The Langlands-Weissman Program for Brylinski-Deligne extensions

TL;DR

This paper extends the Langlands program to Brylinski-Deligne covering groups, constructing an L-group extension, exploring its properties, and proposing a local Langlands correspondence for certain representations.

Contribution

It introduces a conjectural extension of the Langlands program for Brylinski-Deligne covering groups, including the construction of an L-group extension and a framework for automorphic L-functions.

Findings

Construction of an L-group extension for Brylinski-Deligne covers

Description of a local Langlands correspondence for split tori

Definition of automorphic L-functions for genuine automorphic representations

Abstract

We describe an evolving and conjectural extension of the Langlands program for a class of nonlinear covering groups of algebraic origin studied by Brylinski-Deligne. In particular, we describe the construction of an L-group extension of such a covering group (over a split reductive group) due to Weissman, study some of its properties and discuss a variant of it. Using this L-group extension, we describe a local Langlands correspondence for covering (split) tori and unramified genuine representations, using work of Savin, McNamara, Weissman and W.W. Li. Finally, we define the notion of automorphic (partial) L-functions attached to genuine automorphic representations of the BD covering groups.