rt.representation theory - Orbits of a symplectic group on its Lie algebra in the finite field case

It won't tell you everything about the orbits, but the decomposition of $mathfrak{sp}_{2n}textbf{F}_p$ as an $text{Sp}_{2n}textbf{F}_p$-representation is known. This can be found in Hogeweij, "Almost-classical Lie algebras. I." Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 4, 441-452, but it is hard to extract the answer from that paper, so I'll briefly give the argument.

If $p$ is odd, then $mathfrak{sp}_{2n}textbf{F}_p$ is irreducible with highest weight $2omega_1$, where $omega_1,ldots,omega_n$ are the fundamental weights for $text{Sp}_{2n}textbf{F}_p$.

If $p=2$, we proceed as follows. Let $Happrox textbf{F}_p^{2n}$ be the standard representation of $text{Sp}_{2n}textbf{F}_p$. Note that as a subspace of $mathfrak{gl}_{2n}textbf{F}_papprox H^*otimes H$, the condition defining $mathfrak{sp}_{2n}textbf{F}_p$ describes exactly the subspace $text{Sym}^2 H$ inside $Hotimes Happrox H^*otimes H$, so we are looking at orbits of $text{Sp}_{2n}textbf{F}_p$ on $text{Sym}^2 H$.

In characteristic 2 we have an embedding of $H$ into $text{Sym}^2 H$ by $xmapsto xcdot x$, which is linear since $(x+y)^2 = x^2+y^2$ (in general this twists by Frobenius but we are over $textbf{F}_2$). Since $xcdot y=ycdot x=-ycdot x$, the quotient $text{Sym}^2 H/H$ is isomorphic to $bigwedge^2 H$. Now $bigwedge^2 H$ has two invariant subrepresentations. One is trivial, spanned by the vector $omega=a_1wedge b_1+cdots+a_nwedge b_n$. The other is the kernel $K$ of the contraction $bigwedge^2 Hto textbf{F}_2$, defined by $a_iwedge b_imapsto 1$, $a_iwedge a_jmapsto 0$, $b_iwedge b_jmapsto 0$, and $a_iwedge b_jmapsto 0$. Note that under this contraction $omega$ is taken to $nin textbf{F}_2$; thus $omega$ is contained in $K$ iff $n$ is even. Finally, $K$ is irreducible when $n$ is odd, and $K/langleomegarangle$ is irreducible when $n$ is even.

If I'm not mistaken, this means the invariant subrepresentations are thus just $H$, $langleomegarangle$, $Hoplus langleomegarangle$, and $H+K$ (the kernel of the contraction $text{Sym}^2 Hto textbf{F}_2$).