George Lusztig

Born: 20 May 1946 in Timişoara, Romania

George Lusztig's name, in Romanian, was Gheorghe Lusztig. He was born into a Jewish family and brought up in Timișoara, a city in Romania where Hungarian and Romanian are both spoken. He said [5]:-
I'm not religious but being Jewish played a role in my choice of mathematics, which seemed beyond the reach of politics, as well as the fact that it was an area where I had the best possible chance to be judged objectively.
He attended High School in Timișoara and represented Romania in the International Mathematical Olympiad in 1962 and 1963, being awarded a Silver Medal on both occasions. In fact it was mathematics competitions at school which made him realise that he was talented in mathematics and gave him confidence in himself. He studied mathematics at the University of Bucharest and, while still an undergraduate, he began publishing papers. His first two papers were A model of plane affine geometry over a finite field (Romanian) (1965) and Construction of a universal bundle over arbitrary polyhedra (Romanian) (1966). In the latter paper he extended the construction of a universal bundle over denumerable polyhedra given by John Milnor in 1956 to the case of arbitrary polyhedra. His method used the Kelley topologies. He graduated with a Licenta in Matematica in 1968, and three more of his papers (one written jointly with Henri Moscovici) were published in that year; they were all written in French.

At this stage of his career he was told that he could not get a position higher than a high school teacher in Sibiu. However, Dan Papuc, a professor at the University of Timișoara, offered him a university position in Timișoara which he gladly accepted. He realised, however, that there was no future for him in Romania and decided that when the first opportunity arose he would leave the country. He was allowed to attend a conference in Italy, and after that he went to England visiting Warwick University and spending two months at Oxford University during the autumn of 1968. At Oxford, during his first meeting with Michael Atiyah and Isadore Singer they asked him a question about the Kervaire semicharacteristic. He was able to answer the question but they had asked F P Peterson, who was at the Massachusetts Institute of Technology, the same question a couple of months earlier and it was soon discovered that he had found a similar solution. Although they had discovered that results independently, Peterson and Lusztig wrote a joint paper Semi-characteristics and cobordism which was published in Topology in 1969. Michael Atiyah had just been appointed professor of mathematics at the Institute for Advanced Study in Princeton. He invited Lusztig to Princeton and, after he had returned to Timișoara, Lusztig applied for permission to go to Princeton. He had also received an invitation to attend a conference in Bonn, Germany, so he also sought permission to spend a week in Bonn. He was granted permission to attend the conference in Bonn but was refused permission to go to Princeton.

In 1969 he went to the Bonn conference but, after the conference ended, did not return to Romania as he was supposed to do. Instead he went to the Institute for Advanced Study in Princeton where he studied with Michael Atiyah for two years. He was also registered as a Ph.D. student at Princeton University where he worked for his Ph.D. advised by William Browder. He was awarded a doctorate by Princeton in May 1971 for his thesis Novikov's Higher Signature and Families of Elliptic Operators. The thesis was published under the same title in the Journal of Differential Equations in 1972. After the award of his doctorate, Lusztig was appointed as a Research Fellow at Warwick University in England. After spending the academic year 1971-72 as a Research Fellow he was appointed as a Lecturer in Mathematics and, two years later in 1974 as Professor of Mathematics at Warwick University. Roger Carter writes [1]:-

Lusztig's exceptional mathematical ability became evident at an early stage of his career at Warwick. He gave a remarkable 30 lecture M.Sc. course on the modular representation theory of the general linear group in which, during the second half, he was working out the theory while giving the course. There were a few occasions when he was apologetic that the lecture lasted only 40 rather than 50 minutes because he had not made sufficient progress since the previous lecture! His early experience as a mathematician was not without certain difficulties. There was a period during which, for financial reasons, he preferred to live in a tent outside the Mathematics Research Centre houses at Warwick University rather than in the houses themselves. He also experienced problems with the passport authorities in a number of countries so that, for a brief period, he had no entitlement to live in any country. However these difficulties did not prevent him from developing rapidly as a mathematician.
In his response to receiving the Steele Prize in 2008, Lusztig spoke of the change in his research interests around the time he took up his Research Fellowship at Warwick [4]:-
Around the time of my Ph.D., I switched from being a topologist with a strong interest in Lie theory to being a representation theorist with a strong interest in topology. (The switch happened with some coaching by Michael Atiyah and later by Roger Carter.) After that most of my research was concerned with the study of representations of Chevalley groups over a finite field or used the experience I gained from groups over a finite field to explore neighbouring areas such as p-adic groups (which can be viewed as groups over a finite field that are infinite dimensional) or quantum groups (which can be viewed as analogues of the Iwahori-Hecke algebras, familiar from the finite group case).
In 1972 he married Michal-Nina Abraham; they had two daughters. This marriage
ended in 2000 and he married Gongqin Li in 2003. The year 1974 was an important one for Lusztig from a mathematical perspective. In that year his major paper On the modular representations of the general linear and symmetric groups written jointly with Roger Carter was published. In the spring of that year he spent time in France at the Institut des Hautes Études Scientifiques during which time he worked with Pierre Deligne. He reported on this work in the lecture On the discrete series representations of the classical groups over a finite field which he gave to the Algebraic Groups section at the International Congress of Mathematicians held in Vancouver in August 1974. He had originally been invited to address the Algebraic Topology section but had requested the change to the Algebraic Groups section. While at the Institut des Hautes Études Scientifiques in the spring of 1974 he wrote the paper Divisibility of projective modules of finite Chevalley groups by the Steinberg module (1976) and returned to IHES for further visits in December 1974 and December 1975. Out of these visits came the paper On the finiteness of the number of unipotent classes (1976) and his joint paper with Pierre Deligne Representations of reductive groups over finite fields. Also in 1976 he published another major work with Roger Carter, continuing the work they had begun in the 1974 paper mentioned above, namely Modular representations of finite groups of Lie type.

