# George Lusztig

Wolf Prize Laureate in Mathematics 2022

**George Lusztig**

#### Affiliation at the time of the award:

**Massachusetts Institute of Technology, USA**

#### Award citation:

**“for Groundbreaking contributions to representation theory and related areas”.**

#### Prize share:

**None**

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 groundbreaking 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 realize 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 specialization 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.