Fonctions dont les int\'egrales orbitales et celles de leurs transform\'ees de Fourier sont \`a support topologiquement nilpotent

TL;DR

This paper investigates functions on the Lie algebra of a p-adic group whose orbital integrals and Fourier transforms are supported on topologically nilpotent elements, revealing their relation to character-sheaves and endoscopy.

Contribution

It characterizes a subspace of functions with nilpotent orbital integrals and Fourier transforms, demonstrating their stability under endoscopic transfer.

Findings

The subspace is related to characteristic functions of cuspidal nilpotent character-sheaves.

The subspace behaves well under endoscopy.

Fourier transform normalization links orbital integrals to nilpotent support.

Abstract

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