傳播數學知識．促進數學教育

Interview with Prof. George Lusztig

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewers:** Shun-Jen Cheng (SJC), Wei-Qiang Wang (WQW), David Vogan (DV)

**Interviewee:** George Lusztig (GL)

**Date:** January 6th, 2016

**Venue:** Institute of Mathematics, Academia Sinica

Prof. George Lusztig was born in Romania in 1946. He graduated from the University of Bucharest in 1968 and received his PhD in mathematics from Princeton in 1971. Currently, he is an Abdun Nur Professor at the Massachusetts Institute of Technology. He is a world leading expert in representation theory. He was elected as a fellow of the Royal Society in 1983, to the National Academy of Sciences in 1992, and was awarded the Shaw Prize in 2014.

SJC: First of all, George, thank you for allowing us to do this interview with you. Well, I have a couple of questions here that the editor-in-chief of MathMedia asked me to ask you. So let me do it one by one. The first question is when did you know you were interested in mathematics? At what age? What time?

GL: 14, maybe. I was interested in mathematics before but at 14 I decided to be really a mathematician.

SJC: Oh, really. So before you went to university, you already knew you wanted to be a mathematician.

GL: Yes.

SJC: I see. I was wondering because in Chinese culture, maybe you know that, the parents usually are…

GL: No, but this was because in 8th grade there was an Olympiad, this kind of national Olympiad which I had no idea about. I didn’t expect to do well, but I did. I decided to really spend lots of time on it.

WQW: When you were in junior high school? 14?

GL: That was the first year of high school.

SJC: Yes, I see. That’s right, first year of high school. The other question I have is about when you went to the University of Bucharest. How does its mathematics department compare to the mathematics department in the West? I mean, how is the program in the university?

GL: Well, that’s hard to compare. It’s very different. First, this undergraduate school was 5 years. But the last two years you have to specialize in something, which was like a master’s degree program maybe.

SJC: But I mean when you entered the university, do you only have to take mathematics courses, or do you have to take others?

GL: Yes, you have to go to mathematics department. You have to pass some entrance examination. You have to decide exactly which university and which subject you like.

SJC: So you don’t have to take other classes outside mathematics like in the U.S., where you have to take….

GL: No, you have to take Marxism, for example.

SJC: Marxism?

GL: Yes, every year, some kind of Marxism. Maybe in two years of the five.

SJC: What did you learn? I mean these courses, what do they teach…?

WQW: We have a similar course when we were in China. We learned the history of Chinese Communist Party.

SJC: Oh, really.

GL: Yes, there is something called dialectical materialism (辯證唯物主義). You know that word? That was some Marxist philosophy. And there’re some political economy or something. Some of these subjects, but also Marxism. Also, all the time we had to take some course on how to teach.

SJC: How to teach?

GL: Yes, because this department was basically preparing school teachers, so we had a class on how to teach. Otherwise, in the first year we had physics, and we had also mechanics.

SJC: So this was the department of mathematics and mechanics, I see. So it’s like the Russian style. I see. That answers the second question here: If the educational system was influenced by the one in the Soviet Union, but obviously the answer is yes in this university.

GL: But one impact was that lots of Russian mathematics books were available for very low price, so I could easily buy. And they translated everything from the West. Bourbaki, all the volumes were translated. I could easily afford them, more easily than now.

SJC: Now, none of us can afford, especially Elsevier journals.

WQW: Did you read Russian?

GL: Yes.

SJC: But I suppose you have to take Russian in university or in high school?

GL: We started in the 4th grade until 11th grade studying Russian and also the first two years in the university.

SJC: That’s quite a bit.

WQW: Russian is the official second language?

GL: No, we have to take two foreign languages. But Russian was compulsory and the second one was a choice of English, French or German.

SJC: Oh, I see. So which language did you choose?

GL: No, that was in theory, but it was the question of what are available. So not every school has all the choices. I took French actually.

SJC: So, George, during your undergraduate days in the University of Bucharest, were there any professors who have influenced you most?

GL: Yes, there is one called Teleman. There’re quite a few of them but they are all in the same family. They are all related to each other. So this professor had 4 brothers, all mathematicians and all descendants of mathematicians, and one of them was my professor.

