Let $B= \Bigg\{\begin{bmatrix}
* & *&* \\
0 & *&*\\
0&0&*
\end{bmatrix}
\Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$?
Do we use these facts:
Borel fixed point theorem.
Algebraic actions of unipotent groups are cells $U_-\cdot x_0=U_-/H$?
I actually need a detailed solution because I don’t have enough background in this subject and I have to understand it! Any solutions or comments are highly appreciated as well.
I also need to know the basic definitions and examples in flag manifolds which are essential to the solution. Particularity, in matrices case.
anon gave a good reference. It suffices to read it!
Let $V$ be a complex vector space of dim $n$ and $\mathcal{B}$ be a
basis. A complete flag $V\in\mathcal{F}$ of $\mathbb{C}^n$ is:
$\{0\}=V_0\subset V_1\subset\cdots V_n=\mathbb{C}^n$ where
$dim(V_i)=i$. Note that $GL_n$ acts of $\mathcal{F}$ in a natural way.
If $V_S$ is the standard flag associated to $\mathcal{B}$, then $f\in
Gl_n$ satisfies $f(V_S)=V_S$ iff $f\in T_n$ (the upper triangular
invertible matrices); moreover if $V,W\in \mathcal{F}$, then there is
$f\in GL_n$ s.t. $f(V)=W$. Thus the variety $\mathcal{F}$ is the
homogeneous space $GL_n/T_n$ of dimension $n^2-n(n+1)/2=n(n-1)/2$ over
$\mathbb{C}$ and $n^2-n$ over $\mathbb{R}$; that is the same set than
your space $SL_n/B_n$. We can see also $V_{i+1}$ as $V_i\bigoplus
u_{i+1}$ where $u_{i+1}$ is a unitary vector orthogonal to $V_i$. Note
that $u_i$ is defined up to a mult. with an element of $S^1(\mathbb{R})$. Then the
standard flag associated to the orthonormal basis $\mathcal{B}$ is in $U(n)$ and its
stabilizer is the set of diagonal matrices in $(S^1)^n=\tau_n$, the $n$-torus.
Finally our set is the compact real homogeneous space $U(n)/\tau_n$; we find again that its dimension is $n^2-n$ over $\mathbb{R}$.
When $n=2$, we obtain $P^1(\mathbb{C})$ which is homeomorphic to $S^2(\mathbb{R})$.
When $n=3$, we obtain $\{(u,v)\in P^2(\mathbb{C})\times P^2(\mathbb{C})|u\perp v\}$ that has dimension $4+4-2=6$ over $\mathbb{R}$.
EDIT. In other words, when $n=3$, a complete flag $F$ is given by $V_1,V_2$ or by an orthonormal basis $(u_1,u_3,u_3)$ of $\mathbb{C}^3$ s.t. $u_1\in V_1,u_2\in V_2$. Then a representative of $F$ is $A=[u_1,u_2,u_3]\in U(3)$. If $B$ is another representative of $F$, then $B=Adiag(e^{i\theta_1},e^{i\theta_2},e^{i\theta_3})$.