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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?

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I guess that $x=\gamma$. I think your first question is not precise enough. What type of function $\phi$ are you interested in ? If $\phi$ is e.g. the characteristic function of ${\rm GL}({\mathfrak o}_F )$, then you have computations of the integral made by Langlands, Kottwitz, Rogawsky in very particular cases. I don't think that one can expect a nice general formula. But I may be wrong. –  Paul Broussous Nov 29 '11 at 17:18
I edited the question. In fact, I need the computation only slightly more general then for the characteristic function of $K$. Can you give some papers, which contain the computations> I guess with Kottwitz you meant his paper "on Tamagawa numbers", right? –  plusepsilon.de Nov 30 '11 at 7:27
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2 Answers

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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.

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The coset space $G/{\rm GL}(n,{\mathfrak o}_F)$ may be seen as the set of vertices of the extended Bruhat-Tits building of ${\rm GL}(n,F)$. There is a second point of view that amounts to consider this set as the set of $k_F$-rational points of a variety defined over $k_F$ (the residue field of $F$) : an affine flag variety. This is the point of view used by Ngo Bao Chau to prove the Fundamental Lemma.

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Is this a typo, you mean $G_\gamma \backslash G / GL(n,o)$? I am not familiar with Bruhat-Tits buildings, extended or not. Is there a intro specifically dealing with $GL(n)$? What you mention with the flag structure over $k_F$ seems really interesting to me, and also fits within my concept. Would you care to include a reference for this, with page if possible? Thanks a lot for your help. –  plusepsilon.de Nov 30 '11 at 7:25
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