Minimal proximal definable flows over the $p$-adics

Abstract

Let $G$ be a definable group in an NIP theory. We prove that every minimal proximal definable $G$-flow is strongly proximal. Consequently, the universal minimal proximal definable $G$-flow $\Pi^{\mathrm{def}}(G)$ coincides with the minimal strongly proximal definable $G$-flow $\Pi^{\mathrm{def}}_{\mathrm{s}}(G)$. Furthermore, for a $p$-adic definable group $G$, we can compute $\Pi^{\mathrm{def}}(G)$ explicitly. We show that $\Pi^{\mathrm{def}}(G)$ is exactly $\Pi^{\mathrm{def}}(S)$ where $S$ is the semisimple part of the definably amenable-semisimple decomposition of $G$. In addition, $\Pi^{\mathrm{def}}(S)\cong S^*_\mathcal{F}(\mathbb{Q}_p)$, the space of types of full dimension on $\mathbf{F}$, where $\mathcal{F}=\mathbf{F}(\mathbb{Q}_p)$ for a flag variety $\mathbf{F}$ constructed from $S$.