Conference on Differential Geometry, Mathematical Physics, and Mathematics and Society

摘要：在前人与Kroetz的联合研究中，作者构造了一个特定的定义域，证明了半单李群的不可约酉表示及其相关自同构函数的大多数矩阵系数具有正则全纯延拓。对于在相关黎曼对称空间的Plancherel密度中出现的那些表示，他们使用他们的构造来识别这些域中的自然Kahler结构。我们将尝试通过在先前构造的域上研究向量束上的Kahler结构来扩大与复杂微分几何相关的酉表示的类别。这样一个复杂的微分几何公式的成功的部分的幺正对偶应该有有趣的应用于调和分析。在我们之前工作的基础上，我们将尝试通过分析自同构函数的全纯延拓在该域的区分边界上的边值来获得自同构函数的傅里叶系数的估计。另一个不同的项目是与Slupinski的联合工作，我们提出获得半单李群的幂零共伴轨道的非常详细的拓扑结构和微分几何。我们已经确定了一类与相应李代数的5阶相关联的轨道，并在将其与保形几何的扩展版本联系起来方面取得了实质性进展。这些研究的可能成果包括低维流形上异常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nxn矩阵都可以写成两个矩阵的和，其中一个是可对角的，另一个是幂零的，即矩阵乘以自身若干次是零矩阵。可逆矩阵群通过共轭作用（即前乘矩阵后乘逆矩阵）作用于所有复nxn矩阵的向量空间。由于可对角矩阵和幂零矩阵的集合是保留的，因此对每个矩阵从这些类中寻找一个具有一致可识别形式的代表是合理的。对于可对角化的类，对角矩阵是一种自然的选择，并且被普遍使用。另一方面，幂零元素只有有限的可能性，导致线性代数中熟悉的标准形式约旦块。这些有限多的固定块的可能性似乎在几何结构上比那些半简单类更丰富，但却很少被理解。本文给出了这类幂零矩阵的微分几何的详细描述。有些令人惊讶的是，这些课程所带来的几何学包括弦理论物理学家和几何学家目前感兴趣的几何学。我们工作的一个应用是呈现这样的几何空间，并检查在接近空间边界时几何可能的退化。我们的描述的另一个可能的应用是在矩阵空间上识别与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（Penrose）多年前在相对论中使用一种称为旋量的代数物体提出的构造有许多接触点。事实上，高维旋量的使用对我们的研究至关重要。

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.