DOUBLE AFFINE HECKE ALGEBRAS

双仿射赫克代数

• 批准号: 1101535
• 负责人: Ivan Cherednik
• 金额: $ 17万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101535&HistoricalAwards=false
• 关键词: DOUBLE; AFFINE; HECKE; ALGEBRAS

The aim of the project is to study double affine Hecke algebras introduced by PI. They proved to be very useful in the representation theory and found many applications in mathematics and physics. The area of the algebraic analysis and its applications is the major theme of the project, including the harmonic analysis on symmetric spaces, the theory of hypergeometric, spherical and Whittaker functions. The theory of the difference counterparts of these functions, called ``global functions" due to their universality and excellent analytic properties, is one of the greatest applications of double affine Hecke algebras obtained by PI, Stokman and other researches. The global q-Whittaker functions have and expected to have multiple applications, including the Givental-Lee theory and the quantum Langlands program; the corresponding theory of nil-DAHA is very fruitful.The theory of DAHA is a breakthrough development in the geometric, analytic and physically-inspired representation theory, which demonstrates the power of p-adic methods and tremendous potential of the q-functions. The directions of the project are mainly grouped around PI's theory of global difference spherical and Whittaker functions. There are important relations (known and expected) of these functions to the theory of homogeneous spaces of loop groups and (hopefully) to the geometric quantum Langlands program. The first results concerning the behavior of these functions at some special cases are an indication of their significance for the Number Theory. Also, these functions are expected to serve recent physics theories employing the integrability of the various quantum many-body problems.

该项目的目的是研究PI引入的双仿射Hecke代数。事实证明，它们在表示理论中非常有用，并在数学和物理学中发现了许多应用。代数分析及其应用的领域是该项目的主要主题，包括对对称空间的谐波分析，超代，球形和惠特克功能的理论。 The theory of the difference counterparts of these functions, called ``global functions" due to their universality and excellent analytic properties, is one of the greatest applications of double affine Hecke algebras obtained by PI, Stokman and other researches. The global q-Whittaker functions have and expected to have multiple applications, including the Givental-Lee theory and the quantum Langlands program; the corresponding theory of nil-DAHA is very DAHA的富有成效的理论是几何，分析和物理启发的代表理论，证明了P-Adic方法的力量和Q-功能的巨大潜力，该项目的方向主要围绕PI的全球差异和惠特克的关系（这些范围）的关系。 （希望）几何量子兰兰斯计划。同样，这些功能有望使用各种量子多体问题的整合性来服务最近的物理学理论。