# learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig)

Edit: You may assume knowledge of representation theory of finite groups (as in Serre), algebraic groups and étale cohomology.

• In order of increasing theoretical complexity, I know the following books: Carter's "Finite groups of Lie type", Digne-Michel's "Representations of finite groups of Lie type" (shorter, but uses more homological algebra which may or not be a problem depending on background) and Cabanes-Enguehard "Representation Theory of Finite Reductive Groups", with even more homological algebra. There is also Lecture Notes 764 by B. Srinivasan (closer to Carter's style), and another book whose author I forgot which motivates the general theory using the case of SL_2... – Denis Chaperon de Lauzières Apr 22 '15 at 13:10
• @Denis: The book you mention at the end is by C. Bonnafe, Representations of SL$_2(\mathbb{F}_q)$, Springer, 2011. (This and many other sources are also mentioned in other questions/answers here: search for 'Deligne-Lusztig'.) – Jim Humphreys Apr 22 '15 at 14:20
• @Timo: Srinivasan's early lecture notes may be useful. Probably it's worthwhile to look for Carter's own 1987 lectures following publication of his larger 1985 book (ams.org/mathscinet-getitem?mr=906869), which seem to give a substantial introduction without proofs. However, I haven't seen these notes myself. Most other sources I could cite are much more technical or specialized. – Jim Humphreys Apr 22 '15 at 16:20
• For understanding explicit computations of values, I suggest Lusztig's On the Green Polynomials of Classical Groups. It looks really complicated at first, but after fifty readings it comes together. You get a firm intuition of the characters, and it's a good prelude to the later work of Lusztig and Shoji, even if they are in a different language (so to speak). – Dror Speiser Apr 22 '15 at 22:53
• I found it useful to read Green's original paper on the characters of $GL_n(q)$. It isn't Deligne-Lusztig theory as such but it prefigures it. (And, from what I recall, doesn't need too much background.) – Nick Gill Apr 23 '15 at 12:18

You have given no indication as to your background, so the following imagines you don’t know anything. I have purposely left interesting things out as this is designed to get you from 0 to DL theory. Essentially this is what I would do if I could have my time over again, precisely in this order.

EDIT: Assuming the background specified you can miss out anything labelled with a $\star$. If you wanted to go into less detail then you could also not read Chapter 7 of Carter's book and instead just read Chapter 4 of Geck. However I would advise reading the stuff about Harish-Chandra induction in Digne and Michel before going on to read about Deligne-Lusztig characters. It's good to think of it as generalising HC induction.

Character Theory of Finite Groups

• ($\star$) First 3 chapters of Ledermann’s “Introduction to group characters”. It is short and contains the Mackey formula, which is an important idea later on.
• Chapter 28 of James and Liebeck’s “Representations and characters of groups”. There they compute the generic character table of $\mathrm{GL}_2(q)$ in a completely elementary way. It’s important to have an idea of what you’re trying to achieve with DL theory and it will be a good example to test things against later.

EDIT: As pointed out by Nick Gill, Isaacs' book “Character theory of finite groups” is brilliant if you want to go deeper into character theory but it is more demanding than Ledermann. Serre's “Linear representations of finite groups” is also quite good but again does not go into as much detail as Isaacs and as far as I can remember doesn't do the Mackey formula. I personally like James and Liebeck's book but they again don't do the Mackey formula.

Connected Reductive Algebraic Groups

There are many great books on linear algebraic groups (they almost all bear this name) however I would suggest the following.

• ($\star$) First 2 chapters of Springer’s “Linear algebraic groups (Second Edition)” skipping everything in sections 1.9 and 2.5 (except Lemma 1.9.1, which is very important). The stuff about being defined over an arbitrary field is useful when thinking about finite reductive groups.
• ($\star$) Chapter 3 to 12 of Malle and Testerman’s “Linear algebraic groups and finite groups of Lie type” to get a general overview of the structure theory of connected reductive algebraic groups. It’s reasonably concise and gives you more detail than something like the intro to Digne and Michel or Carter’s book.

Finite Reductive Groups

• Chapter 4 of Geck’s “An introduction to algebraic geometry and algebraic groups” up to section 4.3.10. This is, in my opinion, the most clear and straight forward presentation of Frobenius endomorphisms. It is much less confusing than Chapter 3 of Digne and Michel’s book. You should also see the relationship to Springer’s notions of being defined over an arbitrary ground field. The first chapter of this book is also a more gentle introduction to algebraic geometry than Springer if you wanted it but the more general framework in Springer will serve you well in the future.

Character Theory without Geometry

• Chapters 4 to 9 of Digne and Michel’s “Representations of finite groups of Lie type”. Digne and Michel’s book is often hard work but completely worth it. These sections are very well written and emphasise the Mackey formula, which is very important for the modern view point on the subject. I would, however, advise keeping the errata list close at hand

http://www.lamfa.u-picardie.fr/digne/errata.pdf

• At this point it’s not really necessary but if you are interested in learning more about Harish-Chandra theory then you might want to look at §11 and §67 of Curtis and Reiner’s phenomenal “Methods of representation theory Vol. I and Vol. II“. This will give you a lot of information about the decomposition and behaviour of the Harish-Chandra induction of the trivial character from a maximally split torus. This is the prototype for looking at Deligne—Lusztig characters and the theory of Hecke algebras plays a crucial role in the general theory.

