Double Hecke Algebras and Applications

这个项目的目的是研究双重仿射Hecke代数（DAHA），正如之前介绍的主要研究者。 他们已被证明是非常有用的表示理论，理论的特殊和球函数，共形场理论，和组合。 主要目标包括：（1）DAHA的半单、球、酉和傅立叶单变表示的一般分类;（2）DAHA对一般q的有限维表示的分类，并应用于Macdonald的eta型恒等式、Kac-Moody特征标和退化Bessel函数;（3）DAHA在单位根上的表示理论，与双仿射KZ方程的单值性和椭圆辫群的关系;（4）DAHA的Fourier不变酉表示的分类及其在Gauss-Selberg和与Verlinde代数中的应用;（5）紧和非紧情形下DAHA的半单表示的调和分析，单位根，在椭圆的情况下，这是一个数学领域的项目，被称为表示论。 表示论的本质，以及使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中出现在哪里。 该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，该工具可用于表示论在物理学（共形场论）和组合学中的应用。 组合学是计算可能的安排和组织任何集合的方式的科学（例如，原子、卵石、星星团）。

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).