Homogeneous spaces which are not torus bundle over flag manifolds For a compact semisimple Lie group $G$, what is an example of a homogeneous space of $G$ which is not a torus bundle over a generalized flag manifold of $G$. Examples for $SU(N)$ would be of most interest.
 A: Just to summarize the common feature of both Allen's example and my example: there are many Lie subgroups $H$ of $G$ that are contained in no proper parabolic subgroup.  Some examples arise from the fact that a proper subgroup of a parabolic may have a normalizer that is not contained in the subgroup; the example $H$ that I intended was the normalizer in $SU(N)$ of a maximal torus $U(1)^{N-1}$.  But there are also connected subgroups that are contained in no proper parabolic subgroup, e.g., the intersection of $SU(N)$ with an algebraic subgroup of $\text{GL}_n(\mathbb{C})$.  Another example that I particularly like is the image of the group homomorphism $\rho:SU(2)\to SU(N)$ associated to the $N$-dimensional irreducible representation of $SU(2)$.  You can think of $SU(N)/\rho(SU(2))$ as a parameter space for real rational normal curves in $\mathbb{RP}^{N-1}$.      
A: It is like most of them. Are spheres any good for you?
You need some complex and/or algebraic structure to make it into a torus bundle (cf. Borel-Remmert Theorem). In general, let $T$ be the maximal torus of $G$. Then the flag manifold is $G/T$. If you take $G/H$ where $T$ does not contain $H$, the chances are you will have no fibration of $G/H$ over flags.
A: Any non-trivial torus bundle has Euler characteristic 0. Thus if a homogeneous space $G/H$ is not a generalized flag manifold and has non-zero Euler characteristic, it is not a torus bundle over a generalized flag manifold. This includes all even-dimensional spheres starting with $S^4$, as suggested by Bugs Bunny, and Allen Knutson's ${\rm SU}(3)/{\rm SO}(3)$.
