Endoscopic liftings and parameters of epipelagic representations for classical groups

Abstract

Let $G$ be a $p$-adic classical group (orthogonal, symplectic, or unitary) and $π$ be an epipelagic representation of $G$ in the sense of Reeder-Yu. Using Mœglin's theory of extended cuspidal supports and Bushnell-Kutzko's theory of covering types, we determine explicitly the endoscopic lift of $π$ to the general linear group, whose Langlands dual expresses the dual group of $G$ as a complex matrix group, in terms of the inducing type of $π$ that extends the character of the first Moy-Prasad filtration subgroup defined by a stable functional. We interpret the inducing type of $π$ via Stevens' construction of supercuspidal representations by skew semisimple strata and introduce what we will call epipelagic strata, requiring only that the residual characteristic $p$ be odd. As an application, we reprove M. Oi's results on the endoscopic lifts of simple supercuspidal representations, in the sense of Gross-Reeder, of quasi-split classical groups. Finally, on the Galois side, we show that the epipelagic Langlands parameters of $G$ constructed by Reeder-Yu can be recovered via the self-duality of Bushnell-Henniart's admissible triples and endoscopic embeddings of L-groups. We also establish some related results concerning epipelagic parameters that were previously proved under restrictions on $p$, including a parity result on rectifying characters and an adjoint Swan conductor result related to the Hiraga-Ichino-Ikeda conjecture.