= Unipotent radical count for a symplectic parabolic subgroup
{title2=$|Q|=q^{2k(m-k)+k(k+1)/2}$}
In row-block order $(W',U,W)$ with dual bases, a unit-diagonal symplectic flag stabilizer satisfies $E=J_UD^T$ and $F-F^T=DJ_UD^T$. The matrix $D$ is arbitrary. The latter right side is alternating, so each off-diagonal pair of entries of $F$ gives one free choice and each diagonal entry is free. Consequently
$$
|Q|=q^{2k(m-k)+k(k+1)/2}.
$$
Reversed dual-basis order inserts a reversal matrix but does not change the count. The argument holds in characteristic two without division by two.
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