Take the 2-minute tour ×  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?)  share|improve this
question You are dealing with the integral formula for the Haar
measure on $G$, associated with the Cartan decomposition
$G=KAK$. Since the Plancherel measure lives on the dual object
$\hat{G}$, I don't see any obvious relation bet