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The symplectic group $\mathrm{Sp}_n(q)$ acts on $\mathrm{GL}_{2n}(q)$ by conjugation. All the literature I have found concerning the orbits of action of this kind is "Unipotent conjugacy classes in general linear groups" by Alperin, which is not really about what I am looking for.

Is there any work referring to this action and its orbits?

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    $\begingroup$ Do you mean $Sp_n(q)$ or $Sp_{2n}(q)$? The first is a subgroup of $GL_n(q)$ while the second is a subgroup of $GL_{2n}(q)$. $\endgroup$
    – Michael Giudici
    Commented Jun 11, 2019 at 14:17

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