Deligne-Lusztig theory investigates representations of finite reductive groups through geometric methods, particularly l-adic cohomology. This text focuses on the SL2(Fq) group, which allows it the simplicity required for a complete description of the theory.

Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provides the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects.

The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and Representations of SL2(Fq) is based upon this example, and extends it to modular representation theory. To this end, the author makes use of fundamental results of l-adic cohomology. In order to efficiently use this machinery, a precise study of the geometric properties of the action of SL2(Fq) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups.

At the end of the text, a succinct overview (without proof) of Deligne-Lusztig theory is given, as well as links to examples demonstrated in the text. With the provision of both a gentle introduction and several recent materials (for instance, Rouquier's theorem on derived equivalences of geometric nature), this book will be of use to graduate and postgraduate students, as well as researchers and lecturers with an interest in Deligne-Lusztig theory.

Presents an introduction to ordinary and modular Deligne-Lusztig theory through the detailed study of an example: SL2(Fq) Will serve as a complimentary text to existing titles on Deligne-Lusztig theory Includes Rouquier's theorem on derived equivalences of geometric nature (with some unpublished improvements) but is simple enough to be read by graduate/Ph. D. students Includes supplementary material: sn.pub/extras

Autorentext

Cédric Bonnafé defended his PhD entitled "Lusztig functors for the special linear group over a finite field" in 1996, under the supervision of Jean Michel. After two years as a Dickson Instructor in the University of Chicago, he entered the CNRS (France) as a researcher in Besançon. He is now senior researcher in Montpellier.

Inhalt
Part I Preliminaries.- Structure of SL2(Fq).- The Geometry of the Drinfeld Curve.- Part II Ordinary Characters.- Harish-Chandra Induction.- Deligne-Lusztig Induction.- The Character Table.- Part III Modular Representations.- More about Characters of G and of its Sylow Subgroups.- Unequal Characteristic: Generalities.- Unequal Characteristic: Equivalences of Categories.- Unequal Characteristic: Simple Modules, Decomposition Matrices.- Equal Characteristic.- Part IV Complements.- Special Cases.- Deligne-Lusztig Theory: an Overview.- Part V Appendices.- A l-Adic Cohomology.- B Block Theory.- C Review of Reflection Groups.