Fonctions Dont les Intégrales Orbitales et Celles de Leurs Transformées de Fourier Sont à Support Topologiquement Nilpotent

Abstract

Let F be a p-adic field and let G be a connected reductive group defined over F. We assume p is large. Denote $\mathfrak{g}$ the Lie algebra of G. To each vertex s of the reduced Bruhat–Tits’ building of G, we associate as usual a reductive Lie algebra ${\mathfrak{g}}_{\mathit{s}}$ defined over the residual field ${\mathbb{F}}_{\mathit{q}}$. We normalize suitably a Fourier-transform $\mathit{f}\mapsto \hat{\mathit{f}}$ on ${\mathit{C}}_{\mathit{c}}^{\infty }(\mathfrak{g}(\mathit{F}))$. We study the subspace of functions $\mathit{f}\in {\mathit{C}}_{\mathit{c}}^{\infty }(\mathfrak{g}(\mathit{F}))$ such that the orbital integrals of f and of $\hat{\mathit{f}}$ are 0 for each element of $\mathfrak{g}(\mathit{F})$ which is not topologically nilpotent. This space is related to the characteristic functions of the character-sheaves on the spaces ${\mathfrak{g}}_{\mathit{s}}({\mathbb{F}}_{\mathit{q}})$, for each vertex s, which are cuspidal and with nilpotent support. We prove that our subspace behave well under endoscopy.