Managing Metaplectiphobia: Covering p-adic groups

TL;DR

This paper surveys and extends Brylinski and Deligne's framework for understanding central extensions of p-adic groups via algebraic groups over residue fields, using Bruhat-Tits buildings to describe their structure.

Contribution

It provides a detailed description of central extensions of p-adic groups by finite cyclic groups, building on Brylinski and Deligne's work and utilizing Bruhat-Tits buildings.

Findings

Central extensions are classified via algebraic groups over residue fields.

The structure of extensions is linked to points in the Bruhat-Tits building.

A precise description of some central extensions is achieved.

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.