A classification of the irreducible admissible genuine mod p representations of the metaplectic cover of p-adic SL(2)

TL;DR

This paper classifies all irreducible, admissible, genuine mod p representations of the metaplectic double cover of p-adic SL(2), revealing their structure via a generalized Satake transform and distinguishing between supercuspidal and nonsupercuspidal types.

Contribution

It provides a complete classification of these representations, introduces explicit parameters, and shows the absence of genuine special mod p representations in this setting.

Findings

All such representations are quotients of compact inductions by Hecke operators.

Parameters distinguish supercuspidal from nonsupercuspidal representations.

No genuine special mod p representations exist for this cover.

Abstract

We classify the irreducible, admissible, smooth, genuine mod p representations of the metaplectic double cover of SL(2,F), where F is a p-adic field and p is odd. We show, using a generalized Satake transform, that each such representation is isomorphic to a certain explicit quotient of a compact induction from a maximal compact subgroup by an action of a spherical Hecke operator, and we define a parameter for the representation in terms of this data. We show that our parameters distinguish genuine nonsupercuspidal representations from genuine supercuspidals, and that every irreducible genuine nonsupercuspidal representation is in fact an irreducible principal series representation. In particular, the metaplectic double cover of p-adic SL(2) has no genuine special mod p representations.