Orbital integral in Cluckers and Denef

Question

My questions concerns Definition 1.2 of an orbital integral in the paper Orbital integrals on General Linear Groups by Cluckers and Denef. I will recall the definition below, but my question is: how does Definition 1.2 relate to the usual definition of an orbital integral? By the latter I mean for given a reductive group $G$ over $K$ and a regular element $\gamma$ of $G(K)$ with centralizer $G_\gamma$, the orbital integral of a function $f$ with sufficient decay being $$ O_\gamma(f)= \int_{G_{\gamma}(K)\backslash G(K)}f(x^{-1}\gamma x)dx. $$

Definition 1.2 is long, but I include it here for convenience:

Let $F$ be a number field with ring of integers $\mathcal O_F$. Let $\mathcal A_F$ be the collection of all finite field extensions of non-archimedean completions of $F$. Let $\mathcal B_F$ be the collection of all fields of the form $\mathbb F_q((t))$ which are rings over $\mathcal O_F$.For $K$ in $\mathcal A_F\cup \mathcal B_F$, let $\mathcal O_K$ be its valuation ring, $M_K$ its maximal ideal, $k_K$ its residue field, and $q_K\mathrel{:=}\# k_K$. For $N>0$,let $$\mathcal C_N=\{K\in \mathcal A_F\cup \mathcal B_F\mathrel| \text{char}(k_K)>N\}.$$

Let $G$ be a linear algebraic group over $F$, rationally acting on an absolutely irreducible algebraic variety $X$ over $F$. Suppose that $X$ is a homogeneous $G$-space, that is, the action of $G(\mathbb C)$ on $X(\mathbb C)$ is transitive. For $K$ a field over $F$ and $x\in X(K)$, let $G(K)(x)$ be the orbit of x under the action of $G(K)$.

Let $U\subset X$ be an affine open and let $f, g_i:U\to \mathbb A^1_F$ be regular functions. Let $\omega$ be a volume form on $U$, that is, a degree $n$ rational differential form on $U$ when $X$ is of dimension $n$. For each $K\in \mathcal C_1$ let $W(K)$ be $\bigcap_i g^{-1}_i(U_i)$ with $U_i$ either $\mathcal O_K$ or $M_K$. For each $K\in\mathcal C_1$ and for each $x$ in $X(K)$, under the condition of integrability for all $s>0$, define the orbital integral $$ I_{K,x}(s)=\int_{G(K)(x)\cap W(K)}\lvert f^s\rvert\lvert\omega\rvert_K $$ with $\lvert\omega\rvert_K$ the measure on $U(K)$ associated with $\omega$. If for some $K$ and $x$ this is not integrable, put $I_{K,x}(s) := 0$ for this $K$ and $x$.

Answer 1

My sense is that this paper is actually not (at least, not in any straightforward way) related to the orbital integrals one thinks of in connection with the Fundamental Lemma, for example. You see, the key assumption in the Cluckers-Denef paper is that $X$ is a homogeneous $G$-space, that is, $G({\mathbb C})$ acts on $X(\mathbb C)$ transitively. Then they study the orbits on $X(K)$ under $G(K)$ where $K$ is a p-adic field. So I think the only way it would apply to the orbital integrals in the FL is if we take $X$ to be a single stable orbit (thought of as an affine variety in its own right, maybe endowed with a volume form $\omega$ that comes e.g. from Kirillov's form on co-adjoint orbits), and then the paper tells us something about the integrals on rational orbits. In the orbital integrals one typically thinks about for the Fundamental Lemma, the $f$ in Cluckers-Denef would have to be trivial, and you are right, the $W(K)$ in their notation can be the maximal compact or can be rigged to be the Iwahori subgroup, for example; they leave it flexible. Then the result comes out vacuous: if $f$ is trivial, there's no $s$. Note that they prove that the integral they are looking at is a rational function of $q^{-s}$ but I think the coefficients must be allowed to depend on the field as well (they would be counting points on some varieties over the residue field, roughly speaking). They prove there exists a uniform bound on the degree of denominator in that rational function of $q^{-s}$ (when $f$ is a polynomial).