Lift of the trivial representation to a nonlinear double cover

Let $\tilde{G}$ be the nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In [Ts], we introduce a set of genuine small representations of $\tilde{G}$ with infinitesimal character $\lambda$, denoted ${\prod }_{\lambda }^s(\tilde{G})$. In this paper, we show that ${\prod }_{\rho \mathrm{/}2}^s(\tilde{G})$ is precisely the set of genuine irreducible representations arising from the Kazhdan-Patterson lifting of the trivial representation, when $\tilde{G}$ is simply laced and split.