Lift of the trivial representation to a nonlinear cover

Abstract

Let G be the real points of a simply laced, simply connected complex Lie group, and let G^~ be the nonlinear two-fold cover of G. We discuss a set of small genuine representations of G^~, denoted by Lift(C), which can be obtained from the trivial representation of G by a lifting operator. The representations in Lift(C) can be characterized by the following properties: (a) the infinitesimal character is &rho/2; (b) they have maximal &tau-invariant; (c) they have a particular associated variety O. When G is split and of type A or D , we have a full description for Lift(C). In this case, these representations are parametrized by pairs (central character, real form of O), and exhaust all small representations with infinitesimal character &rho/2 and maximal &tau-invariant.