# Find all Linear spaces on an orbit (by a connected linear algebraic group)

Suppose $V$ is a vector space over $\mathbb{C}$ and $G\subset \textrm{GL}(V)$ is a connected linear algebraic group.

Consider the orbit closure $Y = \overline{G.\mathbb{P} L}$, for a subspace $L\subset V$.

Is there a nice description for all the linear spaces contained in $Y$?

I would be really happy with an answer in terms of the representation theory of $G$ and the stabilizer $G_L$.

I know the answer in the case of the Segre variety (I am pretty sure this is classical), and I guess that the answer for other homogeneous varieties might be known. So I'm really interested when the linear space $L$ is at least 2-dimensional.

• I think in general the answer is not known. Dec 7, 2013 at 15:22
• What was wrong with the answer given by Peter Crooks? Dec 8, 2013 at 2:31
• The linear subspace suggested in the answer is not contained in the orbit closure. Dec 8, 2013 at 20:21
• I guess this is a difficult question in general, so I guess I should keep working with small examples. Dec 9, 2013 at 19:05