Intégrales orbitales sur $GL(N,{\mathbb{F}}_q((t)))$

Abstract

Let $F$ be a non--Archimedean local field of characteristic $\ge 0$, and let $G=GL(N,F)$, $N\ge 1$. An element $\gamma \in G$ is said to be quasi--regular if the centralizer of $\gamma$ in $M(N,F)$ is a product of field extensions of $F$. Let $G_{\mathrm{qr}}$ be the set of quasi--regular elements of $G$. For $\gamma \in G_{\mathrm{qr}}$, we denote by ${\mathcal{O}}_{\gamma }$ the ordinary orbital integral on $G$ associated with $\gamma$. In this paper, we replace the Weyl discriminant $|D_G|$ by a normalization factor ${\eta }_G:G_{\mathrm{qr}}\to {\mathbb{R}}_{>0}$ which allows us to obtain the same results as proven by Harish--Chandra in characteristic zero: for $f\in C_{\mathrm{c}}^{\infty }(G)$, the normalized orbital integral $I^G(\gamma ,f)={\eta }_G^{\frac{1}{2}}(\gamma ){\mathcal{O}}_{\gamma }(f)$ is bounded on $G$, and for $\epsilon >0$ such that $N(N-1)\epsilon <1$, the function ${\eta }_G^{-\frac{1}{2}-\epsilon }$ is locally integrable on $G$.

arXiv comment83 pages, in French