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

TL;DR

This paper explicitly constructs simple supercuspidal representations of symplectic groups over local fields of odd characteristic and relates their reducibility to gamma factors, extending previous methods to positive characteristic fields.

Contribution

It extends the criterion for reducibility of induced representations to fields of positive characteristic and explicitly describes the transfer of simple supercuspidal representations to general linear groups.

Findings

Explicit construction of simple supercuspidal representations of Sp_{2N}(F).

Extension of reducibility criterion to positive characteristic fields.

Relation of reducibility to gamma factors for pairs.

Abstract

Let $F$ be a non-Archimedean local field with odd characteristic $p$. Let $N$ be a positive integer and $G=Sp_{2N}(F)$. By work of Lomel\'i on $\gamma$-factors of pairs and converse theorems, a generic supercuspidal representation $\pi$ of $G$ has a transfer to a smooth irreducible representation $\Pi_\pi$ of $GL_{2N+1}(F)$. In turn the Weil-Deligne representation $\Sigma_\pi$ associated to $\Pi_\pi$ by the Langlands correspondence determines a Langlands parameter $\phi_\pi$ for $\pi$. That process produces a Langlands correspondence for generic cuspidal representations of $G$. In this paper we take $\pi$ to be simple in the sense of Gross and Reeder, and from the explicit construction of $\pi$ we describe $\Pi_\pi$ 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 $\pi$ a simple supercuspidal representation of $G=Sp_{2N}(F)$. It relies on a criterion due to Moeglin on the reducibility of representations parabolically induced from $GL_M(F)\times G$ 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 $\gamma$-factors for pairs.