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 ${\tilde{\mathbf{G}}}^{\vee }$ and an L-group ${}^L{\tilde{\mathbf{G}}}^{\vee }$ 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 $M{p}_{2n}$ and special orthogonal groups.

48 pages