ORBITAL INTEGRALS FOR LINEAR GROUPS

Abstract

For a linear group $G$ acting on an absolutely irreducible variety $X$ over $\mathbb{Q}$, we describe the orbits of $X(\mathbb{Q}_p)$ under $G(\mathbb{Q}_p)$ and of $X(\mathbb{F}_p((t)))$ under $G(\mathbb{F}_p((t)))$ for $p$ big enough. This allows us to show that the degree of a wide class of orbital integrals over $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$ is less than or equal to $0$ for $p$ big enough, and similarly for all finite field extensions of $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$.