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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.

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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.

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  • $\begingroup$ My question might be stupid... are you refering to the semisimple rank ? rather than the (full) rank of the Lévi complement ? $\endgroup$
    – th.ng
    Commented May 27, 2011 at 15:16
  • $\begingroup$ Which type of rank you consider here makes no difference, since a central torus lies in all Borel subgroups. (In my answer, you could factor out the solvable or the unipotent radical.) Similarly, isogeny type doesn't matter. So in rank 1 you get the same flag variety $\mathbb{P}^1$ for the groups $GL_2, SL_2, PGL_2$, over any algebraically closed field. $\endgroup$
    – Jim Humphreys
    Commented May 27, 2011 at 17:07

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