# George Lusztig Extras

We give below information about George Lusztig being awarded (1) the Shaw Prize in Mathematical Sciences for 2014 and (2) the Wolf Prize in Mathematics for 2022.

**1. George Lusztig: Shaw Prize in Mathematical Sciences for 2014.**

**Contribution of George Lusztig.**

The Shaw Prize in Mathematical Sciences for 2014 is awarded to George Luzstig, Abdun-Nur Professor of Mathematics at the Massachusetts Institute of Technology, for his fundamental contributions to algebra, algebraic geometry, and representation theory, and for weaving these subjects together to solve old problems and reveal beautiful new connections.

For more than two hundred years, symmetry groups have been at the centre of mathematics and its applications: in Fourier's work on the heat equation in the early 1800s; in Weyl's work on quantum mechanics in the early 1900s; and in the approach to number theory created by Artin and Chevalley. These classical works show that answers to almost any question involving a symmetry group lie in understanding its realisations as a group of matrices; that is in terms of its representations.

Starting with his early work in the 1970s and 1980s, in part jointly with Deligne, Lusztig gave a complete description of the representations of finite Chevalley groups, these being the building blocks of finite symmetry groups. The Deligne-Lusztig description uses the topology and geometry of Schubert varieties. The latter were introduced in the nineteenth century as a tool to count solutions of algebraic equations.

The vision of this work is that the algebraic subtleties of representation theory correspond perfectly to the geometric/topological subtleties of Schubert varieties. This vision has grown into a broad and powerful theme in Lusztig's work: he has shown that many central problems in representation theory - including those of real and $p$-adic Lie groups, which are the language of applications from number theory to mathematical physics - can be related to topology and geometry by means of Schubert varieties. This idea is at the heart of many exciting recent developments, for example in progress toward the Langlands programme in automorphic forms.

Representations are complicated, as are the Schubert varieties to which they are related. Beginning in a 1979 paper with David Kazhdan, and continuing through his most recent work, Lusztig has found combinatorial tools to describe their topology and geometry. (These tools are easy to describe, but had not been used previously in mathematics.) His ideas have guided and inspired the development of perverse sheaves, a tool for studying the topology of general singular algebraic varieties.

These tools, in the hands of Lusztig and of hundreds of other mathematicians, have made possible a depth of understanding of representations and of Schubert varieties that was unimaginable before his work.

Mathematical Sciences Selection Committee

The Shaw Prize

27 May 2014 Hong Kong

**An Essay on the 2014 Shaw Prize.**

For more than two hundred years, symmetry groups have been at the centre of mathematics and its applications: in Fourier's work on the heat equation in the early 1800s; in the work of Weyl and Wigner on quantum mechanics in the early 1900s; and in the approach to number theory created by Artin and Chevalley. These classical works show that answers to almost any question involving a symmetry group lie in understanding its realisations as a group of linear transformations, that is, in terms of its representations.

Lusztig's work has completely transformed our understanding of representation theory, providing complete and precise answers to fundamental questions that were understood before only in very special cases. What he has done has advanced all of the mathematics where symmetry groups play a role: from Langlands' programme for understanding automorphic forms in number theory, to classical problems of harmonic analysis on real Lie groups.

Here are some hints of the ideas at the heart of Lusztig's work. The most basic symmetry group in mathematics is $GL(n)$, the group of invertible $n × n$ matrices. (The entries in the matrices will be different in different problems.) One of the most important invariants of a matrix is the collection of its $n$ eigenvalues. The eigenvalues do not come with an ordering, so they can be rearranged using the permutation group $S_{n}$ (consisting of the $n!$ permutations of $(1, 2, ..., n)$. A consequence (understood already in the nineteenth century) is that some of the properties of the (large and complicated) group $GL(n)$ can be encoded by the (smaller and simpler) group $S_{n}$.

All permutations of $(1, ..., n)$ may be built up from $n-1$ "simple transpositions": the permutations that exchange $i$ with $i+1$, and leave everything else alone. For example, we can reverse the order of (1, 2, 3) by first exchanging the 1 and the 2, then exchanging the 2 and the 3, and finally exchanging the 1 and the 2 again:

$(1, 2, 3) → (2, 1, 3) → (3, 1, 2) → (3, 2, 1)$.

Elementary results from linear algebra provide a beautiful decomposition of $GL(n)$ into pieces labelled by permutations; these are essentially the Schubert varieties for $GL(n)$. The piece labelled by the trivial permutation consists of the upper triangular matrices, where all problems of linear algebra are easy. The piece labelled by the simple transposition $s_{i}$ consists of matrices with a single nonzero entry below the diagonal, in column $i$ and row $(i + 1)$. As the permutation gets larger, the corresponding matrices become less and less upper triangular, and the corresponding Schubert variety gets more and more complicated.

