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 | A Rapid Review of Etale Cohomology: I will give a rapid review of etale cohomology and work out some examples. For convenience, I will use the language of derived categories, which the previous week's speaker presumably covered comprehensively. |
| 2024.9.24 | Amal Mattoo | Finite Groups of Lie Type: This week, we will talk about finite groups of Lie type. In particular, we will introduce the Lang isogeny, and explore how it can be used to analyze group actions of G. We will also examine the decomposition theory of these groups. |
| 2024.10.1 | Lisa Faulkner Valiente | The Drinfeld Curve: We define the Drinfeld curve and various group actions on it. We then look at isomorphisms from the Drinfeld curve when quotiented by these group actions, as well as fixed points under these actions, all of which will be useful when computing induced characters from Deligne-Lusztig induction. |
| 2024.10.8 | Alan Zhao | The Characters of \(\text{SL}_2(\mathbb{F}_q)\): We can discover the characters of \(\text{SL}_2(\mathbb{F}_q)\) in short formulas as a consequence of simple modifications to the restriction and induction of characters. The goal of this talk will be to spell out these ideas. |
| 2024.10.22 | Vidhu Adhihetty | Deligne-Lusztig Varieties: We construct Deligne-Lusztig varieties for any finite group of Lie type. Then, we discuss some of their geometric properties, and begin to explore how their \(l\)-adic cohomology can by used to construct virtual representations of these groups. |
| 2024.10.29 | Wenqi Li | Deligne-Lusztig Induced Representations: We discuss how the twisted Frobenius used for defining Deligne-Lusztig varieties can be replaced by a standard Frobenius. We then discuss the construction of Deligne-Lusztig representations and why they are independent of varies choices. |
| 2024.11.19 | Amadou Bah | The Character Formula: I will introduce the Deligne-Lusztig character associated with a character \(\theta\) of a maximal torus. The goal is to prove a formula for its trace in terms of \(\theta\) and some Green functions I will define. This so-called character formula implies the independence of the Deligne-Lusztig character vis-a-vis the Borel subgroup used to define it. It has some other interesting consequences I will mention if time permits. |
| 2024.12.03 | Sofia Wood | Disjointness Theorem: We explain the concept of geometric conjugacy for pairs \((T, \theta)\) consisting of an \(F\)-stable maximal torus \(T\) and a character \(\theta\) of \(T\). We show that if \((T,\theta)\) and \((T',\theta')\) are not geometrically conjugate, then \(\langle R^\theta_T, R^{\theta'}_{T'}\rangle = 0\). We also mention that any irreducible representation \(\rho\) of \(G^F\) occurs as \(R^{\theta}\) for some \((T, \theta)\). |