Lift of the Trivial Representation: The Nonsplit Type $D$ Case

Let $\tilde{G}$ be a nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In previous work, we introduced ${\prod }_{\rho \mathrm{/}2}^s(\tilde{G})$, the set of genuine small representations with infinitesimal character $\rho \mathrm{/}2$ in the simply laced setting, and showed that, when $G$ is split, these representations are exhausted by the Kazhdan–Patterson lift of the trivial representation. In this paper, we consider nonsplit real Spin groups of type $D_n$. For each $\rho \mathrm{/}2$-compatible genuine central character, we determine the irreducible constituents of ${\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}}_G^{\tilde{G}}(\mathbb{C})$. We also describe explicitly the relevant genuine small representations in terms of Langlands parameters. The nonsplit case exhibits a new phenomenon: the lift of the trivial representation need not exhaust ${\prod }_{\rho \mathrm{/}2}^s(\tilde{G})$. For the families considered here, equality holds only for $\widetilde{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}(2p+2,2p)$; in the other two families the lift is a proper subset, and in one case it is zero.