Some results on simple supercuspidal representations of GLn(F)

Let F be a non-archimedean local field of characteristic zero with residual characteristic p  . In this paper, we first present a simple proof and construction of the local Langlands correspondence for simple supercuspidal representations of GLn(F)GLn(F), when p∤np∤n. Our proof relies on the existence of the local Langlands correspondence for GLn(F)GLn(F), due to Harris/Taylor and Henniart. We then prove Jacquet's conjecture on the local converse problem for GLn(F)GLn(F) in the case of simple supercuspidal representations, following the strategy of Jiang, Nien and Stevens. Recently, Reeder and Yu constructed a family of epipelagic supercuspidal representations for semi-simple p  -adic groups. In the last part of our paper, we show that the analogous construction for GLn(F)GLn(F) produces only simple supercuspidals. In the process, we conclude that the constructed epipelagic supercuspidals of Reeder and Yu may not necessarily exhaust all epipelagic supercuspidals of a p-adic group.