 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.