Split metaplectic groups and their L-groups

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.