Split metaplectic groups and their L-groups

TL;DR

This paper extends the local Langlands conjectural framework to split metaplectic groups by constructing dual groups and L-groups, proposing a parameterization of genuine representations compatible with known metaplectic phenomena.

Contribution

It introduces a novel construction of L-groups for split metaplectic groups over local fields, enabling a conjectural Langlands parameterization of their genuine representations.

Findings

Constructed dual and L-groups for split metaplectic groups.

Defined Weil-Deligne parameters with values in the new L-group.

Established compatibility with known metaplectic and classical correspondences.

Abstract

We adapt the conjectural local Langlands parameterization to split metaplectic groups over local fields. When $\tilde G$ is a central extension of a split connected reductive group over a local field (arising from the framework of Brylinski and Deligne), we construct a dual group $\mathbf{\tilde G}^\vee$ and an L-group ${}^L \mathbf{\tilde G}^\vee$ as group schemes over ${\mathbb Z}$. Such a construction leads to a definition of Weil-Deligne parameters (Langlands parameters) with values in this L-group, and to a conjectural parameterization of the irreducible genuine representations of $\tilde G$. This conjectural parameterization is compatible with what is known about metaplectic tori, Iwahori-Hecke algebra isomorphisms between metaplectic and linear groups, and classical theta correspondences between $Mp_{2n}$ and special orthogonal groups.