A mod-$p$ metaplectic Montr\'{e}al functor

TL;DR

This paper extends Colmez's functor to the metaplectic cover of GL_2(Q_p), establishing a correspondence between genuine supersingular representations and four-dimensional Galois representations with special invariance properties.

Contribution

It introduces a new functor for metaplectic covers, linking genuine representations to Galois representations with additional structure, expanding the local Langlands correspondence.

Findings

Computed images of irreducible genuine objects.

Established a bijection with twist-invariant Galois representations.

Defined metaplectic Galois representations with extra structure.

Abstract

We extend Colmez's functor defined for $\operatorname{GL}_2(\mathbf{Q}_p)$ to the category of finitely generated smooth admissible mod-$p$ representations of the two-fold metaplectic cover of $\operatorname{GL}_2(\mathbf{Q}_p)$. We compute the images of the absolutely irreducible genuine objects and obtain a bijection between the genuine supersingular representations and four-dimensional irreducible Galois representations invariant under twist by all characters of order two. Restricted to genuine objects, the extended functor naturally takes values in the category of what we call metaplectic Galois representations -- Galois representations with a certain extra structure encoding the aforementioned twist-invariance.