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

仿射Hecke代数和量子群表示论中的几何方法

• 批准号: 9732805
• 负责人: George Lusztig
• 金额: $ 56.85万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9732805&HistoricalAwards=false
• 关键词: Geometric; Methods; Representation; Theory; Affine

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.

9732805 Lusztig 该奖项支持对仿射 Hecke 代数所附的周期 W 图的继续研究，并调查它们与正特征半简单李代数的无限制表示的可能联系。 首席研究员还将从反常滑轮的角度研究量化包络代数的规范基。 他将继续研究半简单 p 进群的单能表示，以及（可能不相连的）还原群上的字符滑轮。 李代数和李群的表示理论最初发展于世纪之交，研究对称性的结构及其实现。 该理论涵盖了数学的许多领域，并且在理论物理中具有基础应用。 在过去的二十年里，对一类特殊仿射李代数和李群的表示论的研究在数学和理论物理领域带来了许多意想不到的新发现，并综合了这两个学科中许多已建立的领域。 特别是，它产生了最简单的量子场论的纯数学描述——基本相互作用的原型理论。 预计新的无限维李代数和群的表示论可能会在未来十年内大大加深我们对数学和理论物理的理解。