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

设G为实约李群，K为极大紧子群。分别写出复化李代数的g和k，设X是一个（g, k）模。研究者的研究目标是确定X的k类型集合的非组合描述，首先在X是离散级数表示的情况下，其次在X是上同调诱导模的情况下。这将分三步完成：1)精确定义k型集合的“边”的概念；2)利用上同调方法识别在某条边上的k型；3)利用边缘k型的凸包来描述整个k型集合。研究者的方法是基于D. Vogan关于上同调的某种限制映射的猜想。如果q=l+u是一个具有Levi分量l的稳定抛物线，Vogan猜想指出，从u上同调到u的限制映射的像与k上同调相交，参数化了位于k的Weyl室壁中不包含的边缘上的k型。与该猜想相关的思想提供了接近所提出目标的主要技术。从全局的角度来看，这个项目的动机是世界上对称性的存在。这种对称性以多种形式表现出来，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，一个特别强大的数学工具被称为表示理论。从广义上讲，表征理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，但仍然存在许多难题。用于研究这些遗留问题的一个重要技术是检查表征的k类型。从广义上讲，k类型集合是对复杂表示的略微简化，使其更易于管理。事实证明k类型集合是一种非常有效的方法既可以构造表征也可以推断表征的特殊性质。研究者的研究旨在提供k类型集合的几何描述，这将使更详细的表征分析成为可能。

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.