We work over the reals. Fix a dimension $n$ and a symplectic form $\Omega$. This is a $2n \times 2n$ matrix s.t. $\Omega^2 = -1$. A symplectic matrix is a matrix $S$ such that $ S^T \Omega S = \Omega$.
Given a positive definite $2n \times 2n$ matrix $M$ Williamson's theorem we may always find symplectic matrix s.t. $S^T M S = \Gamma$, where $\Gamma$ is a diagonal matrix of the form $\Gamma = \text{diag}\{\nu_1, \nu_1,\nu_2,\nu_2,\ldots,\nu_n,\nu_n\}$, with the $\nu$ all real. We call the entries of $\Gamma$ the symplectic eigenvalues of $M$.
Useful fact: The symplectic eigenvalues are precisely the absolute values of the usual eigenvalues of the matrix $i \Omega M$
Q: What can be said about the distribution of the symplectic eigenvalues of Wishart matrices? It seems like this should follow straightforwardly from the PDF for the Wishart matrices and the above fact, but I've had no luck.