SJC: I see. One of the questions is about the reason why you moved to the West. And how did you move to the West? So you went directly to the U.S. after you left Romania?

GL: No, in fact I went to the West once and then I returned to Romania. In 1968 I went to Italy to some summer school, then to England, which was not really allowed but I asked permission afterwards. Then I went to the embassy asking for the retroactive permission. Then I went back, and half a year later I went to Bonn. Then I went to the US. In fact, I had an invitation to Princeton because of Atiyah, whom I met in Oxford. That was the first time.

SJC: Yeah, that’s the next question. What was it like to study with Michael Atiyah?

GL: But of course I was not officially his student. Of course he is a very great mathematician and also extremely nice person. He helped me a lot. And we actually talked, talked probably every day, I think. It’s quite….

SJC: I thought probably not everybody knows that Atiyah at that time was in Princeton.

GL: The first time I met him it was in Oxford in 68 and he was moving to Princeton. As he’s moving, he could recommend some young people to the (Institute of) Advanced Study. So he recommended me to work with him. So 69 he was already in Princeton. But I didn’t go to Princeton directly, he asked me to go to England.

SJC: No, I thought you told me that you went to Italy first.

GL: No, that’s another trip, the first trip abroad. First Italy and then England and back to Romania. Then I received an invitation from Princeton which I got already when I was in Oxford. When I returned I applied for permission to go to Princeton, and it was refused.

SJC: It was refused! So what did you do?

GL: But I also got an invitation to Bonn, Arbeitstagung, you know Arbeitstagung?

SJC: Yes.

GL: That one was approved because that was short, that was 1 week. And that one was approved sort of at the local level. Long trip has to be approved in Bucharest. So simultaneously I got one that was approved and the other was not approved. So I went to Bonn and from there, not directly, because I didn’t have a U.S. visa.

SJC: But you have an invitation to Princeton?

GL: Yes, so I actually went to Canada first. There was a conference there and I applied for U.S. visa in Canada and stayed two months, then went to Princeton.

WQW: Then you spent one year in Princeton before going to Warwick?

GL: Two years.

DV: He had to get a PhD.

WQW: Yes, to have the right to teach. But your thesis was not quite on Lie theory, yet, right?

GL: No.

WQW: How did you slowly or quickly move into Lie theory, or finite groups of Lie type?

GL: Of course Atiyah he knew some Lie theory, and he knew a lot of classical group representation very very well. But he had low opinion of the exceptional groups; they are not important and can be ignored. So all I knew was in this index theory. Michael Atiyah is a topologist actually but then because of a lecture by Quillen, Quillen was also at Princeton at that time, he solved some problems in topology, using a method of representation of finite groups, in fact, using something called Brauer lifting of modular representation of finite group to a virtual representation, with complex coefficient, something like that was done by Green. Actually a very deep theorem, no other topologist was aware of such a thing. So he was quite amazing. So he used that and solved something in topology, then I was quite interested in that. Then I asked Atiyah whether this Brauer lifting was known more explicitly. Because he started with the natural representation of $GL_n(F_q)$ over a finite field, and in some way attached some complex representation, virtual representation of it, and I asked him if the components of the virtual representation are known? Atiyah told me that the characters are known, but the actual representations are not known. And the characters are known from Green’s work.

WQW: But in type A?

GL: That’s just standard representation of $GL_n(F_q)$. One of these components is a discrete series representation. And that was not known, only its character. I got interested. So I learned what I needed basically. Then I was more and more interested in representation theory.

WQW: At that time, was Green’s theory on characters for $GL_n(F_q)$ available already?

GL: Yes, that was (known since) 55. And the interesting part was that at the end of my stay, I got an offer to go to Warwick which was the place where Green was located. So Green was actually at Warwick. So I thought that it would be a good place as I wanted to solve this problem. But I was wrong because he was actually not in good health. He was almost never coming to the department. Did I answer your question?

WQW: Yes, but then you still continued working on that in Warwick?

GL: Oh yes, but Carter was there, so I talked to him and he explained various algebraic groups and we wrote some joint papers. Eventually, I could solve the problem.

