Symplectic parabolic subgroup

The stabilizer of a totally isotropic $k$-space in a finite symplectic space also stabilizes its orthogonal complement. Its block decomposition has a unipotent normal subgroup $Q$ and a block-diagonal complement $L\cong {\mathrm{GL}}_k(q)\times {\mathrm{Sp}}_{2m-2k}(q)$. The paired $k$-space carries the contragredient action. A row-vector convention gives upper-triangular blocks; a column-vector convention transposes the arrangement. The unipotent radical count for a symplectic parabolic subgroup yields its order.

In row-block order $(W^{\prime },U,W)$ with dual bases, a unit-diagonal symplectic flag stabilizer satisfies $E=J_U{D}^T$ and $F-F^T=D{J}_U{D}^T$. 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. ConsequentlyReversed dual-basis order inserts a reversal matrix but does not change the count. The argument holds in characteristic two without division by two.