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Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers.

Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, where $(A,B)\in SO(2n)\times SO(2n)$ and $D$ is a $2n\times 2n$ matrix.

Is it known what the principal orbit (an orbit of maximum dimension) of this action and the corresponding stabilizer are?

It seems that the answer is easy over the real numbers, see e.g.

  • Kyle Czarnecki, R. Michael Howe and Aaron McTavish, On the orbits of an orthogonal group action, Involve 2 (2009), No. 5, 495–509, doi:10.2140/involve.2009.2.495.

Thank you very much.

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  • $\begingroup$ I'm sorry to be skeptical, but this looks like a homework problem. Have you tried to check whether the proof over the real numbers continues to work over the complex numbers? Where does it break down (if it does), and (if so) can you replace the crucial step with something that works over the complex numbers? (By the way, the answer over the reals is very, very classical and ought to be in every linear algebra textbook when the notion of 'singular values' is introduced.) $\endgroup$
    – Robert Bryant
    Commented Dec 26, 2020 at 17:49

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