Distributions invariantes sur les groupes reductifs quasi-deployes

Abstract

Let $F$ be a nonarchimedean local field, and $G$ the group of $F$-points of a c onnected quasisplit reductive group defined on $F$; in this paper, we will study the distributions on $G$ which are invariant by conjugation, and the vector spa ce of their restrictions to the Hecke algebra $\mathcal{H}$ of the functions on $G$ with compact support and biinvariant by a given Iwahori subgroup $I$. It is first shown that such a distribution on $\mathcal{H}$ is entirely determined by its restriction to the finite-diimensional subspace of $\mathcal{H}$ containing the elements with support in the union of the parahoric subgroups of $G$ contai ning $I$; this property is then used, by establishing similar results on finite groups, to show, with some conditions on $G$, that this space is generated both by some semisimple orbital integrals and by unipotent integrals.