DOUBLE AFFINE HECKE ALGEBRAS

本项目旨在研究PI引入的双仿射Hecke代数。它们被证明在表示理论中非常有用，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，包括对称空间的调和分析，超几何理论，球面和惠特克函数。这些函数的差分对应物的理论，由于其通用性和优异的解析性质而被称为“全局函数”，是PI、Stokman等人所获得的双仿射Hecke代数的最大应用之一。全局q-Whittaker函数已经并有望有多种应用，包括Givental-Lee理论和量子Langlands计划；相应的无- daha理论是非常富有成果的。DAHA理论是几何、解析和物理表示理论的突破性发展，它展示了p进方法的力量和q函数的巨大潜力。项目的方向主要围绕PI的全局差分球理论和惠特克函数。这些函数与环群的齐次空间理论和（希望）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初步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.