Let $G$ be $SL(n, F)$ 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 $G$?


Let $d i$ denote the Haar measure on $I$. We have 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
$w\mapsto \mu(IwI)/mu(I)$.


Perhaps one suggestion:
Naively, one could argue that the function is multiplicative by the following computation
$$ \mu(Iw\_1w\_2I)/\mu(I) = \mu(Iw\_1Iw\_2I)/\mu(Iw\_1I) \cdot \mu(I w\_1 I) / \mu( I )$$ 
$$ = \mu(Iw\_2I)/\mu(I) \cdot \mu(I w\_1 I) / \mu( I ),$$
which is wrong. What is the correct relation? $\geq$, if $w\_1$ satisfies $w\_1^2 = 1$?



What is this measure on the affine Weyl group?
It is well-known that $\mu(IwI)/\mu(I)=q^{l(w)}$, where $l(w)$ is the length of $w$ with respect to the affine simple reflections corresponding to $I$. The multiplicative formula holds so long as $l(w\_1w\_2)=l(w\_1)+l(w\_2)$.


Take a look at Iwahori and Matsumoto's original paper on the subject (vol. 25 of IHES Publications) or page 43 of my [PhD thesis](http://www.imsc.res.in/~amri/summaries.html#thesis).


