紧群丛的 Peter-Weyl 定理与相对遍历紧扩张的几何表示
A Peter-Weyl theorem for compact group bundles and the geometric representation of relatively ergodic compact extensions
AI总结：
本文证明相对遍历保测扩张具有相对离散谱等价于可由紧齐性空间丛作斜积表示，并通过拓扑动力学分类及紧群丛的 Peter-Weyl 型定理完成推广。
摘要：
我们证明：保测动力系统的一个相对遍历扩张具有相对离散谱，当且仅当它可以表示为由紧齐性空间丛构成的斜积。我们的结果对作用群或底层概率空间没有任何限制。这推广了 Mackey、Zimmer、Ellis、Austin 以及第二作者与 Tao 此前的工作，并受到不可数群作用的 Furstenberg–Zimmer 结构理论和 Host–Kra 结构理论启发。我们的方法将遍历论问题转化为拓扑动力学问题，并在其中建立相应分类：拓扑动力学中的扩张具有相对离散谱，恰在它容许由紧齐性空间丛给出的斜积表示时。论证的关键一步是建立紧群丛的 Peter-Weyl 型定理，该定理本身可能具有独立意义。
英文摘要：
We show that a relatively ergodic extension of measure-preserving dynamical systems has relative discrete spectrum if and only if it can be represented as a skew-product by a bundle of compact homogeneous spaces. Our result holds without restrictions on the acting group or the underlying probability spaces. This generalizes previous work by Mackey, Zimmer, Ellis, Austin, and the second author and Tao, and is inspired by the Furstenberg-Zimmer and Host-Kra structure theories for actions of uncountable groups. Our approach translates the ergodic-theoretic question into topological dynamics, where we establish a corresponding classification: an extension in topological dynamics has relative discrete spectrum precisely when it admits a skew-product representation by bundles of compact homogeneous spaces. A key step in our argument is establishing a Peter-Weyl-type theorem for bundles of compact groups which might be of independent interest.
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