Let $G$ be an $SL(n, F)$ for a non archimedean local field $F$ with Iwahori subgroup $I$. Let $\mu$ be the Haar measure of $G$. 

What are the properties of the function $w\mapsto \mu(IwI)/mu(I)$ for $w$ being an element of the normalizer $N$ of the maximal Levi subgroup of $G$?

Note that for $d i$ being the Haar measure of $I$, that
$$ \int\limits_G f(g) d \mu(g) = \sum\limits_{x \in N / N \cap I} ( \mu(IwI)/mu(I) )^{-1} \int\limits_I \int\limits_I f(i_1 w i_2) d i_1  d i_2 .$$

I would like to have a reference for the above fact and perhaps also the value of $mu(IwI)/mu(I) $ at least for $n=2$ and also for similar situation of $PGL(n)$.

I suspect that the fact that $(I,N)$ is a $BN$ pair and that the Iwahori decomposition is available for $I$ should give everything and also give some nice properties for the function
$w\mapsto \mu(IwI)/mu(I)$.