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arXiv · 0811.0835

The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n)

Mark Colarusso
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Abstract

In recent work Bertram Kostant and Nolan Wallach ([KW1], [KW2]) have defined an interesting action of a simply connected Lie group $A$ isomorphic to \mathbb{C}^{n\choose 2} on gl(n) using a completely integrable system derived from Gelfand-Zeitlin theory. In this paper we show that an analogous action of \mathbb{C}^{d} exists on the complex orthogonal Lie algebra so(n), where d is half the dimension of a regular adjoint orbit in so(n). In [KW1], Kostant and Wallach describe the orbits of $A$ on a certain Zariski open subset of regular semisimple elements in gl(n). We extend these results to the case of so(n). We also make brief mention of the author's results in [Col1], which describe all $A$-orbits of dimension {n\choose 2} in gl(n).

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Mark Colarusso. 2008-11-06. The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n). https://arxiv.org/abs/0811.0835

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