Let $(V,Q)$ be a non-degenerate quadratic space of dimension $n$ over an algebraically closed field of characteristic zero. Let $W$ be a subspace of $V$ of dimension $m < \frac12 n$ which is totally isotropic for $Q$. What is the structure of the subgroup of $O(Q)$ consisting of orthogonal transformations of $(V,Q)$ which fix the subspace $W$? I'm looking for some (sort of) concrete description...

Let $(V,Q)$ be a non-degenerate quadratic space of dimension $n$ over an algebraically closed field of characteristic zero. Let $W$ be a subspace of $V$ of dimension $m < \frac12 n$ which is totally isotropic for $Q$. What is the structure of the subgroup of $O(Q)$ consisting of orthogonal transformations of $(V,Q)$ which send $W$ to itself? I'm looking for some (sort of) concrete description...