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?)



What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$?
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. 


