Let $G$ be an algebraic group. Let $B$ be a Borel subgroup. Let $(\rho,V)$ be a one-dimensional $B$-representation.
Let $B$ act on $V$ by $b\cdot v= \rho(b)v$ and
Let $B$ act on $G$ by $b\cdot g =gb^{-1}$
Then $B$ acts on $G\times V$ by $b\cdot(g,v) = (gb^{-1},\rho(b) v)$.
Let $G\times_B V=G\times V/B$ be the space of orbits under this action by $B$.
Let $\pi: G\times_B V\to G/B$ be defined by $(g,v)\mapsto gB$. I want to show this is a vectorbundle, but I don't know how to give trivializations.
I am not even sure what open subsets I can take to trivialize from. Any ideas?
Since G obviously acts transitively on the base space $G / B$, it's sufficient to give a local trivialisation around the image of the identity in $G/B$. So you want an open neighbourhood of $[1]$ in $G / B$ over which things are "nice". The natural choice is what's usually called the "big cell" $U = (\overline{B} B) / B$, where $\overline{B} = w B w^{-1}$ is the opposite Borel (here $w = $ the long Weyl element).
This is an open subvariety of $G / B$ which is affine, and is isomorphic to the unipotent radical $\overline{N}$ of $\overline{B}$, since $\overline{B} \cap B$ is a Levi subgroup of $\overline{B}$. Thus any orbit in $U$ is represented by a unique $\overline{n} \in \overline{N}$, and we can trivialise $G \times_B V$ over $U$ by sending $v \in V$ to the section over $U$ whose value at $\overline{n}$ is $v$.