Reducibility points and characteristic <i>p</i> local fields I: simple supercuspidal representations of symplectic groups

Abstract Let be a non‐Archimedean local field with odd characteristic . Let be a positive integer and . By work of Lomelí on ‐factors of pairs and converse theorems, a generic supercuspidal representation of has a transfer to a smooth irreducible representation of . In turn, the Weil–Deligne representation associated to by the Langlands correspondence determines a Langlands parameter for . This process produces a Langlands correspondence for generic cuspidal representations of . In this paper, we take to be simple in the sense of Gross and Reeder, and from the explicit construction of we describe explicitly. The method we use is the same as in a previous paper, where we treated the case where is a ‐adic field. It relies on a criterion due to Mœglin on the reducibility of representations parabolically induced from for varying positive integers . We extend this criterion to the case when has any positive characteristic. The main new feature consists in relating reducibility to ‐factors for pairs.