Description of $GL_3/U$ Let $U$ be the set of unipotent upper triangular matrices and $B$  the upper triangular matrices of $GL_3$. How could I describe $GL_3/U$ ? Using coordinates, in a projective or an affine space.
For example, I already know the identification of $SL_2/U$ with $\mathbb{A}^2 \setminus (0,0)$ and the identification of $SL_2/B$ with $\mathbb{P}^1$ ($B,U$ the standard Borel of $SL_2$ and its unipotent radical).
Thank you.
 A: Let $V$ be the basic (3-dimensional) representation of $GL(3)$. Then $SL(3)/U$ is the set of all pairs $x\in V, y\in V^*$ where $x$ and $y$ are non-zero and $(x,y)=0$. 
The quotient $GL(3)/U$ is non-canonically product of the above by $C^{\times}$. Canonically,
you need to choose non-zero $x_i\in \Lambda^i(V)$ (for $i=1,2,3$) such that 
$x_i\wedge x_j=0$ for all $i$ and $j$ (note that if $x_3$ is fixed then
$\Lambda^2(V)$ is canonically the same as $V^*$). This description generalizes immediately
to any $GL(n)$ (and in fact to any $G$).
A: As already mentioned, the quotient $GL_n(\mathbb{C})/B$ is the space $Fl$ of all flags ($0\subset$ a line $\subset$ a 2-plane $\subset\cdots\subset \mathbb{C}^n$). There are line bundles $L_i , i=1,\ldots, n$ on this space where $L_i$ is obtained by pulling back the tautological bundles from Grassmannians of $i$- and $i-1$-planes in $\mathbb{C}^n$ and quotienting one by the other.
Let $L_i^0$ be the total space of $L_i$ minus the zero section and set $L^0$ to be the fibered product $$L_1^0\times_{Fl}L_2^0\times_{Fl}\cdots\times_{Fl}L^0_n.$$ The group $GL_n(\mathbb{C})$ acts transitively on the space $L^0$ and each stabilizer is conjugate to the group of unipotent matrices in $GL_n(\mathbb{C})$. So $GL_n(\mathbb{C})/\mbox{unipotent matrices}\cong L_0$.
A: Let $G$ over a field $k$ be a reductive group and $P$ be parabolic subgroup with Levi decomposition $P =MU$ (over $k$), then 
$$G / U  = \amalg_{w \in N(M) /M} UMwU,$$
where $N_G(M)$ is the normalizer of $M$ in $G$.
Since $MU = UM$ and $Mw = wM$ and $wM \cap M = wU \cap U = \emptyset$ for $w \neq 1$, we encounter that
$$ G/U = M \amalg \; \coprod\limits_{w\neq 1} UMw$$
and
$$G/P = 1 \amalg \;\coprod\limits_{w\neq 1} Uw.$$
I hope this is clearer now.
