Local theta correspondences between epipelagic supercuspidal representations

In this paper we study the local theta correspondences between epipelagic supercupsidal representations of a type I classical dual pair $(G,G^{\mathrm{\prime }})$ over $p$-adic fields. We show that, besides an exceptional case, an epipelagic supercupsidal representation $\pi$ of $\tilde{G}$ lifts to an epipelagic supercupsidal representation ${\pi }^{\mathrm{\prime }}$ of ${\tilde{G}}^{\mathrm{\prime }}$ if and only if the epipelagic data of $\pi$ and ${\pi }^{\mathrm{\prime }}$ are related by the moment maps.