Let $F$ be a local field and $G = GL(n,F)$. Let $f$ be an element $C_c^\infty(G)$.

Let $\gamma$ be an elliptic element of $G$ with irreducible characteristic polynomial.

What are strategies to compute $$ \int\limits_{G_\gamma \backslash G} \phi(g^{-1}\gamma g) \mathrm{d} g?$$

Due to a comment of Paul Broussous: Assume that $\phi$ is bi $GL(n,o)$ invariant (respective $O(n)$ o $U(n)$ invariant at real/complex places). Please give a reference.

I know that Drinfeld has computed this for certain functions of specific type for $F$ non archimedean. Can the general computations be deduced from this?

How does the space $G_\gamma \backslash G / GL(2, o)$ look?