Cells decomposition of quotient space

Let $G$ a Compact Lie group and $T$ a maximal torus in $G$. I'd like to write an explicit Bruhat decomposition of $G/T$ and prove in this way that the Bruhat decomposition has $|W|$ cells of even dimensions, where $W:=N(T)/T$ is the Weyl group. Now look at the case in which $G=U(n)$ (the unitary group): the quotient $U(n)/T_{n}$ is the flag variety (why? I haven't understood the reason why $GL(n,\mathbb{C})/B_{n} \simeq U(n)/T_{n}$, where $B_{n}$ is the Borel subgroup of upper-triangular non singular matrix) and the Bruhat decomposition for flag variety implies that there are $|W|$ cells of even dimensions. I tried also, under a good suggest, to prove this claim by induction on $n$. Prop. $U(n)/T_{n}$ si decompone in $|W|$ celle di dimensione pari. ( incorrect / incomplete) Proof Base of induction: $U(1)/S^{1}\simeq pt$: a 0-cell. Inductive hypothesis: we suppose that $U(n-1)/T_{n-1}= e^{0} \cup \cdots e^{2n-1}$. Now we have to prove that $U(n)/T_{n}= e^{0} \cup \cdots \cup e^{2n}$. So we can use the double quotient theorem: $(U(n)/T_{n})/(U(n-1)/T_{n-1}) \simeq S^{2n-1}/S^{1} = \mathbb{C}P^{n}$ (because $U(n)/U(n-1)= S^{2n-1}$ and $T_{n}/T_{n-1}=S^{1}$). But is a well-know fact that $\mathbb{C}P^{n}= e^{0} \cup \cdots \cup e^{2n}$. In this way the induction fails because I don't know what is the behavior of cells in the quotient. So I'm looking for a good idea in order to modify the proof in a good way.

-

Let $G$ be a complex semisimple Lie group, $B$ be a Borel subgroup, $K$ a maximal compact subgroup and $T$ a maximal torus, then $G/B$ is diffeomorphic to $K/T$. (I think it is a consequence of the Iwasawa decomposition in the complex case). But now, the complete flag variety is given by the case $G/B$ where $G= SL(n,C)$. But recall that for this $G$, $K$ is given by $U(n)$.
the Bruhat decomposition for flag variety implies that there are $|W|$ cells [...] Inductive hypothesis: we suppose that $U(n−1)/T_{n-1}=e^0 \cup ... \cup e^{2n-1}$.
At first glance the inductive hypothesis seems wrong : in this case isn't the Weyl group given by the symmetric group $S_{n-1}$ ?