Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

13-17 August 2018

University of Oregon

Eugene, OR

For this WARTHOG workshop to be effective, we decided that students should be familiar with the **basics** of algebraic groups
and reductive groups before the workshop began. Similarly, we hope students are familiar with the **basics** of Coxeter groups. **Do not worry** if you do not have this prerequisite **yet**, it should not be difficult
to read up before the workshop begins. Below we give suggested readings. For these two topics, the workshop will recall the needed facts and give examples — but we cannot expect anyone who's never seen an algebraic group to be able to follow most of the lectures.

The expected background for attendees of this year's WARTHOG is:

The following topics are advantageous to know, but not necessary:

In particular, these last two topics will be useful primarily for the final day of lectures, when the workshop branches out into modular representation theory.

For the purpose of acquiring a prior knowledge of algebraic groups and Coxeter groups, we recommend skimming through **one of** the following chapters/surveys:

[MT] G. Malle, D. Testerman, Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011.

We will be setting up a Googlegroup for this workshop, and we hope this will be a useful forum to ask questions or find online study groups.

Even though the topic of this workshop is inherently geometric (Deligne-Lusztig varieties), we will not need much algebraic geometry. We may point out some beautiful theorems using the language of algebraic geometry (e.g. smoothness, affineness) but we will not use them, so it is not worth trying to give yourself a crash course. In particular, we will not use the language of sheaves.

It will probably help having seen beforehand the definition of the Iwahori-Hecke algebra associated with a Coxeter group. However, there is no need to learn about Kazhdan-Lusztig theory. One might try sections 7.1-7.4 of [H] (see below).

Of course, it is always advantageous to have seen in advance the more specific aspects of the workshop (e.g. one might read about flag varieties, Deligne-Lusztig varieties, the Lefschetz trace formula, etcetera). However, we will not be assuming any background on these topics. We recommend trying to read ahead only after acquiring the expected background outlined above.

The content of the lectures will be based on the following papers (but you will not need to read them):

These books contain a lot of material on Deligne--Lusztig theory, but we will not really use them:

The classic reference on Coxeter groups is: