Deligne-Lusztig Theory
at the University of Sydney
Seminar meeting: Wednesdays 10-12
Exercise sessions: Wednesdays 13-14
First meeting: February 26 2025
at the University of Sydney
Seminar meeting: Wednesdays 10-12
Exercise sessions: Wednesdays 13-14
First meeting: February 26 2025
In their groundbreaking 1976 paper, entitled "Representations of Reductive Groups Over Finite Fields", Deligne and Lusztig (DL) gave a method to construct representation of reductive groups using l-adic cohomology, generalising parabolic induction. Later, this theory was used by Lusztig to find all representations of all finite simple groups of Lie type.
This seminar will focus on learning about the main ingredients and recipes to construct and describe these representations: reductive groups over finite fields, Frobenius maps, character theory, Harish-Chandra induction, l-adic cohomology, flag varieties, DL varieties, DL induction, character formulas, Lusztig series and more.
The seminar will meet on Wednesdays from 10-12 in the SMRI seminar room. There will be a weekly exercise session on Wednesdays 1-2pm, again in the SMRI seminar room.
The following is a tentative schedule. The rough plan: first half of the course we will try to understand the basics of reductive groups over finite field and their representation theory. In the second half we will study the main contents of Deligne and Lusztig's paper and also further topics. The focus will always be on examples, especially SL(2,q).
February 26: Introduction to Deligne-Lusztig Theory (Charlotte Chan, University of Michigan) (Notes by Joe Newton)
March 5: Algebraic Groups over Finite Fields (Finn Klein)
March 12: Character Theory and Harish-Chandra Induction (Joe Newton)
March 19: l-adic Cohomology (Chris Hone)
March 26: Flag Varieties and Deligne-Lusztig Varieties (Sam Jeralds)
April 2: Deligne-Lusztig Induction (Nick Bridger)
April 9: Parameterisation of Irreducible Representations of Finite Simple Groups of Lie Type (Tom Goertzen)
April 16: Character Sheaves (Tasman Fell)
Week five (Hints for the last exercise can be found in section 4 of these notes)
Representations of Reductive Groups Over Finite Fields (1976 original paper by Deligne and Lusztig)
Representations of Finite Groups of Lie Type (2nd edition, 2020, Digne and Michel)
The Character Theory of Finite Groups of Lie Type (2020, Geck and Malle)
Representations of SL(2,Fq) (2009, Bonnafé)
Finite groups of Lie type (1985, Carter)
In case you missed the introductory lecture by Charlotte, she gave a short undergraduate level introductory course at the IAS last year (click link for the first of four videos)
Two more introductory lectures on DL-Theory for SL(2,Fq) by Bonnafé can be found here and here.
Bonnafé gave a course in 2022 at Tsinghua University, you can find the lecture notes and videos here.
See also this nice mathoverflow thread for more interesting resources
Disclaimer: This website contains links to external sites for your convenience. We do not endorse or take responsibility for the content, accuracy, or privacy practices of these sites. Use them at your own discretion.