What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$? Consider $G = GL_n( \mathbb{Q}_p)$ and $K = GL_n( \mathbb{Z}_p)$, or more favorable replace $ \mathbb{Q}_p$ by a non archimedean field and $\mathbb{Z}_p$ by the ring of integers. 
Note that the space $G //K$ is discrete, since $K$ is open and closed in $G$.
Given a Haar measure on $G$, I can prove that there exists a unique (discrete) measure on $G//K$, such that 
$$ \int\limits_{G} f(g) d g  = \sum\limits_{x \in G//K} w(x) \int\limits_{K \times K} f(k_1 x k_2) d k_1 d k_2 .$$ 
How can $w$ be expressed, if we pick a representative $x = diag( w^{k_1}, \dots, w^{k_n})$ for a uniformizer $w$? 
Perhaps easier, but equivalent what is the ratio: $vol_G (K xK)/ vol_G(K)?$
(More out of curiousity: How is the Plancherel measure related to this?)
 A: The measure of $KxK$ is a classical computation that may be found in: Macdonald "Symmetric Functions and Hall Polynomials" (Oxford Mathematical Monographs), more precisely in Chapter V: The Hecke ring of ${\rm GL}(n)$ over a local field. 
A: Depending on taste, one might also find appealing or helpful the description of this in terms of the Iwahori-Hecke algebra, with affine Weyl group and affine cartan decomposition $G=\bigcup_w BwB$ where $B$ is the Iwahori. Among other features, this does give a way to inductively determine the measure of $BwB$, once the measure of $B$ is normalized, because there is a precise cell-multiplication rule $BwB\cdot BsB=BwsB$ when the length of $ws$ is strictly greater than that of $w$, and $s$ is one of the affine reflections generating $W$. That is, the length in $W$ is equivalent to knowing the measure of the Iwahori coset.
(Inevitably, surely MacDonald's discussion does something equivalent to this, but I don't remember, and I don't have a copy accessible to me.)
