A Wild Local Shimura Correpsondence

Abstract

For the $n$-fold cover of a simply-laced simply-connected Chevalley group $G$ over a $p$-adic field $F$, where GCD$(n,p)=1$, Savin proved a correspondence between certain genuine representations of the $n$-fold cover of $G$ and the Iwahori-spherical representations of the group $G/Z_{n}$, where $Z_{n}$ is the $n$-torsion of the center of $G$. In this paper we prove the analogous result when $n=2$ and $F=\mathbb{Q}_{2}$. In particular, GCD$(n,p)\neq 1$