Geometric Methods in the Representation Theory of Affine Hecke Algebras and Quantum Groups

这一奖项支持了对仿射Hecke代数上的周期w图的继续研究，以及它们与半单李代数的正特征无限制表示的可能联系的研究。首席研究员还将从反常束的角度研究量子化包络代数的规范基。他将继续研究半单p进群的单能表示，以及约化群上（可能不相连的）的字符束。李代数和李群的表示理论，最初是在世纪之交发展起来的，研究对称的结构及其实现。这一理论涵盖了数学的许多领域，并在理论物理中有基本的应用。在过去的二十年中，对一类特殊仿射李代数和李群的表示理论的研究，在数学和理论物理领域取得了许多意想不到的新发现，并对这两个学科的许多既定领域进行了综合。特别是，它产生了一个最简单的量子场论的纯数学描述——一个基本相互作用的原型理论。在未来十年内，无限维李代数和群的新类别的表示理论可能会大大加深我们对数学和理论物理的理解。

9732805 Lusztig This award supports the continued study of periodic W-graphs attached to affine Hecke algebras and the investigation of their possible connections with unrestricted representations of semisimple Lie algebras in positive characteristic. The principal investigator will also study canonical bases of quantized enveloping algebras from the point of view of perverse sheaves. He will continue the study of unipotent representations of semisimple p-adic groups, as well as character sheaves on (possibly disconnected) reductive groups. Representation theory of Lie algebras and Lie groups, initially developed at the turn of the century, studies the structure of symmetries and their realizations. This theory encompasses many areas in mathematics and has fundamental applications to theoretical physics. In the past twenty years, the study of representation theory of a special class of affine Lie algebras and Lie groups led to many new unexpected discoveries in mathematics and theoretical physics, as well as to a synthesis of many established areas in both disciplines. In particular, it yielded a pure mathematical description of a simplest quantum field theory -- a prototype theory of fundamental interactions. It is expected that representation theory of new classes of infinite-dimensional Lie algebras and groups might substantially deepen our understanding of mathematics and theoretical physics during the next decade.