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 $G={\mathrm{SO}}_{2n}(\mathbb{C})$ (resp. ${\mathrm{Sp}}_{2n}(\mathbb{C})$) to that of genuine representations of ${\mathrm{Spin}}_{2n}(\mathbb{C})$ (resp. ${\mathrm{Spin}}_{2n+1}(\mathbb{C})$). 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.