In the first half of 1978 I [EFR] spent five months at the Warwick Symposium on Infinite Groups and Group Rings at Warwick University. I was there with my family and we rented a house in Kenilworth owned by George Lusztig who, at that time, was visiting the Massachusetts Institute of Technology. While we were there Lusztig made the decision to accept the offer of a professorship at MIT and I was asked to show perspective buyers round his Kenilworth house. Lusztig has remained as a professor at MIT being Norbert Wiener Professor from 1999 to 2009 and is now Abdun Nur Professor.

An excellent overview to Lusztig's contributions are given in the citation for the award of the American Mathematical Society's Leroy P Steele Prize for Lifetime Achievement (2008).

Let us look briefly at the books that Lusztig has published. In 1974 he published the monograph The discrete series of GLn over a finite field. As the title indicates this deals with the general linear group but it raises the problem of treating other classical groups. In fact by the time the monograph was published, Lusztig had solved these too and described them in the invited lecture he gave at the International Congress of Mathematicians in Vancouver in 1974 (mentioned above). The lectures he gave on Representations of finite Chevalley groups at the CBMS Regional Conference held at Madison, Wisconsin in August 1974 were published by the American Mathematical Society in the following year. In 1984 he published Characters of reductive groups over a finite field. Bhama Srinivasan writes in a review:-

This book is a major piece of work in the theory of representations of finite groups of Lie type. ... Considering the complexity of the subject matter, this book has been written carefully and with regard to details. However, by its very nature, since it draws from various deep and rich theories such as algebraic geometry, intersection cohomology and enveloping algebras, it requires a great deal of effort and commitment on the part of the reader. The reader who perseveres with the book will be richly rewarded at seeing the unexpected connections of these branches of mathematics with the representation theory of finite groups, which had hitherto been regarded as a somewhat isolated field.
His next major book was Introduction to quantum groups (1993). Jie Du writes:-
The book under review summarizes and updates the author's main contributions to quantum group theory over the last 5 or 6 years. Quantum groups came up originally in physics, but soon became a new and very rapidly developing field in mathematics. Extensive connections have already been found between quantum groups and various mathematical areas such as Lie theory, low-dimensional topology, group theory, noncommutative geometry and so on. ... the book presents many topics, together with detailed arguments for most of them, in quantum group theory. It is written to a very high standard and perhaps is an introductory book for experienced researchers and experts.
In 2003 Lusztig published Hecke algebras with unequal parameters. Götz Pfeiffer writes in a review:-
An introduction to Coxeter groups and their associated Iwahori-Hecke algebras need not be long to be comprehensive. In this book the theory is developed from scratch, in a purely combinatorial setup, through a sequence of 13 short chapters over 58 pages, complete with proofs and examples, from basic properties of Coxeter groups to Kazhdan-Lusztig cells and the definition of the so-called a-function. ... This book leads its reader through a carefully chosen sequence of short pieces, which are more or less easy to digest, from the definitions of the basic concepts right into the middle of an important part of the theory of algebraic groups and their representations. It summarizes much of the current knowledge and adds many new results. It sets the stage for a field of research where many questions are waiting to be answered, in the form of general theorems as well as by concrete calculations for specific cases; the book contains examples of both. The second half of the book is a major research article which assumes some acquaintance with the representation theory of finite or p-adic groups. In its first half the book presents an accessible self-contained introduction to the theory of cells and Kazhdan-Lusztig polynomials.
Lusztig has received major honours for his outstanding contributions. In addition to the invitation to address the International Congress of Mathematicians in Vancouver in 1974, he was invited to give a plenary address at the International Congress of Mathematicians held in Kyoto, August 1990. He gave the address Intersection cohomology methods in representation theory and the American Mathematical Society produced a video of his lecture. Bhama Srinivasan writes:-
The speaker covers a lot of ground in this one-hour talk. He manages to convey the flavour of the deep results that he describes, by giving concrete examples. In many cases he describes the results for special linear or general linear groups in concrete geometric terms. The viewer will benefit from reading the speaker's published paper cited above, which gives more details than the talk as well as references to the cited results. The videotape and the paper complement each other, and give a fine exposition of some of the important developments in representation theory that have occurred in the last decade.
The prizes he had been awarded include the London Mathematical Society's Berwick Prize (1977), the American Mathematical Society's Cole Prize in Algebra (1985), the Dutch Mathematical Society's Brouwer Medal (1999), the Romanian Academy's Diploma of Academic Merit (2007), and the American Mathematical Society's Leroy P Steele Prize for Lifetime Achievement (2008). The citation begins [4]:-
The work of George Lusztig has entirely reshaped representation theory and in the process changed much of mathematics.

He was elected a fellow of the Royal Society (1983), a fellow of the American Academy of Arts and Sciences (1991), and a member of the National Academy of Sciences (1992).

We should note Lusztig's interest in yoga. He said [5]:-

I am interested in yoga. I began to practice in 1994 to relax (in the previous year I had been working too hard - with serious consequences for my health). I did very well and I decided to study yoga seriously. In the summer of 1995 I took a course from which I received a yoga instructor license. For me, yoga is not related to mathematics, rather it is complementary to it.
We end this biography by quoting Roger Carter [1]:-
Lusztig's work is characterized by a very high degree of originality, an enormous breadth of subject matter, remarkable technical virtuosity, and great profundity in getting to the heart of the problems involved. It can be no exaggeration to say that George Lusztig is one of the great mathematicians of our time.

Article by: J J O'Connor and E F Robertson

July 2011
MacTutor History of Mathematics