Irreducible admissible mod-p representations of metaplectic groups

TL;DR

This paper classifies smooth, irreducible, admissible mod-$p$ representations of the metaplectic cover of symplectic groups, extending known results and providing criteria for irreducibility and induction preservation.

Contribution

It adapts Herzig's method to the metaplectic setting and classifies representations in terms of supercuspidal representations of Levi subgroups.

Findings

Classification of irreducible mod-$p$ representations of metaplectic groups.

Irreducibility criteria for principal series representations.

Irreducibility preservation under parabolic induction.

Abstract

Let $p$ be an odd prime number, and $F$ a nonarchimedean local field of residual characteristic $p$. We classify the smooth, irreducible, admissible genuine mod-$p$ representations of the twofold metaplectic cover $\widetilde{\textrm{Sp}}_{2n}(F)$ of $\textrm{Sp}_{2n}(F)$ in terms of genuine supercuspidal (equivalently, supersingular) representations of Levi subgroups. To do so, we use results of Henniart--Vign\'{e}ras as well as new technical results to adapt Herzig's method to the metaplectic setting. As consequences, we obtain an irreducibility criterion for principal series representations generalizing the complete irreducibility of principal series representations in the rank 1 case, as well as the fact that irreducibility is preserved by parabolic induction from the cover of the Siegel Levi subgroup.