Reducibility points and characteristic $p$ local fields I- Simple supercuspidal representations of symplectic groups

Abstract

Let $F$ be a non-Archimedean local field with odd characteristic $p$. Let $N$ be a positive integer and $G=S{p}_{2N}(F)$. By work of Lomelí on ${\hat{I}}^3$-factors of pairs and converse theorems, a generic supercuspidal representation $\ddot{I}€$ of $G$ has a transfer to a smooth irreducible representation ${\hat{I}}_{\ddot{I}}€$ of $G{L}_{2N+1}(F)$. In turn the Weil-Deligne representation $\hat{I}{\pounds }_{\ddot{I}}€$ associated to ${\hat{I}}_{\ddot{I}}€$ by the Langlands correspondence determines a Langlands parameter $\ddot{I}{•}_{\ddot{I}}€$ for $\ddot{I}€$. That process produces a Langlands correspondence for generic cuspidal representations of $G$. In this paper we take $\ddot{I}€$ to be simple in the sense of Gross and Reeder, and from the explicit construction of $\ddot{I}€$ we describe ${\hat{I}}_{\ddot{I}}€$ explicitly. The method we use is the same as in our previous paper arXiv:2310.20455, where we treated the case where $F$ is a $p$-adic field, and $\ddot{I}€$ a simple supercuspidal representation of $G=S{p}_{2N}(F)$. It relies on a criterion due to Moeglin on the reducibility of representations parabolically induced from $G{L}_M(F)\times G$ for varying positive integers $M$. We extend this criterion to the case when $F$ has any positive characteristic. The main new feature consists in relating reducibility to ${\hat{I}}^3$-factors for pairs.

8 pages