Borel subgroups contained in a fixed parabolic subgroup - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:06:34Z http://mathoverflow.net/feeds/question/66174 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66174/borel-subgroups-contained-in-a-fixed-parabolic-subgroup Borel subgroups contained in a fixed parabolic subgroup th.ng 2011-05-27T11:55:23Z 2011-05-27T20:49:09Z <p>The question is asked in the context of (connected) reductive groups.</p> <p>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) :</p> <blockquote> <p>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.</p> </blockquote> <p>Is there a simple way to prove this ? </p> <p>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. </p> http://mathoverflow.net/questions/66174/borel-subgroups-contained-in-a-fixed-parabolic-subgroup/66184#66184 Answer by Jim Humphreys for Borel subgroups contained in a fixed parabolic subgroup Jim Humphreys 2011-05-27T12:58:17Z 2011-05-27T13:14:44Z <p>There is some confusion in the way the question is set up. You have to begin with a fixed Borel subgroup <code>\$B\$</code> in order to speak about a "simple" reflection in the Weyl group. Then there is a unique "minimal" parabolic subgroup <code>\$P \supset B\$</code> corresponding to the specified simple root/reflection. This in turn has a Levi subgroup of rank 1, while <code>\$P/B\$</code> is naturally isomorphic to the projective line. Of course, <code>\$P\$</code> itself contains many other Borel subgroups besides <code>\$B\$</code>, but now the problem has shifted. All such Borels contain the radical of <code>\$P\$</code>. 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.</p>