Comparing Iwahori Decompositions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:15:30Z http://mathoverflow.net/feeds/question/8837 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8837/comparing-iwahori-decompositions Comparing Iwahori Decompositions Peter McNamara 2009-12-14T03:55:04Z 2009-12-14T05:30:02Z <p>Let G be a p-adic group, U a (n appropriate) unipotent subgroup and I an Iwahori subgroup. Then there are Iwahori decompositions I\G/I=U\G/I=W where W is the affine Weyl group. I suspect that $$Uw_1Iw_2I=Uw_1w_2I$$ whenever $\ell(w_1)+\ell(w_2)=\ell(w_1w_2)$. Is this true?</p> http://mathoverflow.net/questions/8837/comparing-iwahori-decompositions/8845#8845 Answer by Ben Webster for Comparing Iwahori Decompositions Ben Webster 2009-12-14T05:23:21Z 2009-12-14T05:30:02Z <p>I believe this is true in any building (which I\G is). Equivalently, this is one of the usual properties of BN-pairs. The way you can think about it is that I is generated by $I\cap w_2 I w_2^{-1}$ and $I\cap w_1^{-1}Uw_1$ (since no positive root space can be sent to a negative one by both $w_2$ and $w_1^{-1}$).</p>