# Steele Prize citation for Lusztig

In 2008 the American Mathematical Society awarded George Lusztig their Leroy P Steele Prize for Lifetime Achievement. What follows is the citation for the award:

The work of George Lusztig has entirely reshaped representation theory and in the process changed much of mathematics.

Here is how representation theory looked before Lusztig entered the field in 1973. A central goal of the subject is to describe the irreducible representations of a group. The case of reductive groups over locally compact fields is classically one of the most difficult and important parts. There were three more or less separate subjects, corresponding to groups over $\mathbb{R}$ (Lie groups), $\mathbb{Q}_{p}$ (

*p*-adic groups), and finite fields (finite Chevalley groups).

Lusztig's first great contribution was to the representation theory of groups over finite fields. In a 1974 book he showed how to construct "standard" representations - the building blocks of the theory - in the case of general linear groups. Then, working with Deligne, he defined standard representations for all finite Chevalley groups. This was mathematics that had been studied for nearly a hundred years; Lusztig and Deligne did more in one paper than everything that had gone before.

With the standard representations in hand (in the finite field case), Lusztig turned to describing irreducible representations. The first step is simply to get a list of irreducible representations. This he did almost immediately for the "classical groups", like the orthogonal groups over a finite field. The general case required deep new ideas about connections among three topics: irreducible representations of reductive groups, the representations of the Weyl group, and the geometry of the unipotent cone. Although some key results were contributed by other (great!) mathematicians like T Springer, the deepest new ideas about these connections came from Lusztig, sometimes in work with Kazhdan.

Lusztig's results allowed him to translate the problem of describing irreducible representations of a finite Chevalley group into a problem about the Weyl group. This allowed results about the symmetric group (like the Robinson-Schensted algorithm and the character theory of Frobenius and Schur) to be translated into descriptions of the irreducible representations of finite classical groups. For the exceptional groups, Lusztig was asking an entirely new family of questions about the Weyl groups, and considerable insight was needed to arrive at complete answers, but eventually he did so.

Lusztig's new questions about Weyl groups originate in his 1979 paper with Kazhdan. The little that was known about irreducible representations first becomes badly behaved in some very specific examples in $SL(4, \mathbb{C})$. Kazhdan and Lusztig noticed that their new questions about Weyl groups first had nontrivial answers in exactly these same examples (for the symmetric group on four letters). In an incredible leap of imagination, they conjectured a complete and detailed description of singular irreducible representations (for reductive groups over the complex numbers) in terms of their new ideas about Weyl groups. This (in its earliest incarnation) is the Kazhdan-Lusztig conjecture. The first half of the proof was given by Kazhdan and Lusztig themselves, and the second half by Beilinson-Bernstein and Brylinski-Kashiwara independently.

The structure of the proof is now a paradigm for representation theory: use combinatorics on a Weyl group to calculate some geometric invariants, relate the geometry to representation theory, and draw conclusions about irreducible representations. Lusztig has used this paradigm in an unbelievably wide variety of settings. One striking case is that of groups over

*p*-adic fields. In that setting Langlands formulated a conjectural parametrization of irreducible representations around 1970. Deligne refined this conjecture substantially, and many more mathematicians have worked on it. Lusztig (jointly with Kazhdan) showed how to prove the Deligne-Langlands conjecture in an enormous family of new cases. This work has given new direction to the representation theory of

*p*-adic groups.

There is much more to say: about Lusztig's work on quantum groups, on modular representation theory, and on affine Hecke algebras, for instance. His work has touched widely separated parts of mathematics, reshaping them and knitting them together. He has built new bridges to combinatorics and algebraic geometry, solving classical problems in those disciplines and creating exciting new ones. This is a remarkable career and as exciting to watch today as it was at the beginning more than thirty years ago.

George Lusztig began his response as follows:

When writing a response it is very difficult to say something that has not been said before. Therefore, I thought that I might give some quotes from responses of previous Steele Prize recipients which very accurately describe my sentiments.

"What a pleasant surprise!" (Y Katznelson, 2002). "I feel honored and pleased to receive the Steele prize - with a small nuance, that it is awarded for work done up to now" (D Sullivan, 2006). "I always thought this prize was for an old person, certainly someone older than I, and so it was a surprise to me, if a pleasant one, to learn that I was chosen a recipient" (G Shimura, 1996). "But if ideas tumble out in such a profusion, then why aren't they here now when I need them to write this little acceptance?" (J H Conway, 2000).

Now, I thank the Steele Prize Committee for selecting me for this prize. It is an unexpected honor, and I am delighted to accept it. I am indebted to my teachers, collaborators, colleagues at MIT, and students for their encouragement and inspiration over the years.

Last Updated July 2011