The subject says it all. I would like to know if Proposition 3.1 in
Arthur-Clozel's book on the trace formula holds for local fields of positive
characteristic.


Thanks!


EDIT: Here is Prop 3.1 of Arthur-Clozel: (Notation will be explained after the statement)


### Proposition 3.1


Assume $\phi\in C\_c^\infty(GL\_n(E))$. Then there exists $f\in C\_c^\infty(GL\_n(F))$ such
that,
for regular $\gamma\in GL\_n(F)$, $O\_\gamma(f) = 0$ if $\gamma$ is not a norm and
$O\_\gamma(f) = TO\_{\sigma\delta}(\phi)$ if $\gamma = N\delta$.


### Notation


* $E/F$ is an finite unramified extension of local fields (hence cyclic)
of degree $r$. Let $\sigma\in Gal(E/F)$ be a generator.
* Then there is a *norm* map
$N : \sigma\text{-conjugacy classes in } G(E)\to \text{conjugacy classes in }
G(F)$, where we say that $\delta, \delta'\in G(E)$ are *$\sigma$-conjugate* if there is $h\in G(E)$ such that $\delta' = h^{-1}\delta\sigma(h)$. The map $N$ is defined by sending the class of
$\delta$ to the class of $\delta\sigma(\delta)\cdots\sigma^{r-1}\delta$.
* $O\_\gamma(f)$ is an orbital integral: $O\_\gamma(f) = \int\_{GL\_n(F)\_\gamma\backslash GL\_n(F)}f(g^{-1}\gamma g)\ dg$, where $GL\_n(F)\_\gamma\subset GL\_n(F)$ is the centralizer of $\gamma$.
* $TO\_{\delta\sigma}(\phi)$ is the *twisted orbital integral*:
$TO\_{\delta\sigma}(\phi) = \int\_{GL\_n(E)\_{\delta\sigma}\backslash GL\_n(E)}\phi(h^{-1}\delta \sigma(h))\ dh$. Here $GL\_n(E)\_{\delta\sigma}\subset GL\_n(E)$ is the *twisted centralizer*
of $\delta$: $GL\_n(E)\_{\delta\sigma} = \{h\in GL\_n(E) : h^{-1}\delta\sigma(h) = \delta\}$.



Arthur-Clozel Prop 3.1 for Function Fields?
This can be found in Laumon Cohomology of Drinfeld modules fourth chapter.


