Consider $G = GL_n( \mathbb{Q}_p)$ and $K = GL_n( \mathbb{Z}_p)$, or more favorable replace $F$ 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 $w$?

(More out of curiousity: How is the Plancherel measure related to this?)

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?)