My paper Nilpotent elements and reductive subgroups over a local field (2020) has been published online in Algebras and Representation Theory.

Here is the official version.

Abstract:

Let 𝒦 be a

local field– i.e. the field of fractions of a complete DVR 𝒜 whose residue field 𝓀 has characteristicp> 0 – and letGbe a connected, absolutely simple algebraic 𝒦-groupGwhich splits over an unramified extension of 𝒦. We study the rational nilpotent orbits ofG– i.e. the orbits ofG(𝒦) in the nilpotent elements of Lie (G)(𝒦) – under the assumptionp> 2h− 2 wherehis the Coxeter number ofG.

A reductive group

Mover 𝒦 isunramifiedif there is a reductive model ℳ over 𝒜 for whichM= ℳ_{𝒦}. Our main result shows for any nilpotent elementX_{1}∈ Lie (G) that there is an unramified, reductive 𝒦-subgroupMwhich contains a maximal torus ofGand for whichX_{1}∈ Lie (M) isgeometrically distinguished.

The proof uses a variation on a result of DeBacker relating the nilpotent orbits of

Gwith the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated withG.