Unitary Dual of Real Groups

Salamanca-Riba博士打算研究Lie群表示理论中的各种问题。第一个问题是对亚普勒群的所有真酉化表示进行分类的问题。Jeff Adams和Dan Barbasch得到了这些表示与亚普勒群的某些内部形式的一组么正表示之间的对应关系。PI计划研究这种对应关系，看看这些表示是否会进入这个映射下有趣的Zuckerman函子模块。此外，PI还将与David Vogan合作开展一个已经在进行中的项目。他们想要解决的问题是这样一个猜想：李群G的最低K型与固定参数相关的李群G的酉表示集与与同一参数相关的具有最低K型的特殊子群的酉表示集之间存在双射。李群在应用数学的许多领域都有联系，如材料科学、量子场论、粒子物理、控制论以及机器人和生物学。例如，控制论和机器人学与某些紧李群的表示有关。此外，李群及其表示也与纯数学的其他领域有联系，如微分方程组、调和分析、拓扑学、几何学和遍历理论。例如，研究分析中的问题，如微分方程解的性态、偏微分方程解的性态、积分方程解的性态等，自然导致这些问题不仅在欧氏空间中，而且在可微流形上。这些流形的许多例子是经典的矩阵李群。这对于在经典力学和物理中出现的这类问题，以及在李群和齐次空间的现代问题中尤其如此。现代分析，当它超越局部结果时，就变成了对可微流形和李群的分析。

AbstractSalamanca-Riba Dr. Salamanca Riba intends to investigate various problems in the representation theory of Lie groups. The first problem is the question of classifying all genuine unitary representations of the metaplectic group. Jeff Adams and Dan Barbasch have obtained a correspondence between these representations and a set of unitary representations of certain inner forms of the metaplectic group. The PI plans to study this correspondence and see if these representations go to interesting Zuckerman functor modules under this map. In addition, the PI will also work on a program already in progress in collaboration with David Vogan. The problem they want to solve is the following conjecture: There is a bijection between the set of unitary representations of a Lie group G, whose lowest K type is associated to a fixed parameter, and the set of unitary representations of a special subgroup with lowest K types associated to the same parameter.Lie groups have connections in many areas of applied Mathematics like Materials Science, Quantum Field Theory, Particle Physics, Control Theory and Robotics and Biology. For example, Control Theory and Robotics is related to some representations of certain compact Lie groups. In addition, Lie groups and their representations have also connections with other areas of pure Mathematics as well, like Differential Equations, Harmonic Analysis, Topology, Geometry and Ergodic Theory. For example, the study of problems in Analysis such as the behavior of solutions of differential equations, partial differential equations, integral equations, etc. leads naturally to formulating these problems not only in Euclidean space, but on differentiable manifolds. Many examples of these manifolds are classical Lie groups of matrices. This is particularly true of problems of this type arising in classical mechanics and physics and in modern problems of Lie groups and homogeneous spaces. Modern Analysis, when it goes beyond local results, becomes analysis on differentiable manifolds and Lie groups.,