When and where: Tuesday 5pm - 6pm, in 622
Deligne-Lusztig theory uses \(\ell\)-adic cohomology to construct representations of certain finite groups of Lie type. We will explore the theory through the lens of the Drinfeld curve \(xy^q-yx^q = 1\) and how it is acted on by \(\text{SL}_2(\mathbb{F}_q)\). This fundamental example echoes the development of the subject, and provides a concrete example which contains the central ideas of Deligne-Lusztig theory.
We will be following the book Representations of \(\operatorname{SL}_2(\mathbb{F}_q)\) by Cédric Bonnafé.
There are some prerequisites for the book and the theory in general. The most notable of these from the geometric side is étale cohomology. The main text has a brief appendix on the subject which should suffice for our purposes. Another geometric prerequisite of less importance is the theory of derived categories. The book Representation Theory of Finite Reductive Groups has a short, introductory appendix on the topic which should again suffice for our purposes. Note that this text also serves as a suitable reference for the seminar and Deligne-Lusztig theory in general.
On the representation theoretic side, the essential prerequisite is block theory. The main text contains another concise appendix on the subject which introduces the necessary background. A working knowledge of the character theory of finite groups is recommended, for which there are many resources. A fairly concise reference is the following set of notes by Aaron Landesman from a course he gave at Harvard Notes on Representations of Finite Groups.
| Date | Speaker | Abstract |
|---|---|---|
| 2024.9.3 | N/A | Initial meeting |
| 2024.9.10 | Rafah Hajjar Munoz | An Overview of Derived Categories: This talk introduces the concept of derived categories, focusing on their standard construction as a localization of the homotopy category of an abelian category. We will discuss the notion of derived functors and their key properties, and if time allows, we will see some examples involving the derived category of sheaves on a topological space, which will be the main application of this theory throughout the seminar. |
| 2024.9.17 | Matthew Hase-Liu | TBA |
| 2024.9.24 | Fan Zhou | TBA |