Managing metaplectiphobia: covering 𝑝-adic groups

Abstract

Brylinski and Deligne have provided a framework to study central extensions of reductive groups by K2 over a field F. Such central extensions can be used to construct central extensions of p-adic groups by finite cyclic groups, including the metaplectic groups. Particularly interesting is the observation of Brylinski and Deligne that a central extension of a reductive group by K2, over a p-adic field, yields a family of central extensions of reductive groups by the multiplicative group over the residue field, indexed by the points of the building. These algebraic groups over the residue field determine the structure of central extensions of p-adic groups, when the extension is restricted to a parahoric subgroup. This article surveys and builds upon the work of Brylinski and Deligne, culminating in a precise description of some central extensions using the Bruhat-Tits building.