. Broué
  • G. Malle
  • J. Michel

DOI: 10.1007/BF01237357

Cite this article as:
Broué, M., Malle, G. & Michel, J. Transformation Groups (1999) 4: 157. doi:10.1007/BF01237357


We present a formalization, using data uniquely defined at the level of the Weyl group, of the construction and combinatorial properties of unipotent character sheaves and unipotent characters for reductive algebraic groups over an algebraic closure of a finite field. This formalization extends to the case where the Weyl group is replaced by a complex reflection group, and in many cases we get families of unipotent characters for a mysterious object, a kind of reductive algebraic group with a nonreal Weyl group, the “spets”.

In this first part, we present the general results about complex reflection groups, their associated braid groups and Hecke algebras, which will be needed later on for properties of “spetses”. Not all irreducible complex reflection groups will give rise to a spets (the ones which do so are called “spetsial”), but all of them afford properties which already allow us to generalize many of the notions attached to the Weyl groups through the approach of “generic groups” (see [BMM1]).

Copyright information

© Birkhäuser 1999

Authors and Affiliations

  • M. Broué
    • 1
  • G. Malle
    • 2
  • J. Michel
    • 3
    • 4
  1. 1.Institut Universitaire de FranceUniversité Paris 7, CNRS UMR 7586, Case 7012Paris Cedex 05France
  2. 2.Fachbereich Mathematik/InformatikUniversität KasselKasselGermany
  3. 3.UMR 7586 du CNRSUniversité Paris 7, Case 7012Paris Cedex 05France
  4. 4.LAMFA, CNRS UPRES-A 6119Amiens Cédex 01France