Une mesure de Radon invariante sur les $F$-strates unipotentes

TL;DR

This paper constructs a canonical Radon measure on unipotent strata in reductive groups over non-Archimedean fields, enabling convergence results for orbital integrals and generalizing previous work to broader settings.

Contribution

It introduces a canonical invariant Radon measure on unipotent strata, extending Deligne-Ranga Rao's construction to new contexts and ensuring orbital integral convergence.

Findings

Defined a positive invariant Radon measure on unipotent strata.

Proved convergence of orbital integrals under certain conditions.

Generalized existing measures to all characteristics and to nilpotent strata.

Abstract

Let $F$ be a non-Archimedean locally compact field and $G$ a connected reductive group defined over $F$. To any unipotent element $u$ in $G(F)$, we have associated in [L] an $F$-stratum $\boldsymbol{\mathfrak{Y}}_{F,u}$ which is a (possibly infinite) union of unipotent $G(F)$-orbits. We define here a "canonical" non-zero positive $G(F)$-invariant Radon measure on $\boldsymbol{\mathfrak{Y}}_{F,u}$. Under additional assumptions, we deduce the convergence of the orbital integral associated to the $G(F)$-orbit of $u$. The construction, valid in any characteristic, generalizes the one of Deligne-Ranga Rao [RR] and also applies to nilpotent strata in $\textrm{Lie}(G)(F)$.