Hecke Algebras, MacDonald Polynomials, and Applications

该奖项为一个研究仿射和双仿射Hecke代数的项目提供资金。后一类代数(最近由主要研究者介绍)为解决表示论、特殊函数、调和分析、保形场理论、组合学、拓扑学和数论中的各种问题提供了一种新的途径。主要目标是(1)作用于多项式、theta函数和各种推广的微分和差分算子的理论，(2)双仿射Hecke代数在单位根的表示，与双仿射KZ方程和椭圆辫子群的单行的联系，(3)模群对Macdonal多项式的作用，(4)与椭圆型量子群、Kac-Moody代数、矩阵模型和W-代数的关系，以及(5)在调和分析中的应用。这项研究属于组合学的一般领域。组合学试图找到有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。这项研究还涉及到了在光传输线设计中有用的正交多项式。

This award provides funds for a project to study affine and double affine Hecke algebras. The latter algebras (recently introduced by the principal investigator) lead to a new approach to various problems in representation theory, special functions, harmonic analysis, conformal field theory, combinatorics, topology, and number theory. The main objectives are (1) the theory of differential and difference operators acting on polynomials, theta functions and various generalizations, (2) representations of double affine Hecke algebras at roots of unity, connections with the monodromy of the double affine KZ equations and elliptic braid groups, (3) the action of the modular group on the Macdonal polynomials, (4) relations to quantum groups of elliptic type, Kac-Moody algebras, matrix models, and W-algebras, and (5) applications to harmonic analysis. This research is in the general area of Combinatorics. Combinatorics attempts to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. This research also deals with orthogonal polynomials, which are useful in designing optical transmission lines.