Suppose $\omega(u,v) = \left\langle Ju, v\right\rangle$ is the canonical nondegenerate symplectic form on $\mathbb{R}^{2N}$. Given a random left-orthogonal matrix $P\in\mathbb{R}^{2N\times 2n}$ (distributed uniformly, say) where $2n<2N$, what can be said about the expected degeneracy of the bilinear form $\bar{\omega}(\bar{u},\bar{v}) = \omega(P\bar{u},P\bar{v}) = \left\langle P^\intercal JP\bar{u}, \bar{v}\right\rangle$ on $\mathbb{R}^{2n}$?
It is easy to cook up examples where $\bar{\omega}$ is degenerate, such as when $P^\intercal$ is projection onto the first $2n\leq N$ coordinates. On the other hand, I simulated 1000 instances of $P^\intercal JP$ for left-orthogonal $P=U_{2n}$ generated through the SVD of random matrices (where $U_{2n}$ is the first $2n$-columns of $U$), and the result was never singular.
Can you help me understand what is going on here?
I think this may have nothing to do with the fact that $J$ is a symplectic matrix. Particularly, the set of left-orthogonal $P$'s with $\mathrm{det}(P^\intercal J P)\neq0$ is a Zariski-open subset which is nonempty since, e.g., $P = (e_1, ..., e_n, e_{N+1}, ... e_{N+n})$ satisfies $P^\intercal J P = J$ on $\mathbb{R}^{2n}$. So, the set of singular $P$'s has measure zero, which explains why I can simulate 1000+ instances and never see this issue.
I would still appreciate a response if anyone can provide more information.