“Nilpotent elements and reductive subgroups over a local field” (manuscript)

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 $\mathcal{A}$ 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 $\mathrm{Lie}(G)(\mathcal{K})$ – 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 $\mathcal{A}$ for which $M={\mathcal{M}}_{\mathcal{K}}$. Our main result shows for any nilpotent element $X_1\in \mathrm{Lie}(G)$ that there is an unramified, reductive $\mathcal{K}$-subgroup $M$ which contains a maximal torus of $G$ and for which $X_1\in \mathrm{Lie}(M)$ 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 $G$.