The unitary group, U(n), acts transitively on the Grassmann manifold X = Gr(2, C^n).  The isotropy group is H = U(2)xU(n-2) = the group elements leaving some x fixed.  What are the dimensions of the orbits of H in X?  The brute force calculation gets messy for some of the orbits, so I am wondering if these results are published somewhere.  The generic orbit is codimension 2, and there are the special orbits of {x} and x_perp = Gr(2,C^{n-2}), but there are other orbits, too.