I want to apologize in advance if my question is too elementary as I am not an expert in Lie theory. I have posted it before on stackexchange without receiving an answer.
Let $G$ be a compact Lie group and $T$ be a maximal torus. I would like to call $G/T$ a flag manifold since for $G = U(n)$ this quotient is isomorphic to the manifold of complete flags in $\mathbb{C}^n$. The 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 cellular 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?
The Bruhat decomposition gives almost this result. More precisely: As explained for example in the book on Lie groups by Bump, $G$ has a complexification $G_{\mathbb{C}}$ and this has a Borel subgroup $B$. In the case $G = U(n)$, we have $G_{\mathbb{C}} = GL_n(\mathbb{C})$ and $B$ is the group of upper triangular matrices. We have $G/T\cong G_{\mathbb{C}}/B$. The Bruhat decomposition is the decomposition of $G_{\mathbb{C}}$ into the double cosets $BwB$, where the $w$ are lifts of the Weyl group to $G$. Thus, we obtain a decomposition of $G_{\mathbb{C}}/B$ into the $[Bw]$, which are all known to be isomorphic to $\mathbb{C}^m$, where $m$ is the length of the Weyl group element; in fact they are $T$-equivariantly isomorphic to a specific $T$-linear representation $V_w$. The open cells $[Bw]$ are often called Schubert cells. (Most of this is explained in Bump's book in the setting of complexifications of compact Lie groups. The rest is contained in Borel's book on Linear Algebraic Groups in Section 14.12 in the setting of reductive groups; it is my understanding that complexifications of compact Lie groups are reductive.) Thus, we have decomposed $G/T \cong G_{\mathbb{C}}/B$ into open cells. Borel claims that this gives a cell decomposition in the sense of algebraic topology. But to get this, we need to answer the following question:
Identifying the cells $[Bw]$ with an open unit ball $\mathring{D}$ in the representation $V_w$, can we extend the map $\mathring{D} \to G/T$ to a $T$-equivariant map from the closed unit disk in $V_w$?
Answering this would yield a positive answer to the first question. Even non-equivariantly and in the case of $G = U(n)$, I find this non-obvious.
