find a canonical form for $P$. First consider $\langle Ju,v\rangle$ for orthonormal vectors $u,v\in P$. This must have a maximum, so let $\cos\theta_1=\langle Ju,v\rangle$ be this maximum realised by some orthonormal vectors $u,v\in P$, where $0\leq\theta_1\leq\frac{\pi}{2}$.

Suppose that $w\in P$ is a unit vector orthogonal to $\operatorname{Span}\{u,v\}$, where $\cos\theta_1=\langle Ju,v\rangle$. The function $$f_w(\theta)=\langle Ju,\cos\theta v+\sin\theta w\rangle$$ has a maximum at $\theta=0$ so $f_w'(0)=\langle Ju,w\rangle =0$. Similarly we have that $\langle Jv,w\rangle =0$, and thus $w\in\operatorname{Span}\{u,v,Ju,Jv\}^{\perp}$.

We then have two cases. If $\theta_1=0$ then $v=Ju$ so we can set $u=e_1,v=Je_1$ and see that $P=\operatorname{Span}\{e_1,Je_1\}\times Q$ where $Q$ is a $2(k-1)$-plane in $\C^{n-1}=\operatorname{Span}\{e_1,Je_1\}^{\perp}$. If $\theta_1\neq 0$ we have that $v=\cos\theta_1 Ju+\sin\theta_1 w$ where $w$ is a unit vector orthogonal to $u$ and $Ju$, so we can let $u=e_1$, $w=e_2$ and see that $P=\operatorname{Span}\{e_1,\cos\theta_1 Je_1+\sin\theta_1 e_2\}\times Q$ where $Q$ is a $2(k-1)$-plane in $\C^{n-2}=\operatorname{Span}\{e_1,Je_1,e_2,Je_2\}^{\perp}$.

Proceeding by induction we see that we have an oriented basis $\{e_1,Je_1,\ldots,e_n,Je_n\}$ for $\C^n$ so that $$P=\operatorname{Span}\{e_1,\cos\theta_1 Je_1+\sin\theta_1 e_2,\ldots,e_{2k-1},\cos\theta_k Je_{2k-1}+\sin\theta_k e_{2k}\},$$ where $0\leq\theta_1\leq\ldots\leq\theta_{k-1}\leq\frac{\pi}{2}$ and $\theta_{k-1}\leq\theta_k\leq\pi-\theta_{k-1}$.

Since we can write $\omega=\sum_{j=1}^n e^j\wedge Je^j$ we see that $\frac{\omega^k}{k!}$ restricts to $P$ to give a product of $\cos\theta_j$ which is certainly less than or equal to $1$. Moreover, equality holds if and 