Let $G$ be a compact Lie group and $T$ be a maximal torus. We call $G/T$ a generalized flag manifold since for $G = U(n)$ this quotient is isomorphic to the manifold of complete flags in $\mathbb{C}^n$. The generalized flag manifold has still a left $T$-action by left multiplication. My question is whether $G/T$ has a $T$-equivariant cell structure in the following sense:
Under what conditions on $G$ is there a $T$-equivariant filtration $F_{\bullet}$ on $G/T$ such that $F_k/F_{k-1}$ is $T$-equivariantly homeomorphic to a wedge of complex representation spheres, i.e. one-point compactifications of complex $T$-representations.
I believe that in the case $G = U(n)$ such filtration can be built by Schubert cells. As described e.g. in Fulton's Young tableaux, Section 10.2, the flag manifold $GL_n(\mathbb{C})/B \cong U(n)/T$ decomposes into subspaces homeomorphic to complex vector spaces (and this decomposition is $T$-equivariant) and these should be the open cells in a CW-decomposition as above: The subspaces are indexed by elements in the Weyl group $\Sigma_n$, with the complex dimension being the length of the element. Taking the union of this open cells for elements of length at most $k$ should provide the filtration step $F_k$ of $U(n)/T$.
Thus my question is how much this picture generalizes to general compact Lie groups.
