Let $G$ be the real points of a simply laced, simply connected complex Lie group. We discuss a set of small genuine representations of the nonlinear double cover 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 one fourth of the sum of positive roots; (b) they have maximal tau-invariant; (c) they have a particular associated variety $O$. When $G$ is split, we will show that all representations in Lift($C$) are parametrized by pairs (central character, real form of $O$) by examples.