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

Sepanski设G是实约化李群，K是极大紧子群。对复化李代数分别记g和k，设X是(g，K)模。研究人员的研究目的是首先在X是离散级数表示的情况下确定X的K型集合的非组合描述，然后在X是上同调诱导模的情况下确定X的非组合描述。这将通过三个步骤来完成：1)精确地定义K-型集合的“边”的概念；2)使用上同调方法识别位于边缘上的K-型；以及3)使用边K-型的凸壳来描述整个K-型集合。研究者的方法是基于D.Vogan关于上同调的某个限制映射的一个猜想。如果q=L+u是具有Levi分量L的theta稳定抛物线，则Vogan猜想指出，从u上同调到u相交k上同调的限制映射的映象将位于K的Weyl室墙中不包含的边上的K型参数化。与这一猜想相关的思想提供了接近所提出的目标的主要技术。从全球来看，这个项目的动力来自于世界对称性的存在。这种对称性以各种形式表现出来，例如化学中的某些晶体的晶格结构，高能物理中的亚原子粒子的行为，或者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被称为表示理论。从广义上讲，表象理论是对给定系统的所有可能对称性的研究。希望对所有的对称性进行分类，并提供关于每一种对称性的详细信息。虽然朝着这一目标迈出了很大的一步，但仍然存在许多难题。用来研究这些剩余问题的一个重要技术是检查一个表示的K型。从广义上讲，K类型集是将复杂的表示形式略微简化为更易于管理的形式。结果表明，K-型集是构造表示和推导表示特殊性质的一种非常有效的方法。研究人员的研究旨在提供K型集合的几何描述，这将使更详细的表示分析成为可能。

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.