This course will be about the representation theory of SL2(Fq). More precisely, we introduce the representation theory of finite groups of Lie type, and in particular the theory of Deligne–Lusztig, in the specific example of SL2(Fq). Along the way, we will learn a little étale cohomology and modular representation theory, among other things.
Instructor: Patrick Allen
Email:
Time and location: MWF, 2:35 – 3:25 in 1234 BURN.
Office hours: MF 1:30 – 2:30 in 1107 BURN.
Prerequisites: Graduate level algebra. Some experience with algebraic geometry and homological algebra will be useful.
Evaluation: There will be homeowork and a final presentation.
We will closely follow Cédric Bonnafé's book, Representations of SL2(Fq).
The orginial article of Deligne and Lusztig is Representations of reductive groups over finite fields. For a quick overview of the above article, Serre gave a Bourbaki seminar on it, Représentations linéaires des groupes finis «algébriques».
For basics in representation theory, the book of Fulton and Harris, Representation Theory, a first course is a clasic. For algebraic geometry, two of my favourite references are The rising sea by Vakil and Algebraic geometry Robin Hartshorne (these are both overkill for what we need, but you should probbably learn algebraic geometry anyway). For étale cohomology, James Milne has lecture notes, Lectures on Étale cohomology, and a book Étale cohomology.
I'll add more references if/when I think it might be useful.
Homework 1. Due October 8.
Aug 31: Introduction. Notes.
Sept 2: Subgroups of SL2(Fq). Notes.
Sept 4: Conjugacy classes in SL2(Fq). Notes.
Sept 9: Sylow subgroups of SL2(Fq). The Drinfeld curve. Notes.
Sept 11: Quotient of varieties by finite group actions. Notes.
Sept 14: Quotient of the Drinfeld curve. Notes.
Sept 16: More quotients of the Drinfeld curve. Notes.
Sept 18: The Grothendieck group and operations with bimodules. Notes.
Sept 21: Harish Chandra induction. (Unfortunately, the notes weren't saved properly. Sorry!)
Sept 23: Finishing up Harish Chandra induction. Recap on cohomology. Notes.
Sept 25: Cohomology of sheaves, sites, and étale maps. Notes.
Sept 28: Étale cohomology. Notes.
Sept 30: Properties of étale cohomology. Notes.
Sept 30: Properties of étale cohomology. Notes.
Oct 2: Étale cohomology examples. Starting Deligne–Lusztig induction. Notes.
Oct 7: Deligne–Lusztig induction. Notes.