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

TL;DR

This paper extends the Gelfand-Zeitlin integrable system action from gl(n) to so(n), showing an analogous complex group action on the orthogonal Lie algebra and analyzing its orbit structure.

Contribution

It introduces an analogous action of a complex vector group on so(n) based on Gelfand-Zeitlin theory, extending previous results from gl(n).

Findings

Existence of a complex group action on so(n) similar to gl(n).

Description of A-orbits on a Zariski open subset of regular semisimple elements in so(n).

Extension of Kostant and Wallach's orbit results to the orthogonal Lie algebra.

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).