THE LANGLANDS-WEISSMAN PROGRAM FOR BRYLINSKI-DELIGNE EXTENSIONS

摘要

We describe an evolving and conjectural extension of the Langlands program for a class of nonlinear covering groups of algebraic origin studied by Brylinski and 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.We then define the notion of automorphic (partial) L-functions attached to genuine automor-phic representations of the covering groups of Brylinski and Deligne.Finally,we see how the L-group formalism explains certain anomalies in the representation theory of covering groups and examine some examples of Langlands functoriality such as base change.