WQW: At that stage, you were still just working on type A?

GL: No, so my first thing was I found some way to construct this discrete series of $GL_n(F_q)$, by some explicit model using Dirichlet sums. But then I immediately started to ask the same question for classical groups because they also have a natural representation. You cannot take Brauer lifting, so that I also found somewhat less explicitly. But by working on this, I started to understand some of the representation theory by looking at the Brauer lifting.

DV: So those particular representations of the classical groups are… I mean in terms of the technology at the time, they were inaccessible. But on the other hand, in terms of what you did over the next 10 years, these were extremely easy somehow. If you were thinking about how complicated the problems turned out to be, these were misleadingly simple. Is that…

GL: No, basically what was happening for classical groups in this Brauer lifting, so for symplectic, there are 2n components. You have 2n dimensional representation in modular setting and you lift this 2n dimensional representation. And one of them is a discrete series corresponding to Coxeter torus actually. But the other one is also difficult. They are not discrete series but you have to decompose using discrete series. In $GL_n$, all the other one are simply induced. They are not decomposed at all. In other classical groups, it’s non-trivial to decompose. I have to understand this Hecke algebra with unequal parameters which I have to use… So that was more complicated than $GL_n$. But this method would only construct one series but for classical groups there are several discrete series, several non-isomorphic tori. So in fact I did quite a lot of work and eventually I did something which I never published, I just wrote some very short summary about it, in fact it was quite a nice theory. It was never published because it became obsolete in some sense. But actually all this helped me quite a lot to gain a lot of experience and helped me to do the other things.

DV: So, it sounds like you probably did things like almost writing down character tables for some of these classical groups, and anyway computing very explicitly the characters of a lot of representations and understanding all the conjugacy classes. So do you enjoy that kind of… you know, everyone has to do some computations like that for mathematics, but some people like it more than other people. Do you enjoy that kind of work? You’ve done a lot of it.

GL: Yes, certainly I like computations.

WQW: Yes, those computations are something I’ve never really read, but I learned your braid group action computation. After 20 years, it’s still brutal to me. I taught this a couple of times, I always choose the easier types, but even those were pretty complicated. $G_2$ I’ve never tried, even today.

DV: So you talked about having to take a class and teaching in Bucharest. Did you learn from…?

GL: No.

DV: But you mentioned on the other hand, Teleman. How do you learn to write mathematics well, to do mathematics well? Whom have you followed, tried to be like?

GL: I did write some papers while I was a student there. But I don’t think I learned from Teleman at all. Maybe from Atiyah certainly. I tried to follow his example and even more Deligne I would say, certainly the way he writes.

WQW: Atiyah’s papers I read a few. I don’t remember your styles were very similar, let me put it that way.

GL: No.

DV: By now you’ve read and heard the mathematics of hundreds of people, you know, read papers and seen people lecture. Are there individuals that you think we should all be more like, best examples?

GL: Milnor is No. 1. I was always told he’s the greatest. His books were really absolutely the best. And his lectures were the best. He was my model for this one. I know him a lot, I was doing topology in the Institute and was close to him.

DV: So did you ever think about deliberately moving back towards topology? After you started working on finite groups, reductive groups and so on, it seems that you were solving things so fast but also opening up new problems so fast, from 1970 to 1980s. It’s hard to see why you would go back to do something else, because there’s still more.

GL: That’s still the case. But it’s also the case that, while I was in Princeton, I thought I would be a topologist, I knew I had some contact with Daniel Sullivan. Actually I had the impression that he’s solving all the main problems in topology. So, that was the incentive for me to move away from topology. In fact, I always, actually even now I think representation theory has more interesting problems than topology. I think topology is very useful as a tool.

WQW: They were more interested in classical groups.

GL: But not so interested in as a goal in itself.

DV: I don’t remember who talked about mathematics as some kind of garden, and it was very important to put fertilizer on the ground and so on.

GL: Hironaka.

