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由PI引入的双仿射Hecke代数。事实证明，它们在表示论中非常有用，并在数学和物理中得到了许多应用。代数分析及其应用领域是该项目的主要主题，包括对称空间的调和分析，超几何，球面和惠特克函数理论。这些函数的差对应理论由于其普遍性和优良的解析性质而被称为“整体函数”，是PI，Stokman等研究者对双仿射Hecke代数的最大应用之一。整体q-Whittaker函数有着广泛的应用前景，包括Givental-Lee理论和量子Langlands计划，相应的nil-DAHA理论也取得了丰硕的成果，DAHA理论是几何表示论、解析表示论和物理表示论的突破性发展，充分展示了p-adic方法的强大功能和q-函数的巨大潜力。该项目的方向主要围绕PI的全局差分球面和Whittaker函数理论进行分组。有重要的关系（已知和预期），这些职能的理论齐性空间的循环群和（希望）的几何量子朗兰兹计划。关于这些函数在某些特殊情况下的行为的第一个结果表明了它们对数论的意义。此外，这些功能预计将服务于最近的物理理论采用的各种量子多体问题的可积性。