Covers of Bruhat-Tits trees

Let $G$ be a locally compact group and let $\widetilde{G}$ be a central extension that splits over a maximal compact subgroup $K$ of $G$. We derive an explicit cocycle that lifts the natural action of $G$ on the homogeneous space $G/K$ to an action of $\widetilde{G}$. As an application, for a non-Archimedean local field $F$, we construct a connected locally finite tree on which the metaplectic covers of $\operatorname{GL}_2(F)$ act by automorphisms, providing a geometric analog of the Bruhat--Tits tree of $\operatorname{GL}_2(F)$. Furthermore, under suitable transitivity assumptions, we prove that $(\widetilde{G},\widetilde{K})$ is a Gelfand pair. Finally, we describe the associated parabolic and contraction subgroups with respect to $\widetilde{G}$ from the perspective of the geometry of the constructed tree.