DV: I think that’s right. But there were some parts of mathematics that were the flowers. So maybe the groups and representations are some of the flowers. There are whole lot of really new ideas in mathematics that you’ve been closely involved with creating. The impression I have is that most of them came when there was a problem that you need to solve and there was no way to solve it using mathematics that was around. And so you made something new that was designed to solve an old problem. Is that…do you…ever just find things that look like they will be interesting ideas or, I don’t know. Somehow it feels like the Kac-Moody Lie algebras were just kind of made because after the Serre relations were written down, you could say, “Well, we don’t need this to be a finite dimensional algebra and everything, all the formalism can still be done.” And sort of feels to me like there is not so much reason to go do this mathematics that you can do. But the new things you brought in to the subject they always seem to be designed to tell you about old problems, at least at first.

GL: Actually, I had the impression that I always have some goals to go to, but always happened that I got some luck that things happened. For example, at some point, I really wanted to understand how to classify representations of classical groups. After the paper with Deligne, I wanted to know; that was really an important problem, I thought. Because $GL_n$ was known, (for classical groups) how to classify that was not known at all. I didn’t come to up with any new methods, I already knew I had this technique from my paper with Deligne, but then I met these Korean mathematicians, Lee and Chang, anyone knows these two people?

DV: The Lie group guy?

GL: Yes, and there’s another, he had a collaborator, who did character of $G_2$ over finite fields. And I met them in Vancouver and Chang told me that he had a student who just computed Iwahori Hecke algebra of classical type with arbitrary unequal parameters, he had construction and degrees. And this guy actually left mathematics. So that came exactly the right time. So then I studied that and I could see that was somehow what I was missing almost. So I think it went like that several times. I just got something I was missing, just by some accidents, it happened that it was provided to me.

WQW: But at that time if there were not available, probably you would be just forced to work out it yourself.

GL: Possibly. It all came one after another, then I understood these classical groups, and then immediately I found some preprint that somebody found the generic degree for Hecke algebra of (exceptional) type E8.

WQW: You were not a good student of Atiyah….

GL: What? No, all I really wanted to classify…. That I got some idea of how to construct this non-abelian Fourier transform matrix, because they had this list of degrees of E8, you could see they all have some leading terms which have some coefficients, one over 120 and one over 24. So then, I saw some patterns in that. But that came exactly at the right time.

WQW: Those you all computed by hand, right?

GL: No, I didn’t do the computation, someone did that.

SJC: But I am saying that if that were not available, chances are that you probably would compute yourself, no?

DV: Well, you have friends you pushed around. I mean you need all these branching tables for Weyl group representation eventually. You didn’t compute those yourself. But you were responsible.

GL: Yes, and even before that, the first time I used computer was to compute the fake degrees, you know that? I did fake degrees for classical groups in this work. I want to know for exceptional groups. And for these, I found some guy in Warwick and told him the rule to compute, so he did this, that was the first time I used computer.

WQW: I think I know the name. Beynon? He’s not a mathematician.

GL: He’s not a mathematician. He was in computer science.

WQW: So for the exceptional type sometimes without computer, even you feel it is hopeless to compute.

GL: Yes, computer made a big difference. Because maybe in the 60s I think, it was considered that classical groups are easy and exceptionals are difficult somehow. Now it’s reversed I think, because of the computer.

DV: I want to ask something that seems completely different but maybe it’s not completely different. So this was the administrator in the math department of MIT, Joan Johnson. Of course, many people in the department travel a lot. So they come and talk to her about places they travel. She told me what she liked best was when you talked to her about places you travelled. Because she said that “He sees everything, and he remembers everything.” Do you know that? I mean obviously you know you see some mathematical things more than other people do, but are you aware of it?

GL: I am surprised!

DV: I think all these things you were saying about fortunate accidents that something was available, the world of what’s available in mathematics is enormous. The question of seeing the right thing that’s available that’s not so easy necessarily.

GL: But actually quantum groups, at some point I got interested in because Borel wrote me a letter. He said there is this work of Michio Jimbo he thought that I might be interested because he had something that looked like Hecke algebra. He knew I was interested in Hecke algebra. He said Jimbo actually had Hecke algebra and it was directly involved. So he pointed out this thing, then I studied and I gave a course.

SJC: When was it? Which year was it?

GL: 86.

