On Iwahori decomposition Let $F$ be a non-archimedean local field with finite field $\mathbb F_q$ of prime characteristic $p$, and let $L$ be the completion of the maximal unramified extension of $F$.
We write $\mathcal O$ for the valuation ring of $L$.
Further, we denote by $\varpi$ a uniformizer of $L$.
Set $G=\mathrm{GL}_n$.
Let $I$ be the inverse image of the subgroup of lower triangular matrices under the map $G(\mathcal O)\rightarrow G(\overline{\mathbb F}_q), \varpi\mapsto 0$.
Then we have the Iwahori decomposition $G(L)=\bigcup_{w\in \tilde{W}}I\tilde{w}I$.
My question is the following: if $w=w_1w_2$ with length($w_1$)+length($w_2$)=length($w$), then we have $Iw_1Iw_2I=IwI$?
 A: So, I was finally able to figure this out and it is wrong in general.
I will give a counterexample for $GL_2$, which can be extended to $GL_n$.
Let $$ w_1=w_2= \begin{bmatrix}
0 & \varpi \\
1 & 0
\end{bmatrix},$$
then $$w=w_1\cdot w_2=\varpi\cdot Id$$
and furthermore
$$l(w_1)=l(w)=0.$$
Thus your condition is met. Now we get an inclusion
$$ IwI\subset Iw_1 I w_2 I$$
but it will be strict: Consider the matrix $$A=\begin{bmatrix}
1 & 1\\
\varpi & 1
\end{bmatrix}$$
which is in the Iwahori subgroup $I$. Then
$$ B=w_1 A w_2=\begin{bmatrix}
\varpi & \varpi^3\\
1 & \varpi
\end{bmatrix}. $$
Thus $B\in Iw_1 I w_2 I$, but it can not be in in $$IwI=w I=\varpi\cdot I,$$ as then every entry in the matrix has to be a multiple of $\varpi$.
Let me put this in a bigger picture: In general for $GL_n$ (you can actually also consider more generally split reductive groups, but then you have to be more careful) we have a short exact sequence
$$ 0\to W_{af}\to \tilde{W}\to \mathbb{Z}\to 0$$
where the first map is the inclusion of the extended Weyl group of $SL_n$ and the second map is the ($\varpi$-)valuation of the determinant of an element in $\tilde{W}$. This sequence is actually split and decomposes $\tilde{W}$ as a semi-direct product. In particular we get a map $s:\mathbb{Z}\to\tilde{W}$ and essentially by definition for any $m\in \mathbb{Z}$, the length of $s(m)$ is $0$.
In the $GL_2$-case, the splitting is given by $s(m)=(w_1)^m$.
