AI 中文总结
本文构造从复特殊线性群等的Harish-Chandra模格罗滕迪克群到对应Spin群真表示格罗滕迪克群的提升算子，明确特殊幂么表示之和的提升，证明其保持酉性。
AI 中文摘要
本文中，我们构造一个提升算子，将$G = \mathrm{SO}_{2n}(\mathbb C)$（对应地，$\mathrm{Sp}_{2n}(\mathbb C)$）的可允许Harish-Chandra模的格罗滕迪克群，映射到$\mathrm{Spin}_{2n}(\mathbb C)$（对应地，$\mathrm{Spin}_{2n+1}(\mathbb C)$）的真表示的格罗滕迪克群。我们明确确定了与任意${}^\vee\mathcal O \subseteq {}^\vee\mathfrak g$相关联的特殊幂么表示之和的提升。特别地，这些提升中出现的表示若非零，则为复Spin群的真幂么表示且是酉的。因此，该提升算子在一大类酉表示上保持酉性。
英文摘要
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.