WQW: The paper was published in 86. I think among us, we probably know a little bit more about your work on quantum group than the early work on finite group Lie type. Even though I browsed your earlier books once or twice, I am still not sure how much I really read.

GL: But I think that the finite groups of Lie type were more exciting in some sense, than the quantum groups because you didn’t know what was the result, there was a mystery.

WQW: Right, those questions in a way are 100-year-old, the finite groups of Lie type. In a way it is an old classical problem. Somehow, can you say that introducing geometric method there is one of the keys for finite groups of Lie type?

GL: No, but certainly without geometric method you cannot do anything.

DV: What’s amazing is as you said Green in the 1950s did make this complete list of the irreducible representations of $GL_n$. So people who work hard enough can… you can do anything somehow. But it was the geometric methods that you and Deligne brought that made some of those results clear, made it clear what had to be true, not just the results of some terrible calculations.

GL: Oh, Green, all his papers were really amazing. He was the only person who knew modular representation very well, so he could use method of Brauer. He was student of (Phillip) Hall. So he knew all about Hall polynomials which he used to define Green polynomials. So he knew all these things. Because before him Steinberg was working on this, he only managed to do $GL_4$, from that work he was then completely exhausted, that of $GL_4$.

WQW: I am sure that the symmetric polynomials, Hall-Littlewood, and all these things, how they get into play, probably, one really need some expertise for those things. But back to quantum groups, for quite a few years from 86 to 94, you were quite focused on quantum groups. It’s a major part of your work.

GL: Yes, almost 10 years. I was doing two theories, there are really two kinds of things. One is quantum groups at a root of unity, trying to understand how they can help to do modular representations, and also this canonical basis.

SJC: That’s the time when we were students there.

WQW: Yes. Those things back then, I didn’t understand.

GL: Pardon?

WQW: The canonical basis. I am sure I took lecture notes from you.

GL: But (now) you know that very well.

WQW: But only now I really need it. The last 5 or 10 years I have been trained very well by my students. Somehow they forced me, different students for different purposes re-taught me.

GL: Actually Borel was correct. He could see that knowing Hecke algebra will help me. I can transport some of the thing to quantum groups, that’s exactly correct.

DV: So you talked about working with Atiyah and Deligne. These people who could think of as in some ways teachers as well as collaborators. So how do you like working with your students? I mean, you know there’re two things you can do. You can write a beautiful book and send it to the publisher, or you can have a student. This is very different kinds of satisfaction. Do you have good experiences?

GL: Yes, I have wonderful experiences. I have some really good students which I was extremely happy to have them. One example is Spaltenstein. Xuhua He is another one. I have several similar students.… But not all of them, some of them were really good.

DV: So have there been students that really changed the way you do mathematics, changed how you think about some of the problems?

GL: Yes, Spaltenstein I’ve learned a lot from him, maybe more than from many other students. He was actually very early, maybe in the very beginning, so it was in England.

SJC: He was your first or second student?

GL: Immediately, I started with three students. Maybe he came one year later. Maybe, he was the second. The first one is de Concini.

SJC: de Concini, that’s right.

WQW: Looking at the publication list, it’s very easy to see that most of your papers are single-authored. Then I am talking about this working style and collaborators. Is that basically the way you choose or is that just the most natural way? How do you feel about it?

GL: Yes, I usually work by myself. There were some exceptions, but I think my natural way would be to write it by myself.

WQW: Even those long computations, you prefer to write it by yourself.

SJC: But George said he likes computations.

DV: I don’t know if collaborating with Deligne spoils you for anybody else.

WQW: I don’t know, probably more people tend to have more collaborators than just working alone all their long career?

DV: I have asked the people of math reviews because they could easily do statistics about whether the number of coauthors on papers is actually increasing. I have the impression, yes, that it is.

GL: Even 4… I see papers of 4 authors that are also becoming quite usual. At least I’ve seen in this conference.

SJC: Yes, that’s right, three or four. That’s quite common because communication becomes so easy.

WQW: That’s true. Also, to be efficient. A single person has single expertise. We don’t learn things that quickly even if we know what to learn. To me, those collaborators, they were amazing, bringing in wonderful, different techniques. Those things for myself would take much longer or forever to learn.

