Generic Blocks of Finite Reductive Groups

To Charlie Curtis Contents 0. Introduction 1. Notation, prerequisites and complements A. Generic nite reductive groups Usual invariants of a generic nite reductive group Class functions on G d{split Levi subgroups B. Generic characters Some consequences of Lusztig's results Generic unipotent characters, uniform functions The generic Deligne{Lusztig induction and restriction 2. d{cuspidality and the uniform theory A. d{cuspidality B. Regular unipotent characters C. Regular characters Introduction: the regular character of E (G F ; 1) The generic formalism D. First application to actual nite reductive groups 3. Generalized Harish{Chandra theory A. The fundamental theorem B. d{Harish{Chandra theory for unipotent characters 4. Generic d-blocks A. More on regular unipotent characters and Mackey formula This work has been motivated by two problems, which interacted with one another: attempts to solve one helped to solve the other one. The rst one originated in Br1], where is presented a conjecture about the structure of any block with abelian defect of any nite group (see Br1], 6.1), which states that such a block should have the same type as the corresponding block of the normalizer of its defect group. One of our goals was to check this conjecture in the case of the nite reductive groups G F (we denote by G a connected reductive algebraic group over an algebraic closure of a nite eld F q , by F : G ! G a Frobenius endomorphism deening a rational structure on this nite eld, and by G F the group of rational points). This is achieved in the present work (see x5.C, 5.24), at least for \large" prime numbers (i.e., in the split case, prime numbers which do not divide the order of the Weyl group of G). By general properties of isotypies (see Br1]), our results imply in particular that, for large prime numbers, conjectures as Alperin's weight conjecture or the Alperin{McKay height conjecture are veriied for unipotent blocks (and so for \almost" all blocks) of all nite reductive groups. The second question which motivated this work was to understand better the \generic" aspects of block theory of nite reductive groups. Throughout the intensive work which has been done recently by many authors about blocks of nite reductive groups (among whom Fong{Srinivasan, Schewe, Cabanes{Enguehard, Hii, Geck, and the authors), it had become gradually clear that, for large prime numbers`, properties of`{blocks of G F do not really depend on the prime number`, but …