Some genuine small representations of a nonlinear double cover

Let G be the real points of a simply connected, semisimple, simply laced complex Lie group, and let G be the nonlinear double cover of G. We discuss a set of small genuine irreducible representations of G which can be characterized by the following properties: (a) the infinitesimal character is ρ/2; (b) they have maximal τ-invariant; (c) they have a particular associated variety O. When G is split, we construct them explicitly. Furthermore, in many cases, there is a one-to-one correspondence between these small representations and the pairs (genuine central characters of G, real forms of O) via the map π → (χπ,AV (π)).