DV: But there’re various jokes. I don’t remember any of them properly. A camel is a horse designed by a committee. There’re problems with collaboration.

WQW: Okay, I am sure that applies to some other collaboration.

DV: Or just to any collaboration with 4 people.

SJC: It’s a committee.

GL: I’ve never been in any team of 4, maybe three was maximum.

DV: You told me once that when you were in school at Bucharest.

GL: Not in school, but in university.

DV: Yes, you were on the crew team that you rowed. So how did that go with studying, with doing mathematics?

GL: Not so much in Bucharest. That was in last two years, actually last two years of high school I did rowing. That was not a contradiction. I went twice a week or something. I was very bad at physical exercise. Somehow I got this. I don’t know how I started but it improved my health and so it was quite good.

DV: But then you stopped.

GL: Then I went to university, and continued in the first year. It’s not so convenient. Actually you had to go far away. I did it for one year but it was not so enjoyable. In my hometown, we have a river and it’s quite convenient, and I liked it.

DV: So have you found other kinds of activities? I mean do you find it necessary to do things other than mathematics in order to do mathematics?

GL: Yes, like yoga, for example, I think I couldn’t function without that.

SJC: But when doing yoga, do you think of mathematics?

GL: No.

SJC: But you found it easy to just shut it off? For example, I mean when you’re ready to go to sleep, you could shut it off?

GL: No, the problem is when I go to sleep, I can. But then I often wake up in the middle of the night, then always some problems come then I cannot sleep for 2 hours. But when I go to sleep, it’s no problem, because I was tired.

DV: So you said you like to read Milnor’s books. Are there non-mathematical books that you find very worthwhile?

GL: Yes.

DV: Love in the Time of Cholera?

GL: Yes, that’s good. I like that. You like it?

DV: Well, I just remember 20 years ago or something that you mentioned this.

GL: Yes, I mentioned this.

DV: See you have to be careful of what you said. We’re all writing things down.

GL: Yes, this particular writer I like, I read several of his books.

SJC: One Hundred Years of Solitude?

GL: Yes.

WQW: I think you (Cheng) probably introduced those books to me.

SJC: Fantastic books. Marquez.

DV: Do you ever read fiction and so on, other than in English now?

GL: Yes, actually I have now read some Balzac’s books, in French, sometimes I read in Romanian, sometimes I read in Italian, but that was less. But in Italian I read quite a few books.

SJC: You said you went to Italy only for a short period.

GL: One year.

DV: But the language is not so distant from Romanian?

GL: Yes, but you have to learn, it’s not…

DV: Yes, I understand it is not a dialect.

GL: But you can also interpolate to some extent.

DV: Did any of your old Russian math books come with you? Do you have them?

GL: No, they came with my sister. My sister came later she brought them, so I got all the books.

DV: Oh, wonderful.

GL: But I can’t read literature in Russian. I’m not at that level… But actually one of the books I read recently which I thought extremely good which I really recommend. Its writer is called Vasily Grossman, some Russian, and he has this book. The subject is the battle of Stalingrad(史達林格勒戰役).

SJC: The battle of Stalingrad. I see.

GL: And it’s just really a wonderful book. It has some similarity with War and Peace. It also talks about some battles in which the existence of the country involved was in question.

DV: So what about music? How much do you have opportunities to listen to music?

GL: My wife Gongqing is playing every day, practicing piano.

SJC: It’s great to have live music at home, I think. Do you read before you go to sleep? Does it sort of relax you?

GL: No, not before.

WQW: You yourself also wrote several books. How did you enjoy that experience, writing books?

GL: They are not really books. The first one was on Discrete Series of $GL_n$, it was not supposed to be a book, it was supposed to be a paper, and I sent it to Annals of Mathematics. They proposed it to be a book instead of a paper. So I did that.

DV: The characters of reductive groups over finite fields?

GL: Also, I considered that is a paper, not a book. Because if it were a book, I should try to make it easy to read, which I didn’t. So I wrote it such that I finished it as soon as possible. So I write everything and don’t care so much about the readers.

