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 \(\mathcal{K}\) be a local field – i.e. the field of fractions of a complete DVR \(\mathscr{A}\) whose residue field \(\mathcal{k}\) has characteristic \(p > 0\) – and let \(G\) be a connected, absolutely simple algebraic \(\mathcal{K}\)-group \(G\) which splits over an unramified extension of \(\mathcal{K}\). We study the rational nilpotent orbits of \(G\)– i.e. the orbits of \(G(\mathcal{K})\) in the nilpotent elements of \(\operatorname{Lie}(G)(\mathcal{K})\) – under the assumption \(p>2h-2\) where \(h\) is the Coxeter number of \(G\).
A reductive group \(M\) over \(\mathcal{K}\) is unramified if there is a reductive model \(\mathcal{M}\) over \(\mathscr{A}\) for which \(M = \mathcal{M}_\mathcal{K}\). Our main result shows for any nilpotent element \(X_1 \in \operatorname{Lie}(G)\) that there is an unramified, reductive \(\mathcal{K}\)-subgroup \(M\) which contains a maximal torus of \(G\) and for which \(X_1 \in \operatorname{Lie}(M)\) 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 \(G\).