Local theta correspondences between epipelagic supercuspidal representations

Abstract

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