Double Hecke Algebras and Applications

这个项目的目的是研究双仿射Hecke代数介绍了博士切雷德尼克这承诺重要的应用到许多其他分支的数学。 具体目标包括研究麦克唐纳多项式的双仿射Hecke代数根的单位，以及诱导和球面表示的代数。 Cherednik博士还打算探索q-球面函数的分析理论，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群的调和分析中的应用。 这是一个被称为代数的数学领域的项目。 在20世纪，代数已经成为所有数学的语言。 研究几何、数论、分析、数学物理、统计学或几乎任何东西的数学家都注意到，在他们的研究中有超越其起源的潜在结构。 在代数中，这些基本结构是以它们的纯形式来研究的。 当这些结构的基本属性被揭示出来时，其含义就会反映到这些结构发生的所有原始环境中。 在早期的研究中，Cherednik博士引入了一种新的代数结构，称为双仿射Hecke代数，它出现在表示论、特殊函数、调和分析、组合学和共形场论中。在这个项目中，他将继续他的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.