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?



Elliptic orbital integral
You have introductions to the building of ${\rm GL}(n)$ in Brown's book "Buildings", or in Paul Garrett's book. The extended building is just the product of the non extended building $X$ with the real line $\mathbb R$, with the action 
$$
g.(x,r)=(\ g.x \ , \ r+val\_F (det(g)) )
$$
It is the geometric realization of a simplicial complex whose vertex set is in equivariant bijection with the set of lattice in $F^n$. In the non extended building the stabilizer of a vertex is conjugate to $F^{\times} {\rm GL}(n,{\mathfrak o})$. In the extended building the stabilizer of a vertex is conjugate to ${\rm GL}(n,{\mathfrak o})$. 


When $\phi$ is the characteristic function of ${\rm GL}(n,{\mathfrak o})$, the value of the orbital integral is related to the number of lattices in $F^n$ fixed by $\gamma$, so to a set of fixed vertices in the extended building.


Examples of calculations of orbital integrals using the building may be found:


-- in the PhD thesis of J. Rogawski that you can find online : <http://www.math.ucla.edu/~jonr/eprints.html>


-- in Langlands's book "base change for ${\rm GL}(2)$".


-- in Kottwitz's PhD thesis : "Orbital integrals on ${\rm GL}\_{3}$" Amer. J. Math. 102 (1980), no. 2, 327–384. 


