Mathematical Sciences: Boundaries of K-Types and Restriction of Cohomology

抽象的Sepanski 设G是真实的约化李群，K是极大紧子群.记g和k分别为复化李代数，设X为（g，K）模。 调查员的研究的目的是确定一个非组合的描述的一套K型的X第一的情况下，X是一个离散的系列表示，第二的情况下，X是一个上同调诱导模块。 这将在三个步骤中完成：1）精确K-类型集合的“边”的概念; 2）使用上同调方法识别位于边上的K-类型;以及3）使用边K-类型的凸船体来描述整个K-类型集合。 研究者的方法是基于D. Vogan关于上同调限制图的一个定理。如果q=l+u是具有Levi分量l的θ-稳定抛物方程， Vogan猜想指出，从u上同调到u相交k上同调的限制映射的图像参数化了位于K的Weyl室壁中不包含的边上的K型。 与这个猜想相关的想法提供了接近所提出的目标的主要技术。 从全球的角度来看，这个项目的动机是世界上存在的对称性。 这种对称性以各种形式表现出来，例如化学中某些晶体的晶格结构，高能物理中亚原子粒子的行为，或工程中描述流体流动的方程。 在所有这些不同的应用中， 其中一个特别强大的数学工具就是表象论。 从广义上讲，表示论是对给定系统的所有可能对称性的研究。 希望能对所有的对称性进行分类，并提供每一种对称性的详细信息。 虽然在实现这一目标方面取得了很大进展，但仍存在许多困难问题。 保持。 用来研究这些遗留问题的一个重要技术是检查被称为表示的K类型。 从广义上讲，K类型集是将复杂的表示稍微简化为更易于管理的东西。 事实证明，K型集合是构造表示和推导表示的特殊性质的一种非常有效的方法。 研究者的研究旨在提供K型集合的几何描述 这将使更详细地分析陈述成为可能。

Abstract Sepanski Let G be a real reductive Lie group with K a maximally compact subgroup. Write g and k for the complexified Lie algebras, respectively, and let X be a (g, K) module. The objective of the investigator's research is to determine a non-combinatorial description for the set of K-types of X first in the case where X is a discrete series representation and second in the case where X is a cohomologically induced module. This will be accomplished in three steps: 1) make precise the notion of an "edge" of the set of K-types; 2) identify the K-types that lie on an edge using cohomological methods; and 3) use the convex hull of the edge K-types to describe the entire set of K-types. The investigator's methods are based on a conjecture of D. Vogan concerning a certain restriction map of cohomology. If q=l+u is a theta-stable parabolic with Levi component l, Vogan's conjecture states that the image of the restriction map from u cohomology to u intersect k cohomology parameterizes the K-types lying on edges that are not contained in a Weyl chamber wall of K. The ideas associated to this conjecture provide the main techniques for approaching the proposed objective. From a global point of view, this project is motived by the existence of symmetry in the world. This symmetry manifests itself in a variety of forms such as the lattice structure of certain crystals in chemistry, the behavior of subatomic particles in high energy physics, or the equations describing fluid flow in engineering. In all these diverse applications, one of the mathematical tools that is particularly powerful is known as representation theory. In a broad sense, representation theory is the study of all possible symmetries of a given system. The hope is to classify all symmetries and to provide detailed information about each one. While great strides have been made towards this goal, many difficult problems remain. An important technique used to study these remaining problems is to examine somethin g known as the K-types of a representation. In a broad sense, the set of K-types is a slight simplification of a complicated representation to something that is more manageable. It turns out that the set of K-types is a very effective way to both construct representations and to deduce special properties of representations. The investigator's research aims to provide a geometrical description of the set of K-types that would make a more detailed analysis of representations possible.