Mathematical Sciences: Microlocal Character Theory for Representations of Classical Lie Groups

本课题在Howe的实约对偶理论背景下，对经典李群的不可约酉表示的特征和矩阵系数进行了微局部研究。我们的目标是理解这些表征的特征、矩阵系数和波前集，这些表征是根据在该理论中自然出现的矩映射来表示的。这些是经典不变理论的矩图。李群理论是为了纪念挪威数学家索夫斯·李而命名的，它一直是20世纪数学的主要主题之一。作为利用系统中固有对称性的数学工具，李群表示理论对数学本身，特别是分析和数论，以及理论物理，特别是量子力学和基本粒子物理产生了深远的影响。这个特殊项目所基于的数学技术在信号处理、数据压缩和密码学中也很有用。

Abstract Prezbinda This project is concerned with a microlocal study of characters and matrix coefficients of irreducible unitary representations of classical Lie groups in the context of Howe's theory of real reductive dual pairs. The goal is to understand the characters, matrix coefficients and wave front sets of such representations in terms of the moment maps which occur naturally in that theory. These are the moment maps of classical invariant theory. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics. The mathematical techniques, on which this particular project is based, are also useful in signal processing, data compression and cryptography.