# does this follow from the Fundamental Lemma of Ngo, Laumon, Waldspurger, …?

Hi,

Does the following follow from FL?

Recall some definitions:

Let $E/F$ be an unramified extension of degree $r$ of local fields of positive characteristic. Let $\theta\in Gal(E/F)$ be a generator. Let $G = GL_{n}/F$. Let $N : G(E)\to G(F)$ be the norm: $N\delta = \delta\theta(\delta) \cdots \theta^{r-1}(\delta)$. It is known that $N\delta$ is conjugate to en element of $G(F)$ and this defines a map: $$N : \{\text{\theta-conjugacy classes in G(E)}\}\to\{\text{conjugacy classes in G(F)}\}.$$ (here $\delta$ and $\delta'$ are $\theta$-conjugate in $G(E)$ if there exists $h\in G(E)$ such that $\delta = h^{-1}\delta'\theta(h)$).

We say that $\gamma\in G(F)$ is a norm if it is conjugate to to $N\delta$ for some $\delta\in G(E)$.

Next, functions $f\in C^\infty_c(G(F))$ and $\phi\in C^\infty_c(G(E))$ are said to be associated if the following condition holds: for every semi-simple $\gamma\in G(F)$ the orbital integral $O_\gamma(f)$ if $\gamma$ is not a norm, and if there exists $\delta\in G(E)$ such that $N\delta = \gamma$ then $$O_\gamma^{G(F)}(f) = TO_{\delta\theta}^{G(E)}(\phi).$$ Here $TO_{\delta\theta}^{G(E)}(\phi)$ is the twisted orbital integral of $\phi$.

Theorem: Suppose that $\phi\in C^\infty_c(G(E))$. Then there exists $f\in C^\infty_c(G(F))$ such that $f$ and $\phi$ are associated.

Thanks!

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Have you checked Arthur-Laumon's book on base change and Laumon "Drinfeld modules, ... " part I chapter IV? –  Marc Palm Jun 14 at 9:27