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 ${\bf G}$ equipped with a finite cyclic group $L$ of order $m$ acting on ${\bf G}$ by algebraic automorphisms defined over $k$. We assume that the Lie algebra of $G$ decomposes into a direct sum of eigenspaces under the action of $L$ (with generator $\theta$); we let ${\frak g}^i$ denote the eigenspace corresponding to eigenvalue $\xi^i$, where $\xi$ is a primitive $m^{th}$ root of unity, that is, $X \in {\frak g}^i$ if and only if $\theta(X)=\xi^iX$. If ${\bf H}$ is a $k$-subgroup of ${\bf G}^{L}$, the group of $L$-fixed points, which contains the neutral component of ${\bf G}^{L}$, then $H={\bf H}(k)$ acts on each eigenspace of ${\frak g}$. Let $r \in \Bbb{R}$. Under mild restrictions on the residual characteristic of $k$, the set of nilpotent $H$-orbits in the $\xi$-eigenspace ${\frak g}^1$ is parametrized by equivalence classes of noticed Moy-Prasad cosets of depth $r$ which lie in ${\frak g}^1$.