If we quotient $U(N)$ by $U(N-1)$ we get the odd dimensional sphere $S^{2N-1}$. (Here the quotient is in the sense of embedding $U(N-1)$ in the bottom right hand corner (with 1 as the (1,1) entry and zero everywhere else) and taking its orbits as the set of new objects.) If we quotient now by $U(1)$ (embedded on the diagonal) we get ${\mathbb CP}^{N-1}$.
More generally, if we quotient $U(N)$ by $U(N-k)$, for some $k < N$ (with an analagous embedding), and then quotient by $U(k)$ (embedded again on the diagonal) we get the $k$-Grassmannian $G_k({\mathbb C}^N)$.
My question is: What is the object we obtain when we quotient by $U(N-k)$? As we saw, it is the sphere for $k=1$. However, I cannot identify it with a familar object for higher $k$.
Also, more generally, if $F$ is a generalised flag manifold of signature $(d_1, \ldots ,d_k)$, then quotienting $U(N)$ by $$ U(N-d_1) \times \cdots \times U(N-d_k), $$ and then by $$ U(d_1) \times \cdots \times U(d_k), $$ gives $F$. What is the object we get from the first quotienting?
