Int\'egrales orbitales sur $GL(N,{\Bbb F}_q((t)))$

TL;DR

This paper generalizes Harish-Chandra's results on orbital integrals for $GL(N)$ over non-Archimedean fields by introducing a normalization factor that ensures boundedness and local integrability of orbital integrals in positive characteristic.

Contribution

It introduces a new normalization factor for orbital integrals on $GL(N)$ over non-Archimedean fields, extending Harish-Chandra's characteristic zero results to positive characteristic.

Findings

Normalized orbital integrals are bounded on $G$.

The function $ ext{eta}_G^{-rac{1}{2}- ext{epsilon}}$ is locally integrable.

Results hold for fields of characteristic $ eq 2$.

Abstract

Let $F$ be a non--Archimedean local field of characteristic $\geq 0$, and let $G=GL(N,F)$, $N\geq 1$. 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 $G_{\rm qr}$ be the set of quasi--regular elements of $G$. For $\gamma\in G_{\rm qr}$, we denote by $\mathcal{O}_\gamma$ the ordinary orbital integral on $G$ associated with $\gamma$. In this paper, we replace the Weyl discriminant $\vert D_G\vert$ by a normalization factor $\eta_G: G_{\rm qr}\rightarrow {\Bbb R}_{>0}$ which allows us to obtain the same results as proven by Harish--Chandra in characteristic zero: for $f\in C^\infty_{\rm c}(G)$, the normalized orbital integral $I^G(\gamma,f)=\eta_G^{1\over 2}(\gamma)\mathcal{O}_\gamma(f)$ is bounded on $G$, and for $\epsilon>0$ such that $N(N-1)\epsilon <1$, the function $\eta_G^{-{1\over 2}-\epsilon}$ is locally integrable on $G$.