DOUBLE AFFINE HECKE ALGEBRAS

本课题的目的是研究PI引入的双仿射Hecke代数。事实证明，它们在表象理论中非常有用，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，包括对称空间上的调和分析，超几何、球面和惠特克函数理论。由于这些函数的普适性和良好的分析性质，这些函数的差对应理论被称为“整体函数”，是PI、Stokman等人研究的双仿射Hecke代数的最大应用之一。整体q-Whittaker函数已经并有望有多种应用，包括Givental-Lee理论和量子朗兰兹计划；相应的Nil-Daha理论是非常富有成果的。Daha理论是几何、解析和物理启发表示理论的突破性发展，它展示了p-adine方法的能力和Q-函数的巨大潜力。该项目的方向主要围绕派的整体差分球函数和惠特克函数理论展开。这些函数与循环群的齐次空间理论和几何量子朗兰兹程序有重要的关系(已知的和预期的)。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，这些函数有望服务于利用各种量子多体问题的可积性的最新物理理论。

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.