Double Affine Hecke Algebras

二重仿射Hecke代数Ivan Cherednik这个项目的目的是研究双仿射Hecke代数(DAHA)及其表示，特别是那些在调和分析中的应用。重点是Daha的多项式表示的结构、它的分解以及一般Q、单位根和么模Q的特殊商。主要例子是从共形场论推广Verlinde代数的多项式表示的有限维商；无限维商和非半单商是自然的下一步。人们期望与仿射Hecke代数的经典理论、Kac-Moody代数理论和现代数学物理有很深的联系。Lie群和Lie代数使特殊函数论、物理学、几何学和组合学中的对称性概念形式化。同样，Daha也是傅里叶变换概念非形式化的候选者。在最简单的一维情况下，它们与著名的sl(2)直接相关。有迹象表明，它们比Lie和Kac-Moody代数更好地服务于傅里叶变换的多维理论。达哈也是新的单位无限维理论的来源，这些理论可能会在数学和物理、特殊函数理论和组合学中找到基本的应用。达哈理论还有其他重要的方向，主要是代数性质的，但么正表示和傅立叶分析仍然是最优先的。

Double Affine Hecke AlgebrasIvan CherednikThe aim of the project is to study double affine Heckealgebras (DAHA) and their representations, especially thosewith applications in harmonic analysis. The focus is thestructure of the polynomial representation of DAHA, itsdecomposition and special quotients for generic q, at rootsof unity and for unimodular q. The main examples are finitedimensional quotients of the polynomial representationgeneralizing the Verlinde algebras from conformal fieldtheory; infinite dimensional and non-semisimple quotientsare a natural next step. Deep connections to the classicaltheory of affine Hecke algebras, the theory of Kac-Moodyalgebras, and modern mathematical physics are expected.Lie groups and Lie algebras formalize the concept of symmetryin the theory of special functions, physics, geometry, andcombinatorics. In a similar way, DAHA are candidates for aformalization of the notion of the Fourier transform. In thesimplest one-dimensional case, they are directly related tothe celebrated sl(2). There are indications that they servethe multidimensional theory of Fourier transform better thanLie and Kac-Moody algebras. DAHA are also a source of new unitaryinfinite dimensional theories, that, presumably, will findfundamental applications in mathematics and physics, thetheory of special functions and combinatorics. There are otherimportant directions of the DAHA theory, mainly of algebraicnature, but unitary representaions and Fourier analysis remainof high priority.