# Massachusetts Institute of Technology: Mathematician George Lusztig receives Wolf Prize

George Lusztig, the Abdun-Nur Professor of Mathematics at MIT, has been awarded the prestigious 2022 Wolf Prize in Mathematics for his work on geometric representation theory and algebraic groups.

The Israel-based Wolf Foundation cited the American-Romanian mathematician “for groundbreaking contributions to representation theory and related areas.” 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.

His contributions include the determination of the character table of finite reductive groups, his development of canonical bases for quantum groups and Hecke algebras, and his introduction of total positivity.

“His 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,” says the Wolf Foundation. “Lusztig’s groundbreaking contributions mark him as one of the great mathematicians of our time.”

Growing up in Romania, Lusztig discovered an early passion for mathematics. In eighth grade, he began representing Romania in the International Mathematical Olympiad, and was awarded silver medals in 1962 and 1963.

After graduating from the University of Bucharest in 1968, he received his MA and PhD from Princeton University in 1971 under Michael Atiyah and William Browder. He was a professor at the University of Warwick from 1974 to 1977, and joined the MIT mathematics faculty in 1978.

His work helped to usher in modern representation theory, with fundamental new concepts that include the character sheaves, the “Deligne-Lusztig” varieties, and the “Kazhdan-Lusztig” polynomials.

Lusztig’s first breakthrough came with Pierre Deligne around 1975, with the construction of Deligne-Lusztig representations. He 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,” says the Wolf Foundation. “To achieve his goal, he developed a panoply of techniques which are in use today by hundreds of mathematicians.”

Such techniques 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, David 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 MIT mathematics professor David Vogan then introduced a variant of the “Kazhdan-Lusztig” algorithm to produce “Lusztig-Vogan” polynomials. These polynomials are said to be fundamental to the 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 Masaki Kashiwara’s parallel theory of crystal bases) has led to deep results in combinatorics and representation theory. Recent progress in representation theory and low-dimensional topology via “categorification” are connected to Lusztig’s geometric categorification of quantum groups via perverse sheaves on quiver moduli.

He was appointed the Norbert Wiener Professor at MIT from 1999 to 2009, and currently holds the Abdun-Nur professorship. In 2014, Lusztig was the recipient of the 2014 Shaw Prize in mathematics. He used a significant part of this prize to help establish the George Lusztig PRIMES mentorships.

Other prizes and honors include the Berwick Prize of the London Mathematical Society, the AMS Cole Prize in Algebra, the Brouwer Medal of the Dutch Mathematical Society, and the AMS Leroy P. Steele Prize for Lifetime Achievement. Lusztig is a fellow of the Royal Society, a fellow of the American Academy of Arts and Sciences, and a member of the National Academy of Sciences.

The Wolf Prize in each field consists of a certificate and a monetary award of $100,000. Lusztig is among 345 scientists and artists to receive this honor since the prize was created in 1978 “for achievements in the interest of mankind and friendly relations amongst peoples” in fields ranging from physics, chemistry, mathematics, and agriculture to painting and sculpture, music, and architecture. MIT professor of mathematics Michael Artin also received the Wolf Prize in 2013.

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