Nilpotent Elements and Reductive Subgroups Over a Local Field

Abstract

Let ${\mathcal {K}}$ be a local field – i.e. the field of fractions of a complete DVR ${\mathcal {A}}$ whose residue field 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 $\text {Lie}(G)({\mathcal {K}})$ – 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 ${{\mathscr{M}}}$ over ${\mathcal {A}}$ for which $M = {{\mathscr{M}}}_{{\mathcal {K}}}$ . Our main result shows for any nilpotent element X1 ∈Lie(G) that there is an unramified, reductive ${\mathcal {K}}$ -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.