Transitivity of $\Omega$ in subspace actions

Let $$G=O(V, Q)$$ be a finite orthogonal group acting naturally on a space $$V\cong \mathbb{F}_q^n$$ equipped with a quadratic form $$Q$$. Assume $$n=\dim V\geq c$$ for some large enough constant $$c$$ in order to get rid of small exceptional examples. Let $$W\subseteq V$$ be a subspace of dimension $$m$$ that is either totally singular or nondegenerate ($$W$$ can also be a nonsingular 1-space when $$q$$ is even). Let $$S$$ be the set of subspaces $$W'\subseteq V$$ isometric to $$W$$ with respect to $$Q$$, i.e., $$W'$$ admits a invertible linear map $$\sigma: W\to W'$$ such that $$Q(\sigma(x))=Q(x)$$ for $$x\in W$$. Then $$G$$ acts on $$S$$ by permuting the subspaces. By Witt's lemma, this action is transitive. The stabilizers $$G_{W'}$$ for $$W'\in S$$ belong to Aschbacher's class $$\mathcal{C}_1$$ introduced in Michael Aschbacher: On the maximal subgroups of the finite classical groups.

Let $$\Omega$$ be the normal subgroup of $$SO(V, Q)\leq G$$ of index two. I am curious about whether $$\Omega$$ acts transitively on $$S$$.

In the case that $$V$$ is hyperbolic and $$W$$ is totally singular of dimension $$m=n/2$$, it is known that $$\Omega$$ is intransitive: it has two orbits distinguished by the parity of $$\dim W'\cap W$$. Is this the only exception? If so, how to see it? Thanks.

I am reading the book The Subgroup Structure of the Finite Classical Groups by Peter B. Kleidman and Martin W. Liebeck, and the tables there seem to suggest the above exception is the only one. But I am not so sure.

Yes, I believe that you are correct in saying that the case you describe is the only exception.

By the Orbit-Stabilizer Theorem, the transitivity of the action of $$\Omega$$ on $$S$$ is equivalent to the stabilizer of $$W$$ including elements in each of the cosets of $$\Omega$$ in $$G$$. (There are two such cosets when $$q$$ is even and four when $$q$$ is odd). This information can be read off from the tables in the book by Kleidman and Liebeck.

Looking at the subgroups in class $$\mathcal{C}_1$$ Tables 3.5.D, 3.5.E and 3.5.F, we see that $$\pi = 1$$ in most cases, in which case the $$\Omega$$-conjugacy class of $$S$$ is also a conjugacy class of the full conformal orthogonal group, and so the action of $$\Omega$$ on $$S$$ is transitive in these cases.

Otherwise, we have $$\pi=\pi_1$$ or $$\pi_2$$, and there are two $$\Omega$$-classes that are fused under the conformal group (i.e. $$c=2$$ in the notation of the book).

The case when it is $$\pi_2$$ is the exceptional case when $$\Omega$$ is not transitive on $$S$$. In this case, the two $$\Omega$$-orbits are interchanged by the outer automorphism of $$\Omega$$ that is denoted by $$\gamma$$ (possibly with two dots on it - I find all these dots confusing).

In the case $$\pi=\pi_1$$, that the stabilizer of the $$\Omega$$-class is the kernel of the map $$\tau$$ with two dots on it. This map is defined at the end of Section 2.1 on page 20, and this kernel is equal to the orthogonal group $$\Omega$$, which has index two in the conformal orthogonal group, so the action of $$\Omega$$ on $$S$$ is transitive in that case.

As for how to see all of this, I am afraid I do not know any way other than reading the proofs in the book.