Lifting of characters for nonlinear simply laced groups

Abstract

One aspect of the Langlands program for linear groups is the lifting of characters, which relates virtual representations on a group G G with those on an endoscopic group for G G . The goal of this paper is to extend this theory to nonlinear two-fold covers of real groups in the simply laced case. Suppose G ~ \widetilde G is a two-fold cover of a real reductive group G G . A representation of G ~ \widetilde G is called genuine if it does not factor to G G . The main result is that there is an operation, denoted Lift G G ~ \text {Lift}_G^{\widetilde G} , taking a stable virtual character of G G to a virtual genuine character of G ~ \widetilde G , and Lift G G ~ ( Θ π ) \text {Lift}_G^{\widetilde G}(\Theta _\pi ) may be explicitly computed if π \pi is a stable sum of standard modules.