The classical problem regarding the action of symplectic group on its Lie algebra gives rise to the following question in the finite field case.


Let $\mathbb F\_p$ be a finite field. Then the symplectic group over $\mathbb F\_p$ acts by conjugation on the set of matrices over $\mathbb F\_p$ that satisfy $\Omega A + A^t \Omega = 0$, $\Omega$ is the skew symmetric matrix


$$ \begin{pmatrix} 0 & I \\\\ -I & 0 \end{pmatrix} $$


where $I$ is identity matrix. What are the orbits of this action? 



Orbits of a symplectic group on its Lie algebra in the finite field case
This problem is answered in a paper by Burgoyne and Cushman. I don't have the reference to hand.


This also came up in [Classification of adjoint orbits for orthogonal and symplectic Lie algebras?](https://mathoverflow.net/questions/25901/)