Deligne—Lusztig Characters

• ($\star$)? Chapter 7 in Carter’s “Finite groups of Lie type“ is still, in my opinion, a great reference for learning about the Deligne—Lusztig virtual characters $R_{\mathbf{T}}^{\mathbf{G}}(\theta)$. Comparing with Digne and Michel you should think of the compactly supported cohomology groups being the correct bimodule generalising the Harish-Chandra bimodule.
• Finish reading Chapter 4 of Geck’s book. Similar material is covered there but in a less involved way and he computes the generic character table of ${}^2\textsf{B}_2(q)$ using Deligne—Lusztig characters, which is a great example.
• ($\star$) Another good source is Srinivasan’s “Representations of finite Chevalley groups“. Chapter 5 of this book is a great place to look if you want to understand a little more about the definition of $\ell$-adic cohomology and so forth. She also gives this nice proof in Theorem 5.13 counting the number of F-stable maximal tori using $\ell$-adic cohomology.

Deligne—Lusztig Induction and Restriction

• Chapters 11 and 12 in Digne and Michel’s book. Notice how slick the Mackey formula and Alvis—Curtis duality makes things. For instance, compare the proof of Proposition 12.17 in Digne—Michel to the proof of Theorem 7.5.1 in Carter.
• Chapter 4 of Carter’s book. This will give you a bit more of an idea of duality. This is done quite quickly in Digne—Michel’s book and is one of the more difficult things to wrap your head around, in my opinion.
• Chapter 13 of Digne—Michel to get an idea of Lusztig series and the Jordan decomposition of characters.
• §15.4 of Digne-Michel now gives you a way to construct all the irreducible characters of $\mathrm{GL}_n(q)$ and $\mathrm{U}_n(q)$ as explicit linear combinations of $R_{\mathbf{T}}^{\mathbf{G}}(\theta)$'s and explains the role of the Lusztig series. Look also at the example of $\mathrm{GL}_2(q)$ in §15.9. Go back and compare with the computation in James and Liebeck, especially look at what is going on in Proposition 28.14.

I would say that, at this point, you should really go back and read Deligne—Lusztig’s original paper from 1976. You will be able to truly appreciate just how wonderfully elegant and well written it really is. It is a true highlight of mathematics.

From this point on it’s a bit about what you want to know and what you plan to do with Deligne—Lusztig theory. For information about unipotent characters and their classification look at:

• Carter, “On the representation theory of the finite groups of Lie type over an algebraically closed field of characteristic 0” in Algebra IX. Contains lots of information about the parameterisation of unipotent characters and more about the relationship to intersection cohomology. This is essentially a survey of Lusztig’s orange book.
• Geck, “Finite groups of Lie type”. This is a survey article contained in the book “Representations of reductive groups” which is edited by Carter and Geck. It also contains lots of other nice survey articles. This article by Geck is shorter than that by Carter and gives a bit more in the direction of character sheaves.
• Lusztig, “Representations of finite Chevalley groups“. This is published in the CBMS Regional Conference Series in Mathematics. It comes before the orange book and contains a lot of nice things.

For more hands on stuff with Deligne-Lusztig varieties, cohomology groups and modular representation theory check out:

• Bonnafè, “Representations of $\mathrm{SL}_2(q)$”. This book is great to understand a bit more the geometry of things and get your hands dirty with modules. You have to know a bit to see how it fits in to the general theory, so it’s better to approach it later if you want to understand general stuff. However I highly recommend it! It's also useful as a gateway to understanding the modern aspects of the modular representation theory of these groups, where more emphasis is necessarily placed on the structure of the $\ell$-adic cohomology groups as modules.

For information about values of Green functions and the generalised Springer correspondence have a look at:

• Shoji, “Green functions of reductive groups over a finite field”. This is again in the Arcata Conference proceedings from 1986. This gives a good overview of the relationship between the Springer correspondence and explains how you can compute values of Green functions. With this, together with what you have learnt above, you could compute the generic character table of any finite general linear or finite unitary group. This necessarily involves more geometry.
• Shoji, “Geometry of orbits and Springer correspondence”. This is in Astérisque (1988). This is a good introduction to the generalised Springer correspondence, which is a key ingredient in determining the values of Green functions and, more generally, generalised Green functions (which arise from character sheaves).
• Geck, "Some applications of CHEVIE to the theory of algebraic groups". This is a nice paper considering computational aspects of really computing things. It covers similar material to Shoji in his Arcata conference proceedings article but minimises the geometry and IC's, so is a bit easier to read.

There really isn't a good reference for character sheaves except Lusztig's original articles but this is not bad:

• Lusztig, “Introduction to character sheaves” published in The Arcata Conference proceedings from 1986. This gives a little snapshot of character sheaves, which is quite nice.

In the end, I hope some of this advice helps and good luck!

• That is a brilliant answer. – Nick Gill Apr 23 '15 at 11:25
• In the first section on Character Theory of Finite Groups I'd also recommend Marty Isaacs' book. (Indeed, a general mathematical rule of thumb: always read Marty Isaacs' books.) – Nick Gill Apr 23 '15 at 11:28
• OK, I've given you a reduced road map given your new stipulations. – Jay Taylor Apr 23 '15 at 14:45
• @JayTaylor Great answer! But Serre actually does the Mackey formula in Part II. Just defending French national pride... :) – Daniel Juteau May 19 '15 at 10:51
• @DanielJuteau Thanks! Sorry about that, apologies to Serre and France. I hope you'll except my apology on their behalf ;). – Jay Taylor May 19 '15 at 11:45

As far as I've experienced, having a good pre-image and overview of the topic you are going to learn is so much helpful.It helps you to make a general skeleton of the theory in your mind, so you will be able to put the new technical details in their right positions in the process of learning.

So I think beside the brilliant learning roadmap which Jay Taylor mentioned above, you can watch the two-part lecture of Cedric Bonnafe, Introduction to Deligne-Lusztig Theory, to get a nice overview of the theory. The lecture was delivered at the MSRI Introductory Workshop on Representation Theory of Finite Groups and the videos are available in the website.