Double Hecke Algebras and Applications

本项目旨在研究Cherednik博士提出的双仿射Hecke代数，该代数有望在许多其他数学分支中得到重要应用。具体目标包括研究双仿射Hecke代数在单位根处的Macdonald多项式，以及代数的诱导和球面表示。Cherednik博士还打算探索q-球函数的解析理论，以及差分傅立叶变换，包括高斯积分、Zeta函数和L-函数的q-变形。最后讨论了双仿射Hecke代数在Harish-Chandra理论和Kac-Moody群谐波分析中的应用。这是数学领域的一个项目，叫做代数。在20世纪，代数已经成为所有数学领域的通用语言。研究几何、数论、分析、数学物理、统计学或任何学科的数学家都注意到，他们的研究中存在着超越其起源的潜在结构。在代数中，这些潜在的结构以它们的纯粹形式进行研究。随着这些结构的基本属性被揭示，其含义会回响回这些结构发生的所有原始设置。在早期的研究中，Cherednik博士介绍了一种新的代数结构，称为双仿射Hecke代数，它出现在表示理论、特殊函数、调和分析、组合学和共形场论中。在这个项目中，他将继续研究这个新的代数对象。

The aim of this project is to study the double affine Hecke algebra introduced by Dr. Cherednik which promises important application to many other branches of mathematics. Specific objectives include the study of Macdonald polynomials of double affine Hecke algebras at roots of unity, as well as induced and spherical representations of the algebras. Dr. Cherednik also intends to explore the analytic theory of q-spherical functions, and the difference Fourier transformation including q-deformations of the Gauss integrals, Zeta functions, and L- functions. Finally applications of double affine Hecke algebra to Harish-Chandra theory and the harmonic analysis on Kac-Moody groups will be considered. This is a project in the area of mathematics known as Algebra. In the 20-th century, algebra has become the language of all of mathematics. Mathematicians studying geometry, number theory, analysis, mathematical physics, statistics, or just about anything have noticed that there are underlying structures in their studies that transcend their origins. In Algebra these underlying structures are studied in their pure form. As the fundamental properties of these structures are uncovered, the implications reverberate back to all the original settings in which these structures occur. In earlier research Dr. Cherednik introduced a new structure to algebra known as a double affine Hecke algebra which appears in representation theory, special functions, harmonic analysis, combinatorics, and conformal field theory. In this project he will continue his study of this new algebraic object.