标题： 凯莱变换在表示论中 显示英文标题

标题： The Cayley Transform on Representations

摘要： 经典的凯莱变换是在二次矩阵群与其李代数之间的有理映射，最早由凯莱于1846年发现。 由于其在纯数学和应用数学中的重要作用，经典的凯莱变换已经从各种角度进行了推广。 本文关注经典的凯莱变换的表示论推广。 这项工作的基本思想是经典的凯莱变换的适用性很大程度上取决于李群是如何被表示的。 目标是表征经典的凯莱变换适用的不可约表示。 为此，我们首先建立了适用于一般李群的适用性标准。 如果该群是半单的，我们进一步获得了此类表示的权图上的几何条件。 最后，我们提供了经典复单李群及其紧实形式的完整分类。 除了之前已知的例子外，$\mathrm{Spin}(8)$的自旋表示是我们列表中唯一的一个。 显示英文摘要

摘要： The classical Cayley transform is a birational map between a quadratic matrix group and its Lie algebra, which was first discovered by Cayley in 1846. Because of its essential role in both pure and applied mathematics, the classical Cayley transform has been generalized from various perspectives. This paper is concerned with a representation theoretic generalization of the classical Cayley transform. The idea underlying this work is that the applicability of the classical Cayley transform heavily depends on how the Lie group is represented. The goal is to characterize irreducible representations of a Lie group, to which the classical Cayley transform is applicable. To this end, we first establish criteria of the applicability for a general Lie group. If the group is semisimple, we further obtain a geometric condition on the weight diagram of such representations. Lastly, we provide a complete classification for classical complex simple Lie groups and their compact real forms. Except for the previously known examples, spin representations of $\mathrm{Spin}(8)$ are the only ones on our list.