DV: I was going to ask how many books you’ve written. But I guess it’s an easy calculation that just you write papers and the lengths are normally distributed and any paper of length more than 200 pages is a book.

WQW: But his Introduction to Quantum Group must be a book.

GL: It’s a type of… But …

SJC: It’s an introduction…

DV: See with the paper version, there was no introduction to it.

WQW: But this one you did intend to write as a book from the beginning.

GL: Yes.

DV: And Heckle algebras with unequal parameters?

GL: That was requested because I gave this Borel lecture. They asked me to, but I gave some lectures in MIT so I already had some notes. I just had to expand them. So actually maybe (Introduction to) Quantum Group is probably the only one supposed to be some kind of book, not a paper.

DV: That’s interesting.

WQW: Yes, this Introduction to Quantum Group takes many practices to get into it. I mean it takes introduction to Introduction.

SJC: Yes, I was wondering about the title. I mean you intentionally…the title is called “Introduction to Quantum Groups.”

GL: You think it’s misleading?

DV: This is the title in the tradition of André Weil’s Basic Number Theory. I assume that you believe in the importance of these excellent books for getting the next generation of mathematicians into these problems. You said this most exciting or this extremely exciting work was about the characters of finite Chevalley groups. So Roger Carter wrote an account of that, which I think has been helpful to a lot of people. Do you think about these things? Do you feel that, do you see it’s not your job to write the books like that? I mean the quantum groups’ book is a little bit like that.

GL: I know, but probably the representation theory of finite groups, I feel, I don’t understand it well enough. So there is something which still needs to be done, I will think about that and when they are done I would write some nice exposition, but I don’t feel ready for that. I think it’s not yet complete. But also I don’t think I am good at this.

DV: You just have to learn LaTeX.

WQW: I think George’s using TeX, right?

DV: AMS TeX, yeah.

GL: No, but for example I am terrible at proofreading. So I’ve made mistakes.

DV: In these four books we have talked about there are 7 misprints….

WQW: I think your formulas we can trust. That’s not only my impression. I have used various things in a very concrete way.

DV: Sometime around 1980, you let me photocopy some pages of calculations you‘ve done about nilpotent elements and so on, and exceptional groups. It was wonderful to have these calculations. They were perfect. The most terrible thing about them was that you wrote in this light-blue ink and photocopy machines didn’t like this at all. So it’s very hard to read the photocopies, but they were very reliable.

WQW: Nowadays do you still write most of your work in hand-written notes or do you type up very quickly and throw away hand written notes along the way?

GL: No, I type. I certainly don’t write the whole paper, maybe just do some computations, then if I have enough, I go type them.

WQW: When we learned LaTeX versus the AMSTeX, the advantage of labelling automatically persuades us that LaTeX is the way we want to choose. Somehow this cross reference never bothers you? Because some of us sometimes need to totally change the order, of course, the labels of the equations would be all wrong. Automatically LaTeX would take care of that. How does that work for you in AMSTeX?

GL: Because I have a different way of numbering. Because I have some sections that I number formulae just by A, B, C, D, it doesn’t matter if I change the number of the section, the (labels of) formulae don’t need to be changed. So section number is not included in the formula. It’s only a letter. Then I refer to section 1t then a or something.

WQW: The quantum group book is really marvelous. The more I understand the subject, the more I appreciate it. Also, I almost find no mistakes. That’s one book in the last ten years I got to know reasonably well.

SJC: I think both of us own two copies of it.

WQW: One copy at home, one copy in office.

GL: But I think that quantum group is relatively easy. I think representations of finite groups are much more exciting and interesting. Quantum group is relatively easy.

WQW: That’s why I can understand it a little bit.

SJC: You mentioned about Atiyah and Deligne. Are there any other mathematicians you admire? Besides those two, are there any other you admire?

GL: Gauss, for example.

SJC: Oh, that’s one. We admire him as well. Somebody in Lie theory?

DV: Have you read papers by Lie or Élie Cartan?

GL: Lie certainly not, but Élie Cartan I read something.

DV: And Hermann Weyl?

GL: I have his book on classical groups that I read most of it.

