Kazhdan-Lusztig theory is one of the most important developments in the representations of algebraic groups in the last 40 years. This theory played key roles in the solutions of many important problems, for example, the classification of irreducible characters of finite groups of Lie type, the determination of the characters of some irreducible representations in Lie theory. In the meantime, this theory initiates many active research areas, for instance, Kazhdan-Lusztig polynomials, cells of Coxeter groups, the connections between Coxeter groups and intersection cohomology and K-theory, etc. This talk will briefly introduce the origin, developments, influences of this theory and some unsolved problems.

Professor Nanhua Xi is a Member of Chinese Academy of Sciences, Vice-President of University of Chinese Academy of Sciences, President of Academy of Mathematics and Systems Science of Chinese Academy of Sciences. His main research interests are algebraic groups and quantum groups. He has been awarded with National Natural Science Award (second prize), Shiing-Shen Chern Mathematics Award and Outstanding Young Scholar of National Science Foundation of China, etc.