A first theme in Lusztig's work is that the algebraic subtleties of representation theory correspond perfectly to the topological and geometric subtleties of Schubert varieties. Although this idea is foreshadowed in earlier works, such as that of Borel and Weil in the 1950s on representations of compact groups, Lusztig's results have revolutionised the field. Mathematicians talk about geometric representation theory to distinguish what is possible now from its antecedents.

Representations are complicated, as are the Schubert varieties to which Lusztig's work relates them. A second theme in his work is the creation of combinatorial tools - easy to describe, but almost without precedent in mathematics - to describe in fantastic detail the topology and geometry of Schubert varieties. This theme begins in a 1979 paper with David Kazhdan, and continues through Lusztig's most recent work. The idea is to build complicated Schubert varieties from simpler ones exactly as complicated permutations are built from simpler ones, by multiplication of simple transpositions. This description of permutations has been studied in combinatorics for more than a hundred years, but Lusztig's ideas have opened whole new fields of investigation for them.

A third theme in Lusztig's work is the new field of quantum groups, introduced by Vladimir Drinfeld in the 1980s. Lusztig has said that the test of a new theory is whether you can use it to answer questions that were asked before the new theory existed. He has made quantum groups pass that test in a number of amazing ways. For example, his theory of canonical bases (which can be introduced only using quantum groups) allowed him to extend the classical theory of "totally positive matrices" from GL(n) to all reductive groups.

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

Mathematical Sciences Selection Committee

The Shaw Prize

24 September 2014 Hong Kong

**2. George Lusztig: Wolf Prize in mathematics 2022.**

George Lusztig, Massachusetts Institute of Technology, USA, was awarded the Wolf Prize in mathematics 2022 "for Groundbreaking contributions to representation theory and related areas."

Lusztig is a Romanian-American mathematician, who works on geometric finite reductive groups, representation theory and algebraic groups. 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. Lusztig's ground-breaking contributions mark him as one of the great mathematicians of our time.

His passion for mathematics began at a young age. In fact, it was in math competitions at school which made him realise that he was talented in mathematics. After finishing 10th grade, Lusztig represented Romania in the International Mathematical Olympiad in 1962 and then again, in 1963: being awarded a Silver Medal on both occasions. Lusztig graduated from the University of Bucharest in 1968 and received both the M.A. and Ph.D. from Princeton University in 1971 under the direction of Michael Atiyah and William Browder. He joined the MIT mathematics faculty in 1978 following a professorship appointment at the University of Warwick, 1974-77. He was appointed Norbert Wiener Professor at MIT 1999-2009.

Lusztig is known for his work on representation theory, in particular for the objects closely related to algebraic groups, such as finite reductive groups, Hecke algebras, p-adic groups, quantum groups, and Weyl groups. He essentially paved the way for modern representation theory. This has included fundamental new concepts, including the character sheaves, the "Deligne–Lusztig" varieties, and the "Kazhdan–Lusztig" polynomials.

Lusztig's first breakthrough came with Deligne around 1975, with the construction of Deligne-Lusztig representations. He then obtained a complete description of the irreducible representations of reductive groups over finite fields. Lusztig's description of the character table of a finite reductive group rates as one of the most extraordinary achievements of a single mathematician in the 20th century. To achieve his goal, he developed a panoply of techniques which are in use today by hundreds of mathematicians. The highlights include the use of étale cohomology; the role played by the dual group; the use of intersection cohomology, and the ensuing theory of character sheaves, almost characters, and the noncommutative Fourier transform.

In 1979 Kazhdan and Lusztig defined the "Kazhdan-Lusztig" basis of the Hecke algebra of a Coxeter group and stated the "Kazhdan-Lusztig" conjecture. The "Kazhdan-Lusztig" conjecture led directly to the "Beilinson-Bernstein" localization theorem, which four decades later, remains our most powerful tool for understanding representations of reductive Lie algebras. Lusztig's work with Vogan then introduced a variant of the "Kazhdan-Lusztig" algorithm to produce "Lusztig-Vogan" polynomials. These polynomials are fundamental to our understanding of real reductive groups and their unitary representations.

In the 1990s, Lusztig made seminal contributions to the theory of quantum groups. His contributions include the introduction of the canonical basis; the introduction of the Lusztig form (which allows specialisation to a root of unity, and connections to modular representations); the quantum Frobenius and a small quantum group; and connections to the representation theory of affine Lie algebras. Lusztig's theory of the canonical basis (and Kashiwara's parallel theory of crystal bases) has led to deep results in combinatorics and representation theory. Recently there has been significant progress in representation theory and low-dimensional topology via "categorification"; the roots of this work go back to Lusztig's geometric categorification of quantum groups via perverse sheaves on quiver moduli.

Last Updated December 2023