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.
Let 𝒦 be a local field – i.e. the field of fractions of a complete DVR 𝒜 whose residue field 𝓀 has characteristic p > 0 – and let G be a connected, absolutely simple algebraic 𝒦-group G which splits over an unramified extension of 𝒦. We study the rational nilpotent orbits of G– i.e. the orbits of G(𝒦) in the nilpotent elements of Lie (G)(𝒦) – under the assumption p > 2h − 2 where h is the Coxeter number of G.
A reductive group M over 𝒦 is unramified if there is a reductive model ℳ over 𝒜 for which M = ℳ𝒦. Our main result shows for any nilpotent element X1 ∈ Lie (G) that there is an unramified, reductive 𝒦-subgroup M which contains a maximal torus of G and for which X1 ∈ Lie (M) is geometrically distinguished.
The proof uses a variation on a result of DeBacker relating the nilpotent orbits of G with the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated with G.