Double Hecke Algebras and Applications

这个项目的目的是研究双仿射Hecke代数（DAHA），正如之前由首席研究员介绍的那样。它们已被证明在表示理论、特殊函数和球函数理论、共形场论和组合学中非常有用。主要目标包括：(1)DAHA的半单、球面、酉和傅立叶不变表示的一般分类；(2)一般q的DAHA有限维表示的分类，并应用于Macdonald’s eta型恒等式、Kac-Moody特征和退化贝塞尔函数；(3) DAHA在单位根的表示理论，与双仿射KZ方程和椭圆辫群的单态的联系；(4) DAHA的傅里叶不变酉表示的分类及其在Gauss-Selberg和和Verlinde代数中的应用；(5)在紧、非紧、单位根和椭圆情况下DAHA的半简单表示的调和分析。这是一个在数学领域被称为表征理论的项目。表征理论的本质，以及它在数学和科学的许多领域如此有用的原因，在于它是一种研究和编码对称性的方法，无论它们在自然界或抽象的任何地方出现。本项目的目的是研究和发展一种新的表征理论工具的性质，“双仿射Hecke代数”，可用于将表征理论应用于物理学（共形场论）和组合学。组合学是一门计算任何事物（如原子、鹅卵石、星团）集合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).