The question is asked in the context of (connected) reductive groups.


In the article i'm working on, the author states the following fact (well it's not word to word exact, I simplified it a little) :



> 
> If we choose a parabolic subgroup,
>  determined by a simple reflection $s$
>  in $W$ (the Weyl group, given a
>  maximal torus), then the variety of
>  the Borel subgroups contained in $P$
>  is one-dimensional, moreover it can be
>  identified with the projective line.
> 
> 
> 


Is there a simple way to prove this ? 


What I tried : since the parabolic subgroup $P$ is determined by a simple reflection, I wrote down the Levi decomposition using roots, then I was thinking that the Borel subgroups contained in $P$ are in bijection with the Borel subgroups of the Levi complement. 



Borel subgroups contained in a fixed parabolic subgroup
There is some confusion in the way the question is set up. You have to begin with a fixed Borel subgroup $B$ in order to speak about a "simple" reflection in the Weyl group. Then there is a unique "minimal" parabolic subgroup $P \supset B$ corresponding to the specified simple root/reflection. This in turn has a Levi subgroup of rank 1, while $P/B$ is naturally isomorphic to the projective line. Of course, $P$ itself contains many other Borel subgroups besides $B$, but now the problem has shifted. All such Borels contain the radical of $P$. so after factoring that out you are just looking at the flag variety of a rank 1 group which is the desired copy of the projective line.


