# Arthur-Clozel Prop 3.1 for Function Fields?

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 $\delta$: $GL_n(E)_{\delta\sigma} = \{h\in GL_n(E) : h^{-1}\delta\sigma(h) = \delta\}$.

• The subject doesn't say it all. In particular, it doesn't say what Prop. 3.1 in Arthur-Clozel's book is ... Mar 5 '13 at 23:36