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

摘要：Stanton在以前与Kroetz的合作中，作者构造了一个特定的区域，在这个区域上，他们证明了半单李群的不可约么正表示以及相关的自同构函数的大多数矩阵系数都有正则全纯连续。对于那些出现在伴随黎曼对称空间的Plancherel密度中的表示，他们利用它们的构造来确定这些区域上的一个自然Kahler结构。我们将试图通过研究先前构造的区域上向量丛上的Kahler结构来扩大与复微分几何有关的酉表示的类别。部分酉对偶的复杂微分几何公式的成功应该在调和分析中有有趣的应用。在前人工作的基础上，我们将尝试通过分析自同构函数在区域可分辨边界上的全纯延拓的边界值来得到其傅里叶系数的估计。另一个不同的项目是与Slupinski的合作，在该项目中，我们建议获得半单李群的幂零余共轭轨道的拓扑和微分几何的非常详细的结构。我们已经确定了一类与相应李代数的5分次有关的轨道，并在将其与扩展形式的共形几何联系起来方面取得了实质性进展。这些研究的可能结果包括对低维流形上特殊完整结构的模空间的紧致化以及与这些轨道相关的表示的几何构造。任何复n×n矩阵可以写成两个矩阵的和，其中一个是对角化的，另一个是幂零的，即矩阵本身乘以一定次数是零矩阵。可逆矩阵群通过共轭作用于所有复n×n矩阵的向量空间上(即，预乘于矩阵，后乘以逆)。当对角化和幂零矩阵的集合保持不变时，合理地从这些类中寻找一个具有某种统一可识别形式的代表。对于可对角化的类，对角线矩阵是一种自然而然的选择，并且被普遍使用。另一方面，幂零元素只有有限数量的可能性，导致线性代数中常见的标准形式，称为Jordan块。固定块的这些有限多个可能性在几何结构上似乎比半简单类的几何结构丰富得多，但了解的要少得多。本文对这类幂零矩阵的微分几何进行了详细的刻画。有些令人惊讶的是，这些课程带来的几何学包括目前弦理论理论物理学家和几何学家感兴趣的几何学。我们的工作的一个应用是呈现一个由这样的几何组成的空间，并在接近空间边界时检查几何中可能的退化。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联的新几何。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结构有许多联系，该结构使用了一种名为旋量的代数对象。事实上，使用更高维度的旋子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.