In this paper, we use the theory of algebraic groups to prove a number of new and fundamental results about the orthogonal Gelfand-Zeitlin system. We show that the moment map (orthogonal Kostant-Wallach map) is surjective and simplify criteria of Kostant and Wallach for an element to be strongly regular. We further prove the integrability of the orthogonal Gelfand-Zeitlin system on regular adjoint orbits and describe the generic flows of the integrable system. We also study the nilfibre of the moment map and show that in contrast to the general linear case it contains no strongly regular elements. This extends results of Kostant, Wallach, and Colarusso from the general linear case to the orthogonal case.

中文翻译：

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复杂的正交 Gelfand-Zeitlin 系统

在本文中，我们使用代数群理论来证明关于正交 Gelfand-Zeitlin 系统的一些新的和基本的结果。我们证明矩图（正交 Kostant-Wallach 映射）是满射的，并简化了 Kostant 和 Wallach 的标准，使元素具有强规则性。我们进一步证明了正交 Gelfand-Zeitlin 系统在规则伴随轨道上的可积性，并描述了可积系统的一般流。我们还研究了矩图的零纤维，并表明与一般线性情况相比，它不包含强规则元素。这将 Kostant、Wallach 和 Colarusso 的结果从一般线性情况扩展到正交情况。