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.



Description of $GL_3/U$
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 $G$).


