Borel subgroups contained in a fixed parabolic subgroup - MathOverflow

Borel subgroups contained in a fixed parabolic subgroup

2011-05-27T11:55:23Z

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.

Answer by Jim Humphreys for Borel subgroups contained in a fixed parabolic subgroup

2011-05-27T12:58:17Z

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.