Harmonic Analysis on Lie Groups

李群的调和分析

• 批准号: 0301133
• 负责人: Robert Stanton
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301133&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Lie; Groups

Abstract:StantonIn previous joint work with Kroetz the proposer constructed a specific domain for which they showed that most matrix coefficients of irreducible unitary representations of semisimple Lie groups as well as related automorphic functions have a canonical holomorphic continuation. For those representations occuring in the Plancherel density of the associated Riemannian symmetric space they used their construction to identify a natural Kahler structure on these domains. We shall attempt to enlarge the class of the unitary representations so related to complex differential geometry by studying Kahler structures on vector bundles over the previously constructed domain. The success of such a complex differential geometric formulation of parts of the unitary dual should have interesting applications to harmonic analysis. Also following up on our previous work, we shall attempt to obtain estimates on the Fourier coefficients of automorphic functions by analyzing the boundary values of their holomorphic continuation on the distinguished boundary of this domain. A different project is joint work with Slupinski in which we propose to obtain very detailed structure of the topology and differential geometry of nilpotent co-adjoint orbits of semisimple Lie groups. We have identified a class of orbits associated to 5-gradings of the corresponding Lie algebra and have substantial progress towards relating this to an extended version of conformal geometry. Possible payoff of these investigations include a compactification of the moduli space of exceptional holonomy structures on low dimensional manifolds as well as a geometric construction of representations associated to these orbits.Any complex nxn matrix may be written as a sum of two matrices where one is diagonalizable and the other is nilpotent, i.e. the matrix times itself some number of times is the zero matrix. The group of invertible matrices acts via conjugation (i.e. pre-multiply by the matrix and post-multiply by the inverse) on the vector space of all complex nxn matrices. As the sets of diagonalizable and nilpotent matrices are preserved it is reasonable to seek for each matrix a representative from these classes which is in some uniformly recognizable form. For the diagonalizable class the diagonal matrices are a natural choice and are universally used. On the other hand, the nilpotent elements have only a finite number of possibilities leading to the familiar standard form called Jordan blocks in linear algebra. These finitely many possibilities of a fixed block seem to be richer in geometric structure that those of the semisimple classes but much less understood. The proposed research is give a detailed description of the differential geometry of these class of nilpotent matrices. Somewhat surprisingly, the geometry these classes lead to include ones of current interest to theoretical physicists in string theory as well as geometers. One of the applications of our work is to present a space of such geometries, and to examine possible degenerations in the geometry as one approaches the boundary of the space. Another possible application of our description is towards identifying new geometries associated with certain algebraic structures on the space of matrices. Our approach has many points of contact with a construction proposed many years ago in relativity theory by the physicist Penrose using an algebraic object called spinors. Indeed, the use of higher dimensional spinors is critical to our investigations.

翻译后摘要：StantonIn以前的联合工作与Kroetz的提议者构建了一个特定的域，他们表明，大多数矩阵系数的不可约酉表示的半单李群以及相关的自守函数有一个典型的全纯延续。对于那些出现在相关黎曼对称空间的Plancherel密度中的表示，他们使用它们的构造来确定这些域上的自然Kahler结构。我们将试图扩大类的酉表示，所以有关复杂的微分几何的研究卡勒结构向量丛在先前构建的域。成功的这样一个复杂的微分几何制定的部分酉对偶应该有有趣的应用调和分析。在我们以前工作的基础上，我们将通过分析自守函数在该区域的特殊边界上的全纯延拓的边界值来获得自守函数的Fourier系数的估计。一个不同的项目是联合工作Slupinski在其中我们建议获得非常详细的结构拓扑和微分几何的幂零共同伴随轨道的半单李群。我们已经确定了一类与相应的李代数的5分次相关的轨道，并取得了实质性的进展，这与一个扩展版本的共形几何。这些研究的可能结果包括低维流形上特殊完整结构的模空间的紧化，以及与这些轨道相关的表示的几何构造。任何复n × n矩阵可以写为两个矩阵的和，其中一个是可对角化的，另一个是幂零的，即矩阵乘以自己的某个次数是零矩阵。可逆矩阵群通过共轭（即，预乘矩阵和后乘逆矩阵）作用于所有复nxn矩阵的向量空间。由于可对角化矩阵和幂零矩阵的集合是保持不变的，所以合理的做法是从这些类中为每个矩阵寻找一个具有某种一致可识别形式的代表。 对于可对角化类，对角矩阵是一个自然的选择，并且被普遍使用。另一方面，幂零元素只有有限的可能性，导致熟悉的标准形式称为线性代数中的Jordan块。这些固定块体的许多可能性在几何结构上似乎比那些半单类的更丰富，但理解得少得多。本文对这类幂零矩阵的微分几何进行了详细的描述。有些令人惊讶的是，这些类导致的几何包括弦理论的理论物理学家以及几何学家目前感兴趣的几何。我们的工作的应用之一是提出这样的几何空间，并检查可能退化的几何形状，因为一个接近空间的边界。我们的描述的另一个可能的应用是识别与矩阵空间上的某些代数结构相关的新几何。我们的方法与物理学家彭罗斯多年前在相对论中提出的一种构造有许多联系，该构造使用了一种称为旋量的代数对象。事实上，使用高维旋量是至关重要的，我们的调查。