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 ${\Pi }_{\mathrm{s}}^{\mathrm{def}}(G)$. 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, ${\Pi }^{\mathrm{def}}(S)\cong S_{\mathcal{F}}^{\ast }({\mathbb{Q}}_p)$, 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 $S$.