Unitary Shimura Correspondence for Complex Classical Groups

Abstract: In this paper, we construct a lifting operator from the Grothendieck group of admissible Harish-Chandra modules of $G =\mathrm{SO}<em>{2n}(\mathbb C)$ (resp. $\mathrm{Sp}</em>{2n}(\mathbb C)$) to that of genuine representations of $\mathrm{Spin}<em>{2n}(\mathbb C)$ (resp. $\mathrm{Spin}</em>{2n+1}(\mathbb C)$). We determine the lift of the sum of special unipotent representations attached to any $${}^{{\vee}\mathcal} O \subseteq {}^{{\vee}\mathfrak} g$$ explicitly. In particular, the representations occurring in these lifts, if nonzero, are genuine unipotent representations of complex Spin groups and are unitary. As a consequence, the lifting operator preserves unitarity on a large class of unitary representations.