Parametrizing nilpotent orbits in $p$-adic symmetric spaces using Bruhat-Tits theory

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 $\mathbf{\text{G}}$ equipped with an involution $\theta$ defined over $k\mathrm{.}$ Let $\mathfrak{p}$ denote the $(-1)$-eigenspace in the decomposition of the Lie algebra of $G$ under the differential $d\theta \mathrm{.}$ If $\mathbf{\text{H}}$ is a subgroup of ${\mathbf{\text{G}}}^{\theta }$, the set of $\theta$-fixed points, which contains the connected component of ${\mathbf{\text{G}}}^{\theta },$ then $H=\mathbf{\text{H}}(k)$ acts on $\mathfrak{p}$, which we treat as a symmetric space. Let $r\in \mathbb{R}\mathrm{.}$ Under mild restrictions on $\mathbf{\text{G}}$ and $k,$ the set of nilpotent $H$-orbits in $\mathfrak{p}$ is parametrized by equivalence classes of noticed Moy-Prasad cosets of depth $r$ which lie in $\mathfrak{p}\mathrm{.}$