Unitary Dual of Real Groups

摘要萨拉曼卡-里瓦 萨拉曼卡·里巴博士打算研究李群表示论中的各种问题。第一个问题是对元群的所有真正的酉表示进行分类的问题。Jeff亚当斯和Dan Barbasch已经得到了这些表示与亚格群的某些内部形式的一组酉表示之间的对应。PI计划研究这种对应关系，看看这些表示是否会在这个映射下到达有趣的Zuckerman函子模块。此外，PI还将与大卫沃根合作开展一项已经在进行中的计划。 他们想要解决的问题是以下猜想：李群G的最低K型与一个固定参数相关联的酉表示集与一个特殊子群的最低K型与同一参数相关联的酉表示集之间存在一个双射。李群在许多应用数学领域都有联系，如材料科学，量子场论，粒子物理，控制理论、机器人学和生物学。例如，控制理论和机器人学与某些紧致李群的某些表示有关。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，调和分析，拓扑学，几何学和遍历理论。例如，研究问题的分析，如行为的解决方案的微分方程，偏微分方程，积分方程等自然导致制定这些问题不仅在欧几里德空间，但对可微流形。这些流形的许多例子是经典的李群矩阵。这是特别真实的问题，这种类型的出现在经典力学和物理学和现代问题的李群和齐次空间。现代分析，当它超越局部结果时，成为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.,