A 2025 Seminar organised by Tom Goertzen and Edmund Heng
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 focused 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.
Schedule
The first half of the course was concerned with trying to understand the basics of reductive groups over finite field and their representation theory. In the second half, we studied the main contents of Deligne and Lusztig’s paper and also further topics. The focus was always on examples, especially SL(2,q).
1. Introduction to Deligne-Lusztig Theory
Speaker: Charlotte Chan, The University of Michigan.
26 February 2025
Notes for lecture available for download (Credit: Joe Newton). Link to week one exercises.
2. Algebraic Groups over Finite Fields
Speaker: Finn Klein, University of Sydney
5 March 2025
Notes for lecture available for download. Link to week two exercises.
3. Character Theory and Harish-Chandra Induction
Speaker: Joseph Newton, University of Sydney
12 March 2025
Notes for lecture available for download. Link to week three exercises.
4. l-adic Cohomology
Speaker: Chris Hone, University of Sydney
19 March 2025
Notes for lecture available for download. Link to week four exercises.
5. Flag Varieties and Deligne-Lusztig Varieties
Speaker: Sam Jeralds, University of Sydney
19 March 2025
Notes and exercises for lecture available for download. Hints for the last exercise can be found in section 4 of these notes.
6. Deligne-Lusztig Induction
Speaker: Nick Bridger, University of Sydney
2 April 2025
Notes for lecture available for download. Link to week six exercises.
7. Parameterisation of Irreducible Representations of Finite Simple Groups of Lie Type
Speaker: Tom Goertzen, University of Sydney
9 April 2025
Notes for lecture available for download. Link to week seven exercises.
8. Character Sheaves
Speaker: Tasman Fell, University of Sydney
9 April 2025
Notes for lecture available for download. Link to week eight exercises.
Resources
- 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
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