Comparing Iwahori Decompositions

Question

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?

Answer 1

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 $w_1^{-1}$).