$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Gr{Gr}$I came across a problem that could be formulated in term of Grassmannians, I would be very glad to have your opinion about it.
I have an irreducible representation of $G = \SO(k)$, $k\ge 3$ on some complex vector space $V$ of dimension $n\ge 4$. Assume this is a real representation i.e. there is $U$ real of dimension $n$ invariant by $G$ with $U \oplus \sqrt{-1}U = V$.
I would like to show that the following thing is not possibile.
There is a two dimensional real subspace $\Sigma\subseteq U$ and an $n-2$ dimensional real subspace $\tau$ of $U$ such that if we consider the induced action of $G$ on the Grassmannian $\Gr(2,U)$, $\Sigma\subseteq\tau$ and the $G$ orbit of $\Sigma$ in the Grassmannian $\Gr(2,U)$ is contained in the Plücker hyperplane section of spaces intersection $\tau$ non trivially.
I think I proved this for $\SO(3)$ and $\SO(4)$.
Said differently, can I have two not trivial $G$ invariant orthogonal (with respect to the invariant inner product ) subspaces of $\bigwedge\nolimits^2 U$ such that the image in $P(\bigwedge\nolimits^2 U)$ of both subspaces intersect the Grassmannian?
