Double Hecke Algebras and Applications

这个项目的目的是研究双仿射Hecke代数(DAHA)，这是由首席研究员先前介绍的。它们在表象理论、特殊函数和球函数理论、共形场理论和组合学中被证明是非常有用的。主要目的包括：(1)Daha的半单表示、球面表示、酉表示和傅里叶单变量表示的一般分类；(2)一般q的Daha的有限维表示的分类及其在Macdonald的ETA型恒等式、Kac-Moody特征标和退化的Bessel函数中的应用；(3)Daha在单位根处的表示的理论，与双仿射KZ方程和椭圆辫子群的单行的联系；(4)Daha的傅立叶不变的单位表示的分类及其在Gauss-Selberg和和Verlinde代数中的应用；(5)达哈的半单表示在紧性和非紧性、单位根和椭圆情况下的调和分析。这是数学领域的一个项目，称为表示理论。表示论的本质，以及它在数学和科学的许多领域中如此有用的原因，是它是一种研究和编码对称的方法，无论它们可能发生在哪里，在自然中还是抽象的。本项目的目的是研究和开发一种新的表示论工具--“双仿射Hecke代数”的性质，该工具可用于将表示论应用于物理(保形场论)和组合学。组合学是一门计算任何东西(例如原子、鹅卵石、星团)的可能排列和组织方式的科学。

The purpose of this project is to study double affine Hecke algebras (DAHA), as previously introduced by the principal investigator. They have proved to be very useful in representation theory, the theory of special and spherical functions, conformal field theory, and combinatorics. The main objectives include: (1) A general classification of the semisimple, spherical, unitary, and Fourier-univariant representations of DAHA; (2) the classification of finite dimensional representations of DAHA for generic q, with applications to Macdonald's eta-type identities, Kac-Moody characters, and degenerate Bessel functions; (3) the theory of representations of DAHA at roots of unity, connections with the monodromy of the double affine KZ equations and elliptic braid groups; (4) classificaton of the Fourier-invariant uitary representations of DAHA with applications to Gauss-Selberg sums and Verlinde algebras; (5) harmonic analysis on semisimple representations of DAHA in the compact and noncompact cases, at roots of unity, and in the elliptic case.This is a project in the area of mathematics known as Representation Theory. The nature of representation theory, and what makes it so useful in so many areas of mathematics and science, is that it is a way of studying and encoding symmetries, whereever they may occur, in nature or abstractly. The purpose of this project is to study and develop the properties of a new representation-theoretic tool, "double affine Hecke algebras," that can be used in the application of representation theory to physics (conformal field theory) and combinatorics. Combinatorics is the science of counting the possible arrangements and ways of organizing collections of anything (e.g., atoms, pebbles, star clusters).