Finite order automorphisms and a parametrization of nilpotent orbits in p-adic Lie algebras

Let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field. Let G be the group of k-rational points of a reductive, linear algebraic group G equipped with a finite cyclic group L acting on G by algebraic automorphisms defined over k. We assume that the Lie algebra of G decomposes into a direct sum of eigenspaces, which we denote by gi , under the action of L. If H is a k-subgroup of GL, the group of L-fixed points, which contains the neutral component of GL, then H = H(k) acts on each eigenspace of g , Let r ∈ R . Under mild restrictions on the residual characteristic of k, the set of nilpotent H-orbits in the 1-eigenspace g1 is parametrized by equivalence classes of noticed Moy-Prasad cosets of depth r which lie in g1 .