From January 4 to 8, 2016, the Institute of Mathematics of Academia Sinica hosted "Taipei Conference in Representation Theory V," which was dedicated to George Lusztig on the occasion of his seventieth birthday. The papers in this volume arose from that conference; they offer a sample of the many ways that George's work has transformed representation theory over the past half century.

George Lusztig was born on May 20, 1946 in Timisoara, Romania. His parents were accountants and spoke Hungarian at home; George learned Romanian when he entered kindergarten in 1949. He attended elementary and high school in Timisoara. Around 1960, in 8th grade, he began to participate in mathematical Olympiads, and decided to become a mathematician. The first two International Mathematical Olympiads took place in Romania, in 1959 and 1960. George was a member of the Romanian team in 1962 and 1963. At this time he met Maria Neumann, a mathematician at the University of Timisoara, who discussed with him books on the foundations of geometry and non-Euclidean geometry.

In 1963, George entered the University of Bucharest. He worked particularly with Kostake Teleman, studying Emil Artin's book Geometric Algebra and the Gelfand-Naimark theory of induced representations. He graduated in 1968 with a Diploma de Licenta from the Department of Mathematics and Mechanics.

In the summer of 1968, George met Isadore Singer at a conference on pseudodifferential operators in Italy. Singer suggested that he should meet Michael Atiyah. This he did, and George solved several problems (in topology and complex geometry) around the work of Atiyah and Singer. Atiyah therefore arranged for George to visit him at the Institute for Advanced Study in Princeton. George remained at the Institute from 1969 to 1971. During this time he received a Ph.D. from Princeton University (formally advised by William Browder) for work on Novikov's higher signatures.

From 1971 to 1977, George was at the University of Warwick, first as a Research Fellow and finally as a Professor. He worked with James Alexander Green and Roger Carter on representations of reductive groups. By 1972 he found a construction of the discrete series representations of $GL(n,{\mathbb F}_q)$; previously only the characters of these representations were known (by Green in 1955). This is recorded in his 1974 book The discrete series of $GL_n$ over a finite field.

In 1974, George visited the Institut des Hautes Études Scientifiques, where he began a collaboration with Pierre Deligne. Early in 1975, they had a construction of families of representations of arbitrary reductive groups over finite fields, analogous to the Gelfand-Naimark induced representations (but defined also in the absence of rational parabolic subgroups). Their 1976 Annals of Mathematics paper "Representations of reductive groups over finite fields" provided results analogous to the Cartan-Weyl representation theory for compact Lie groups.

The great difference from the Cartan-Weyl theory is that the Deligne-Lusztig representations (still for finite Chevalley groups) are only generically irreducible. George worked next to determine the irreducible representations completely. By 1977, he had completed this for the classical groups. In the process he introduced and analyzed new combinatorial structure related to classical Weyl groups.

Extending his results to the exceptional groups required a corresponding combinatorial analysis of the exceptional Weyl groups. Particularly in the case of $E_8$, when the Weyl group has 696729600 elements, this analysis was based in part on very serious computer calculations. One of his collaborators was W. Meurig Beynon, a computer scientist at University of Warwick. The work was completed by 1978 (in time to be published in honor of Atiyah's fiftieth birthday). A wonderful complete account is in George's 1984 book Characters of reductive groups over a finite field.

A key to this work was George's definition and explicit calculation in 1978 of what are now called special representations of a Weyl group.

In 1978, George came to MIT, where he remains today. That year he stated (in a letter to Anthony Joseph) a precise conjecture, joint with David Kazhdan, for the parametrization of primitive ideals in the enveloping algebra of a complex reductive Lie algebra.

In 1979, this conjecture became a part of his Inventiones paper with Kazhdan ``Representations of Coxeter groups and Hecke algebras,'' defining and calculating the Kazhdan-Lusztig polynomials $P_{x,y}$ indexed by a pair of Coxeter group elements. They conjectured that $P_{x,y}(1)$ was a coefficient in the character formula of an irreducible highest weight module $L(y)$ of highest weight $-y\rho -\rho$. During that same year, George wrote another paper with Kazhdan, "Schubert varieties and Poincaré duality," proving that $P_{x,y}$ was a Poincaré polynomial for local intersection cohomology groups for a Schubert variety $X_y$ along a smaller Schubert variety $X_x$. Their result is the most difficult step in the proof of the Kazhdan-Lusztig conjectures completed a year later by Beilinson-Bernstein and Brylinski-Kashiwara.

These two papers reshaped representation theory for real and complex reductive groups just as George's earlier work had done for groups over finite fields. The formal character theory of modules like $L(y)$ was introduced by Harish-Chandra in the 1950s, but there had been essentially no progress in computing the characters beyond the work of Weyl (which treated finite-dimensional highest weight modules: the case $y=w_0$, corresponding to the smooth Schubert variety $X_{w_0}$). Kazhdan and Lusztig showed that a way to study such characters was by means of appropriate $q$-analogues, and that the $q$-analogues could be related to computable geometry on flag manifolds. These ideas have been generalized again and again (most often by George!). What used to be called "representation theory" is now often subsumed in "geometric representation theory," which means studying versions of the geometry problems introduced in George's work.

By this time George's ideas were moving in so many different directions that writing a linear account of them seems impossible. We have already omitted his contributions to the representation theory of $p$-adic reductive groups. These began in 1977 with ideas about constructing supercuspidal representations of $p$-adic reductive groups, along the lines introduced in his work on finite Chevalley groups. In the early 1980s he formulated a program for proving some of Langlands' conjectures about $p$-adic groups, which he completed in the 1987 paper "Proof of the Deligne-Langlands conjecture for Hecke algebras" with David Kazhdan.

There are two more themes that have captured much of George's mathematical attention since 1980. One is character sheaves. These are intersection cohomology complexes on a reductive group that encode his work on characters of finite Chevalley groups, and are of tremendous independent interest. Another is canonical bases. The term refers to the method of construction of the original Kazhdan-Lusztig polynomials (by means of a new basis of any Hecke algebra). George has made the idea work in a tremendous range of situations, notably in quantum groups.

What remains characteristic of George's work is that he writes it down, first in research papers and often later (when he has understood the best way to think about a subject) in beautiful books. His 1993 book Introduction to quantum groups provided a wonderful and entirely new way to define these objects, as well as a clear and complete account of more classical definitions. The 2003 book Hecke algebras with unequal parameters provided a complete survey of the (very advanced) state to which he had already brought that subject, as well as a compelling collection of open problems.

This conference was a great success because George has made his mathematics available to all of us. We hope that these papers offer a chance to glimpse some of the mathematical worlds that he has created.

Shun-Jen Cheng, Institute of Mathematics, Academia Sinica.

David Vogan, MIT Mathematics.

Weiqiang Wang, Department of Mathematics, University of Virginia.