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

TL;DR

This paper develops a parametrization of nilpotent orbits in p-adic symmetric spaces using Bruhat-Tits theory, linking algebraic group actions with geometric structures in a non-Archimedean setting.

Contribution

It introduces a new parametrization of nilpotent orbits in p-adic symmetric spaces via Moy-Prasad cosets, extending Bruhat-Tits theory to this context.

Findings

Parametrization of nilpotent orbits by Moy-Prasad cosets.

Applicable under mild restrictions on the group and field.

Provides a geometric approach to orbit classification.

Abstract

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 $\textbf{G}$ equipped with an involution $\theta$ defined over $k.$ Let $\mathfrak{p}$ denote the $(-1)$-eigenspace in the decomposition of the Lie algebra of $G$ under the differential $d\theta.$ If $\textbf{H}$ is a subgroup of $\textbf{G}^{\theta}$, the set of $\theta$-fixed points, which contains the connected component of $\textbf{G}^{\theta},$ then $H=\textbf{H}(k)$ acts on $\mathfrak{p}$, which we treat as a symmetric space. Let $r \in \mathbb{R}.$ Under mild restrictions on $\textbf{G}$ 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 $\mathfrak{p}.$