DV: We are talking about the styles of mathematics that maybe, I don’t know if it is Roger Howe who said about this Weyl book that it has a million dollars’ worth of mathematics all in pennies.

GL: So maybe Chevalley I might mention. In fact, when I was at Warwick University, they had a library, and this Chevalley seminar was available, which was hard to get and it was not officially published somehow.

DV: Yes, there are a bunch of these French seminars you could find in the university libraries and a few places.

GL: So these I studied.

WQW: These are the lecture notes for those seminars?

GL: No, they are not lectures, they are exposé, by Chevalley, some by Borel, some by other people. But the seminar was led by Chevalley, they are about the classification of reductive groups.

WQW: Do you still have any plan to write another book?

GL: No, but if I understand the representations (of the finite groups of Lie type)

SJC: There would be an introduction to it.

DV: Do you know this paper of Macdonald about classifying…rewriting Green’s work about $GL_n(F_q)$ in terms of Weil groups and, well, representations of Weil groups.

GL: No, I don’t know.

DV: He proves that representations of $GL_n(F_q)$ are in one-to-one correspondence with equivalence classes of n-dimensional representations of p-adic Weil group. The p-adic field should have residue field $F_q$. And you just want Weil group representations that are trivial on the Weyl inertia and the equivalence relation he puts on these representations is equivalence when restricted to the inertia sub-group. So essentially, it’s just the representation of the inertia sub-group modulo Weyl inertia, and except that there has to exist the extension to the Weil group. He proves also those were in one-to-one correspondence with $GL_n(F_q)$ representation. Anyway, the reason I asked this is I just learned about this paper a week or 2 ago, and it illuminates from 1980s or something. The formulation makes complete sense for any finite Chevalley group.

GL: No, but it need the special classes which its formulation does not…. So this will not give a correct…

DV: See, that’s apparently what Bill Casselman always said that it can’t possibly be right, but I am not convinced. I mean there are some l-packet stuff that has to be…Maybe it is quite difficult or impossible to describe l-packets, but it seems to me that those ideas could provide a way to say some of the things that you did.

GL: I don’t think so, I think you have to pronounce this very special representation, without that you cannot hope to look at the classification.

DV: I don’t think so. We can talk about it. So what pleases you the most about doing mathematics? If you think about what you are going to do over the next five years, you could hope to have really wonderful students, you could hope to teach a really excellent class, you could hope to solve some serious problems and write great papers, all these things...It’s reasonable for you to hope to do. These are all things you’ve done. What do you aim at?

GL: Teaching, yeah, but I still get most of my fun from writing papers. I don’t think I will solve big problems, I just have some fun doing some smaller things.

WQW: But in general, you yourself are more of a problem solver, or a theorist? Of course, this is very hard to classify. But some people still have tendency to solve problems, outstanding problems. They set some goals.

GL: No, actually I like to find conjectures, finding a pattern of conjectures. Probably that’s more enjoyable than solving….

WQW: In particular, solving your own conjecture?

DV: Gelfand, I think, said that making precisely correct conjectures, not only is more fun, it’s more important. Anybody can write proofs somehow, but only if the conjecture is just exactly right and interesting.

GL: That’s what I like about conjectures.

WQW: Probably they shouldn’t be too easy to prove.

DV: Well, you know you can… sometimes if you formulate the conjecture properly, then it “is” easy to prove. That’s maybe one of the best kind is when you say something in a way that wasn’t said before, so that...

GL: No, but sometimes I find if you find a good conjecture which clearly looks like correct, then I am not even that much interested in. The proof is not so important, I think.

DV: Yes, absolutely. Maybe the Riemann Hypothesis is an extreme example. It’s clear that it’s worthwhile to do number theory, using it to see what that tells you, because it’s very likely to be true. It tells you an enormous amount. I think that was the most of the questions I have on my list.

SJC: Okay, I think that’s about it. So George, thank you very much again.

GL: Thank you.

- Shun-Jen Cheng is a faculty member at the Institute of Mathematics, Academia Sinica.
- Weiqiang Wang is a faculty member at the University of Virginia.
- David Vogan is a faculty member at the Massachusetts Institute of Technology.